MATHEMATICAL AND YOUTH MAGAZINE PROBLEMS May 10, 2018 Abstract. The Mathematics and Youth Magazine, which started in october 1964, is a Vietnamese problem solving journal. The magazine aims to secondary and high school students and teachers. In each issue, the first five problems are “For Secondary Schools”. The problems “For High Schools” are three or four next problems. And the remaining problems are “Toward Mathematical Olympiad”. Here we translate the problems from the Mathematics and Youth Magazine and share to everyone who want to enjoy the Vietnamese mathematical culture. We try to update new problems as soon as the new issue is released. Please follow the link for the updated version which can be recognize by the “blue date” above. This work continues the work of Pham Van Thuan in collecting the MYM Problems before 2002, this collection can be download from here. Statistically, in such a large set of problems some mistakes must occur. We try to reduce their number to a minimum. If you encounter any (for example, a senseless or wrong problem, a language mistake, a typo, etc.), please tell us by writing a message to e-mail address bbt.molympiad@gmail.com or by using contact form. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 2 1. Issue 415 – January 2012 Problem 1. Let 20112011 , 20122012 Which number is greater, A or B?. A= B= 20112011 + 2011 . 20122012 + 2012 Problem 2. Given r A= 6+ q √ 6 + . . . + 6, B = r 3 q √ 3 3 6 + 6 + . . . + 6, where there are exactly n square roots in A and n cube roots in B. Write [x] for the A−B greatest integer not exceeding x. Determine the value of . A+B Problem 3. Find all pairs of natural numbers x, y such that x2 − 5x + 7 = 3y . Problem 4. Prove the inequality 1 1 1 1 + 2 . . . 1 + n < 3. 1+ 2 2 2 Problem 5. Let ABCD be aparallelogram. Points H and K are chosen on lines AB and BC such that triangles KAB and HCB are isosceles (KA = AB, HC = CB). Prove that a) Triangle KDH is also isosceles. b) Triangle KAB, BCH and KDH are similar. Problem 6. In a triangle ABC with a = BC, b = CA, c = AB, A1 is the midpoint of BC; O and I are its circumcenter and incenter respectively. Prove that if AA1 isperpendicular to OI then min{b, c} ≤ a ≤ max{b, c}. Problem 7. The real numbers x, y and z are such that (√ p √ √ x sin α + y cos α − z = − 2(x + y + z) , √ 2x + 2y − 13 z =7 π≤α≤ 3π . 2 Determine the value of (x + y)z. Problem 8. Solve the following system of equations in two variables ( log2 x = 2y+2 p √ . 2 1 + x + xy 4 + y 2 = 0 Problem 9. A collection of prime numbers (each prime can be repeated) is said to be beautiful if their product is exactly ten times their sum. Find all beautiful collections. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 3 Problem 10. Points A, B, C, D, E in clockwise order, lie on the same circle. M, N, P, Q are the feet of perpendicular lines from E onto AB, BC, CD, DA. Prove that M N , N P , P Q, QM are tangent lines to a certain parabole whose focus point if E. Problem 11. The sequence (an ) is defined recursively by the following rules √ 1 a1 = 1, an+1 = − 2, n = 1, 2, . . . . a1 + . . . + an Find the limit of the sequence (bn ) where b n = a1 + . . . + an . Problem 12. Let α and β be two real roots of the equation 4x2 − 4tx − 1 = 0 2x − t where t is a parameter. Let f (x) = 2 be a funtion defined on the interval [α; β], and x +1 let g(t) = max f (x) − min f (x). x∈[α;β] x∈[α;β] π Prove that if a triple a, b, c ∈ 0; 2 are such that sin a + sin b + sin c = 1, then √ 1 1 1 3 6 + + < . g(tan a) g(tan b) g(tan c) 4 2. Issue 416 – February 2012 Problem 1. Find all natural numbers x, y, z such that 2010x + 2011y = 2012z . Problem 2. The natural numbers a1 , a2 , . . . , a100 satisfy the equation 1 1 1 101 + + ... + = . a1 a2 a100 2 Prove that there are at least two equal numbers. Problem 3. Let a, b, c be positive real numbers. Prove the inequality (a + b)2 (b + c)2 (c + a)2 a b c + + ≥9+2 + + . ab bc ca b+c c+a a+b Problem 4. Solve the equation √ 4x2 + 14x + 11 = 4 6x + 10. Problem 5. In a triange ABC, te incircle (I) meets BC, CA at D, E respectively. Let K be the point of reglection of D through the midpoint of BC, the line through K and perpendicular to BC meets DE at L, N is the midpoint of KL. Prove that BN and AK are orthogonal. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 4 Problem 6. Determine the maximum value of the expression mn A= (m + 1)(n + 1)(m + n + 1) where m, n are natural numbers. Problem 7. Triangle ABC (AB > AC) is inscribed in circle (O). The exterior angle bisector of BAC meets (O) at another point E; M, N are the midpoints of BC, CA respectively; F os the perpendicular foot of E on AB, K is the intersection of M N and AE. Prove that KF and BC are parallel. Problem 8. Solve the equation sin2n+1 x + sinn 2x + (sinn x − cosn x)2 − 2 = 0 where n is a given positive integer. Problem 9. Find all polynomials P (x) such that P (2) = 12, P (x2 ) = x2 (x2 + 1)P (x), ∀x ∈ R. n 1 X Problem 10. Let r1 , r2 , . . . , rn be n rational numbers such that 0 < ri ≤ , ri = 1 2 i=1 1 (n > 1), and let f (x) = [x] + x + . Find the greatest value of the expression 2 n X P (k) = 2k − f (kri ) where k runs over the integers Z (the notation [x] means the greati=1 est integer not exceeding x). Problem 11. Suppose that f : R → R is a continuous funtion such that f (x)+f (x+1006) is a rational number if and only if x ∈ R, f (x + 20) + f (x + 12) + f (x + 2012) is itrational. Prove that f (x) = f (x + 2012) for all x ∈ R. Problem 12. Prove the following inequality ma mb mc R + = ≤1+ , ha hb hc r where ma , bb , mc are medians; ha , hb , hc are the altitudes from A, B, C and R, r are the circumradius and inradius, respectively. 3. Issue 417 – March 2012 Problem 1. Which number is bigger, 23100 or 32100 ?. Problem 2. Let ABC be an isosceles triangle with AB = AC. BM is the median from \ = ABM \ . Prove that CM ≥ CN . B. N is a point on BC such that CAN MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 5 Problem 3. Let a, b, c be positive numbers such that |a + b + c| ≤ 1, |a − b + c| ≤ 1, |4a + 2b + c| ≤ 8, |4a − 2b + c| ≤ 8. Prove the inequality |a| + 3|b| + |c| ≤ 7. Problem 4. Solve the equation (x − 2)(x2 + 6x − 11)2 = (5x2 − 10x + 1)2 . Problem 5. Let ABC be a right triangle, with right angle atA, AH is the altitude from A and I, J ae the incenters of triangles HAB and HAC, respectively. IJ cuts AB at M and meets AC at N . Let X and Y be the intersections of HI with AB and HJ with AC; BY , CX cuts M N at P and Q respectively. Prove that AI HP = . AJ HQ Problem 6. Let x, y, z be real numbers such that x2 + y 2 + z 2 = 3. Find the minimum and maximum value of the expression P = (x + 2)(y + 2)(z + 2). Problem 7. In a triangle ABC, let ma , mb , mc be its median lengths, and la , lb , lc be the lengths of its inner bisectors, p is half of its perimeter. Prove the inequality √ ma + mb + mc + la + lb + lc ≤ 2 3p. Problem 8. Let S.ABC be a pyramid where surface SAB is a isosceles triangle at S and √ [ = 1200 , the plane (SAB) is perpendicular to (ABC). Prove that SABC ≤ 3, when BSA SSAC does the equality occur?. (Denote by SDEF the area of triangle DEF ) Problem 9. A natural number n is a good number if it is possible to partition any square into n smaller squares such that at least two of them are not equal. a) Prove that if n is a good number, then n ≥ 4. b) Prove that both 4 and 5 are not good. c) Find all good numbers. Problem 10. A sequence a0 , a1 , . . . , an (n ≥ 2) is defined by 1 1 1 a0 = 0, ak = + + ... + , k = 1, 2, . . . , n. n+1 n+2 n+k Prove the inequality n−1 X eak + (ln 2 − an )ean < 1 n+k+1 k=0 n 1 where e = lim 1 + . n→∞ n MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 6 Problem 11. Find all functions f : R+ → R+ satisfying f (x)f (yf (x)) = f (y + f (x)), x, y ∈ R+ . Problem 12. Given a triangle ABC inscribed in a circle (O, R), with center G and area S. Prove that √ OG2 2 2 2 a + b + c ≥ 4 3 + 2 S + (a − b)2 + (b − c)2 + (c − a)2 . R 4. Issue 418 – April 2012 Problem 1. Given A = 15 + 25 + 35 + . . . + 20115 . Find the last digit of A. Problem 2. Let ABC be an isosceles right triangle with right angle at A. On the halfplane defined by AB containing C draw an isosceles right triangle ABD with right angle at B. Let E be the midpoint of segment BD. Draw CM perpendicular to AE at M . Let N be the midpoint of segment CM , K is the intersection of BM and DN . Find the measure of the angle BKD. Problem 3. Find all minimal value of the expression 1 4x2 y 2 + 2 A= 3 + x + xy + y 3 xy where x and y are positive real numbers satisfying x + y = 1. Problem 4. Find all positive integer solutions of the equation 3x − 32 = y 2 . Problem 5. Let ABC be an acute triangle with orthocenter H. Prove that ABC is an equilateral triangle if and only if AH BH CH = = . BC CA AB Problem 6. Let ABC be a triangle with circumcenter O, and incenter I. BC touches the circle (I) at D. The circle whose diameter is AI meets (O) at M (M 6= A) and cuts the line passing through A parallel to BC at N . Prove that M O passes through the midpoint of DN . Problem 7. Solve the system of equations (p √ √ √ xy + (x − y)( xy + 2) + x = y + y . √ (x + 1)(y + xy + x(1 − x)) =4 Problem 8. Let ABC be an acute triangle. Prove the inequaltiy 1 cos3 A + cos3 B + cos3 C + cos A. cos B. cos C ≥ . 2 MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 7 Problem 9. For each natural number n, let (Sn ) be the sum of all digits of n (in the decimal system). Put Sk (n) = S(S(. . . (S(n)) . . .)) (k times). Find all natural numbers n such that S1 (n) + S2 (n) + . . . Sk (n) + . . . + S223 (n) = n. Problem 10. Does there exist a set X satisfying the following two conditions i) X contains 2012 natural numbers. ii) The sum of any arbitrary elements in X is the k-th power of a positive integer (k ≥ 2). Problem 11. Find all functions f : R → R satisfing yf (x) xf (y) f +f = 4xy, ∀x, y ∈ R. 2 2 Problem 12. Fix two circles (K) and (O), where (K) is inside (O). Two circles (O1 ), (O2 ) are moving so that they always externally touch each other at M . Both also internally touch (O), and externally touch (K). Prove that M belongs to a fixed circle. 5. Issue 419 – May 2012 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 6. Issue 420 – June 2012 Problem 1. Problem 2. Problem 3. Problem 4. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 8 Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 7. Issue 421 – July 2012 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 8. Issue 422 – August 2012 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 9 Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 9. Issue 423 – September 2012 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 10. Issue 424 – October 2012 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 10 11. Issue 425 – November 2012 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 12. Issue 426 – December 2012 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 11 13. Issue 427 – January 2013 9a Problem 1. Let a = 123456789. Which number is greater 20129 a9 or 2013a ?. Problem 2. Let ABC (AB < AC) be a triangle, with two altitudes BD, CE and AB = c, AC = b, BD = hb , CE = hc . Prove that cn + hcn < bn + hnb , ∀n ∈ N∗ . Problem 3. Find all positive integers n such that 2 n +n−5 A= 2 is a prime number, where [a] is the largest integer not exceeding a. Problem 4. Find all postive integer soltuions of the equation √ √ √ x + y − x − y + 2 = 0. Problem 5. Let ABC be a right triangle, right angle at A. The bisectors BD and CE intersect at O. The area of BOC is a. Determine the product BD · CE in terms of a. Problem 6. Solve the system pf equations q q 2 4 x4 + 4 = 1 + 3 |y| q3 q2 . 2 4 y4 + 4 = 1 + 3 |x| 3 2 √ √ Problem 7. The side lengths of √ a traingle ABC are AB = 9, BC = 39, CA = 201. Find a point M on the circle (C; 3) such that the sum M A + M B is the maximum. Problem 8. Prove that in any traingle ABC, s B C B A tan + tan tan + tan 2 2 2 2 s B C A C tan + tan + tan + tan 2 2 2 2 s C A A B + tan + tan tan + tan 2 2 2 2 ≤2(cot A + cot B + cot C). Problem √ 9. Let N = 1 + 10 + 102 + . . . + 104023 . Find the 2013-th digit after the decimal comma of N . Problem 10. Find the maximum and minimum values of the expression a−b b−c c−a + + , P = c a b MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 12 where a, b, c are positive real numbers satisfying the condition 1 min{a, b, c} ≥ max{a, b, c}. 4 Problem 11. Let {Sn (x)} be a sequence of real-valued functions defined by n 1 1 −1 3 3 3 2 Sn (x) = cos x − cos 3x + 2 cos 3 x − . . . + cos3 3n x. 3 3 3 Find all real values of x such that 3 − 3x lim Sn (x) = . n→∞ 4 Problem 12. In a con-cyclic quadrilateral ABCD, let A0 , B 0 , C 0 , D0 be the circumcenters of triangles BCD, CDA, DAB and ABC respectively. Let A00 , B 00 , C 00 , D00 be the centers of the Euler circles of triangle BCD, CDA, DAB, ABC respectively. Prove that the two quadrilateral A0 B 0 C 0 D0 , A00 B 00 C 00 D00 are both convex and inversely similar. 14. Issue 428 – February 2013 Problem 1. Determine all triple of prime numbers a, b, c (not necessarily distinct) such that abc < ab + bc + ca. Problem 2. Let ABC be a right triangle, with right angle at A and AH is the altitude [ = 300 . Construct an equilateral triangle ACD (D and B are in opposite from A, ACB side AC). K is the foot of the perpendicular line from H onto AC. The line through H and parallel to AD meets AB at M . Prove the points D, K, M are colinear. Problem 3. Consider a 6 × 6 board of squares with 4 corner squares being deleted. Find the smallest number of squares that can be painted black given that among the 5 squares in any figure, there is at lease one black. Problem 4. Let a, b, c be real numbers in the interval [1, 2]. Prove the inequality p a2 + b2 + c2 + 3 3 (abc)2 ≥ 2(ab + bc + ca). Problem 5. Let ABC be a non-right triangle (AB < AC) with altitude AH. E, F are the orthogonal projection of point H onto AB and AC respectively. EF meets BC at D. Draw a semicircle with diameter CD on the half-plane containing A with edge CD. The line through B and perpendicular to CD meets the semicircle at K. Prove that DK is tangent to the circumcircle of triangle KEF . Problem 6. Given that the equation ax3 − x2 + ax − b = 0 (a 6= 0, b 6= 0) has three positive real roots. Determine the greatest value of the following expression √ 11a2 − 3 3ab − 13 √ . P = 9b − 10( 3a − 1) MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 13 Problem 7. Solve the following system of equations r r √ 1 1 = 3 x− + y− 4 r 4 r √ 1 1 y− + z− = 3. 16 r 16 r √ 9 9 z− + x− = 3 16 16 Problem 8. Let a, b be real constants such that ab > 0. Let {un } be a sequence where n = 1, 2, 3, . . . given by un+1 = un + bu2n , ∀n ∈ N∗ . u1 = a, Determine the limit lim n→∞ u1 u2 un + + ... + u2 u3 un+1 . Problem 9. Find all positive integers k with the property that there exists a polynomial f (x) with integer coefficients of degree greater than 1 such that for all prime numbers p and natural numbers a, b if p divides (ab − k) then it also divides (f (a)f (b) − k). Problem 10. Given ai ∈ [0, α] (i = 1, n), (α > 0). Prove the inequality ! n n Y X ai (α − ai ) ≤ αn 1 − Si + α i=1 i=1 where Si = n X ak − ai for all i = 1, n. k=1 Problem 11. Point O is in the interior of triangle ABC. The ray Ox parallel to AB meets BC at D, ray Oy parallel to BC meets CA at E, ray Oz parallel to CA meets AB at F . Prove that a) 3SDEF ≤ SABC . b) OD · OE · OF ≤ 27AB · BC · CA. Problem 12. The circle (O) and (O0 ) meet at points A, B. Point C is fixed on (O) and point D is fixed on (O0 ). A moving point P is on the opposite ray of ray BA. The circumcircles of traingles P BC, P DB intersect BD, BC at seconde points E, F respectively. Prove that the midpoint of line segment EF is always on a fixed segment EF is always on a fixed straight line. 15. Issue 429 – March 2013 Problem 1. Find an integer size square whose area is a 4-digit number such that the last rightmost three digits are idnentical. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 14 Problem 2. Determine the values of a1 , a2 , a3 , a4 , a5 , a6 , a7 . Given that 2a1 = 3a2 , 2a3 = 4a4 , 5a4 = 2a5 , 2a5 = 5a6 , 2a6 = 3a7 , 2a7 = 3a1 and a1 + a2 + a3 + a4 + a5 + a6 + a7 = 400. Problem 3. Find all pairs of integers (x, y) such that x2 (x2 + y 2 ) = y p+1 , where p is a prime number. Problem 4. Find the maximum and minimum values of the expression P = xy + yz + zx − xyz where x, y, z are non-negative real numbers satisfying x2 + y 2 + z 2 = 3. [ = 700 , ACB [ = 500 . The Problem 5. Triangle ABC inscribed in circle (O) with BAC points M, N, P, Q and R on circle (O) are such that P A = AB = BR, QB = BC = CM and N C = CA = AN . Let S be the intersection of arc N Q and the diameter P P 0 of (O). Prove that ∆N RS v ∆N QR. Problem 6. Solve the equation x3 − 3x = √ x + 2. Problem 7. Find the measure of the angles of a triangle ABC such that the expression C T = −3 tan + 4(sin2 A − sin2 B) 2 is greatest possible. Problem 8. Let S.ABC be a triangular pyramid where the sides SA, SB, SC are pairwise orthogonal, SA = a, SB = b, Sc = c. H is the foot of the perpendicular from S onto ABC. Prove the inequality √ abc 3 aSHBC + bSHAC + cSHAB ≤ . 2 Problem 9.√Let p be an odd prime number, and x, y are two positive integers such that √ √ x + y ≤ 2p. Find the minimum value of the following expression p √ √ A = 2p − x − y. Problem 10. Does there exist a funtion f : N∗ → N∗ such that f (mf (n)) = n + f (2013m), ∀m, n ∈ N∗ ?. Problem 11. The non-negative real numbers a, b, c are such that max{a, b, c} ≤ 4 min{a, b, c}. Prove the inequality 2(a + b + c)(ab + bc + ca)2 ≥ 9abc(a2 + b2 + c2 + ab + bc + ca). MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 15 Problem 12. Let ABC be a traingle inscribed in circle centered at O, and let I be its incenter. AI, BI, CI intersect (O) at A1 , B1 , C1 ; A1 C1 , A1B1 meet BC at M, N ; B1 A1 , B1 C1 meet CA at P, Q; C1 B1 , C1 A1 meet AB at R, S respectively. Prove that 2 SM N P QRS ≤ SA1 B1 C1 . 3 16. Issue 430 – April 2013 Problem 1. Do there exist natural numbers x, y, z such that 5x2 + 2016y+1 = 2017z ?. Problem 2. Point O is chosen in a right triangke ABC, right angle at A, such that [ = 300 and OA = OC. Point E on side BC such that EOB \ = 600 . Determine the ABO three angles of traingle ABC given that the line CO passes through the midpoint I of the line segment AE. Problem 3. Find all pair of natural numbers x, y such that 5x = y 4 + 4y + 1. Problem 4. Solve the system of equations √ √ 2 x + √ y − 2 + √ 4 − z = y − 5z + 11 y + z − 2 + 4 − x = z 2 − 5x + 11 . z + √x − 2 + √4 − y = x2 − 5y + 11 Problem 5. Let AB = 2a be a line segment with midpoint O. Two half-circles, one with center O and diagonal AB, another with center O0 and diagonal AO are drawn on the same half-plane divided by AB. Point M , different from A and O, moves on the half-circle (O0 ). OM meets the half-circle (O) at C. Let D be the second intersection point of CA and half-circle (O0 ). The tangent line at C of half-circle (O) meets OD at E. Find the position of point M on (O0 ) such that M E is parallel to AB. Problem 6. Let ABCD be a quadrilateral where the diagonals AC, BD are equal and perpendicular. The triangles AM B, BN C, CP D, DQA, similar in order, are constructed outside the given quadrilateral. O1 , O2 , O3 , O4 are the midpoints of M N , N P , P Q, QM respectively. Prove that the quadrilateral O1 O2 O3 O4 is s square. Problem 7. Find a formula counting the number of all 2013-digits natural numbers which are multiple of 3 and all digits are taken from the set X = {3, 5, 7, 9}. Problem 8. Solve for x, log2 x = log5−x 3. Problem 9. The positive integers a1 , a2 , . . . , a2013 , b1 , b2 , . . . b2013 where bk > 1 for all k are chosen from the set X = {1, 2, . . . , 2013}. Prove that there exists a positive integer n satisfying the following two conditions ! 2013 ! 2013 Y Y i) n ≤ ai bi + 1. i=1 i=1 MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 16 ii) ak bnk + 1 is a composite number for every k ∈ X. Problem 10. Let a, b, c ∈ 0, 21 be such that a + b + c = 1. Prove the inequality 9 a3 + b3 + c3 + 4abc ≤ . 32 Problem 11. Let ap be a prime number, p ≡ 1 (mod 4). Determine the sum 2 p−1 X 2k 2 k −2 , p p k=1 where [a] denotes the largest integer not exceeding a. Problem 12. Let ABC be a triangle inscribed inside circle (O). Point M not on lines BC, CA, AB as weel as circle (O); AM , BM , CM intersect (O) at A1 , B1 , C1 ; A2 , B2 , C2 are the circumcenters of triangles M BC, M CA, M AB respectively. Prove that the lines A1 A2 , B1 B2 , C1 C2 meet at a point on the circle (O). 17. Issue 431 – May 2013 Problem 1. Which number is greater 1 1 1 A= 1+ 1+ ... 1 + 2013 20132 2013n 20132 − 1 ?. 20122 − 1 Problem √ 2. Given four points in the plane such that no pair of points has distance less than 2 cm. Prove that there exists two of them having a distance greater than or equal to 2 cm. where n is a positive integer, or B = Problem 3. Find the last two digits of the number .2013 .. 2004 2003 . Problem 4. Find the maximum and minimum value of the expression √ √ P = 27 x + 8 y where x, y are non-negative real numbers satisfying p √ x 1 − y 2 + y 2 − x2 = x 2 + y 2 . Problem 5. Let ABCD be a cyclic quadrilateral, inscribed in circle (O). I and J are the midpoints of BD and AC respectively. Prove that BD is the angle bisector of angle AIC if and only if AC is the angle bisector of angle BJD. Problem 6. Solve the following system of equations ( x3 (1 − x) + y 3 (1 − y) = 12xy + 18 . |3x − 2y + 10| + |2x − 3y| = 10 MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 17 Problem 7. Determine the greatest value of the expression E = a2013 + b2013 + c2013 , where a, b, c are real numbers satisfying a + b + c = 0, a2 + b2 + c2 = 1. Problem 8. Let S.ABC be a triangular pyramid, G is the centroid of the base triangle ABC, O is the midpoint of SG. A moving plane (α) through O meets the edges SA, SB, SC at A0 , B 0 C 0 respectively. Prove that SA02 SAB 0 SC 02 AA02 BB 02 CC 02 + + ≥ + + . AA02 BB 02 CC 02 SA02 SB 02 SC 02 Problem 9. Find all natural numbers n such that h i n+3 n+5 n A= + + + n2 + 3n − 1 4 4 2 is a prime number, where [x] denotes the greatest integer not exceeding x. Problem 10. Consider the real-valued function y = a sin(x + 2013) + cos 2014x where a is given real number. Let M, N be the greatest and smallest values respectively of this function over R. Prove that M 2 + N 2 ≥ 2. Problem 11. Let {an } be a sequence given by an 2 1 , n = 1, 2, . . . a1 = , an+1 = 2 an2 −an +1 a) Prove that the sequence {an } converges to a finite limit and find this limit. b) Let bn = a1 + a2 + . . . + an for each positive integer n. Determine the integer part [bn ] and the limit limn→∞ bn . Problem 12. Given four points A, B, C, D on circle (ABC) and M is a point not on this circle. Let Ti be the triangle whose three vertices are 3 of 4 given points, except point i (i = A, B, C, D). Let Hi be the triangle whose vertices are the feet of the perpendicular drawn from M onto the edges (or extended edges) of triangles Ti (i = A, B, C, D). Prove that a) The circumcenter of triangles Hi (i = A, B, C, D) lie on the same circle, centered at O0 . b) When D moves on the circle (ABC), O0 always lie on a fixed circle. 18. Issue 432 – June 2013 Problem 1. The first 2013 natural numbers from 1 to 2013 are writeen in a line in some order. Substract one from the first number, two from the second ... and 2013 from the 2013th number. Is the product of the resulting 2013 numbers odd or even?. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 18 Problem 2. Let ABC be an acute triangle with orthocenter O. AO meets BC at D. Points E and F are on sides AB and AC respectively such that DE = DB, DF = DC. Prove that DA is the angle bisector of angle EDF . Problem 3. Find all positive integers a, b (a ≥ 2, b ≥ 2) so that a + b is amultiple of 4 and 1 a(a − 1) + b(b − 1) = . (a + b)(a + b − 1) 2 Problem 4. Find x, y such that ( √ √ x x+y y x3 + 2y 2 =2 . ≤ y 3 + 2x Problem 5. Given a circle centered at O, and diameter AB. Point C, different from A and B, is chosen on circle (O). Point P on AB such that BP = AC. The perpendicular from P to AC meet AC at H. The intenal angle bisector of angle CAB intersects circle (O) at E and intersects P H at F . CF meets circle (O) at N . Prove that CN passes through the midpoint of AP . Problem 6. Let a, b, c the positive real number. Prove the following inequality 2 2 2 1 2 3 1 2 3 1 2 3 + + + + + + + + a b+c a+b+c b c+a a+b+c c a+b a+b+c 81 ≥ 2 . a + b2 + c 2 Problem 7. Triangle ABC is inscribed in circle (O), another circle (O0 ) touches AB, AC at P , Q respectively and touches circle (O) at other points M , N . Points E, D, F are the perpendicular feet of point S on AM , M N , N A respectively. Prove that DE = DF . Problem 8. The real numbers a, b, c satisfying the condition that the polynomial P (x) = x4 + ax3 + bx2 + cx + 1 has at least one real root. Determine all triple (a, b, c) such that s2 + b2 + c2 is smallest possible. Problem 9. Let a and B be two real numbers such that ap − bp is a positive integers for all prime number p. Prove that a and b are integers. Problem 10. The sequence {un } is given recursively as follows 1 1 1 1 u1 = , = 2 − + 1, ∀n ≥ 1 1+a un+1 un un where a ∈ R, a 6= −1. Let Sn = u1 + u2 + . . . + un , Pn = u1 u2 . . . un . Determine the value of the following expression aSn + Pn . MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 19 Problem 11. Determine all funtions f : R+ → R+ such that a) f is a decreasing function on R+ . b) f (2x) = 2012−x f (x), ∀x ∈ R+ where R+ = (0, +∞). Problem 12. Let ABC be a triangle inscribed in circle (O). AD is a diameter of (O). Point E belongs to the opposite ray of ray DA. The perpendicular through E o AD meets BC at T . T P is a tangent line to (O) such that P and A are on opposite sides of BC; AP meets T E at Q. M is the midpoint of AQ; T M meets AB, AC at X, Y respectively. Prove that M is the midpoint of XY . 19. Issue 433 – July 2013 Problem 1. Find all positive integers x, y, z such that x2 + y 3 + z 4 = 90. Problem 2. Let ABC be an equilateral triangle whose altitudes are AD, BE and CF . Suppose M is an arbitrary point inside triangle ABC. I, K, L are the perpendicualr feet from M to AD, BE, CF . Prove that the sum AI + BK + CL does not depend on the position of M . Problem 3. The rational numbers a, b satisfy the identity a2013 + b2013 = 2a1006 b1006 . Prove that the equation x2 + 2x + ab = 0 has two rational solutions. Problem 4. Find the minimum value of the expression 1 1 1 4 4 4 P = (x + y + z ) + + , x4 y 4 z 4 where x, y, z are positive real numbers that satisfy x + y ≤ z. Problem 5. Let AH be the altitude from A of right triangle ABC, right angle at A. Point D on the oppostite ray of HA such that HA = 2HD. Point E is the reflection of B through D; I is the midpoint of AC; DI and EI meet BC at M and K respectively. \ =M \ Prove that BDK CD. Problem 6. Solve the equation √ q √ √ 27 2 x + x2 − 1 = (x − 1)2 x − 1. 8 Problem 7. A convex quadrilateral ABCD with area S is inscribed in a circle whose radius is R and AB = a, BC = b, CD = c, DA = d, AC = e. If there exists a circle touching all the opposite rays of the rays BA, DA, CD and CB. Prove that S·e a) R = 2 , p − e2 8SR = 2p2 , where 2p = a + b + c + d. b) a2 + b2 + c2 + d2 + e MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 20 Problem 8. Find the maximum value of the expression α(sin2 A + sin2 B + sin2 C) − β(cos3 A + cos3 B + cos3 C) where A, B, C are three angles of an acute triangle and α, β are two given positive numbers. Problem 9. Find the maximum area of a convex pectagon in the coordinate plane Oxy having the following properties: all interior angles are the same, all vertices have integer coordinates, there exists a side that is parallel to the axis Ox, there are exactly 16 points, including the vertices, with integer coordinates on its boundary. Problem 10. Find all continuous functions f such that (x + y)f (x + y) = xf (x) + yf (y) + 2xy, ∀x, y ∈ R. Problem 11. Let (an ) be a sequence where a1 ∈ R and an+1 = |an − 21−n |, ∀n ∈ N∗ . Find lim an . n→∞ Problem 12. A right triangle ABC with right angle at C is inscribed in circle (O). M is an arbitrary point moving on circle (O), different from A, B, C. Point N is the reflection of M in AB, P is the perpendicular foot of N to AC, M P meets (O) at a second point Q. Prove that the circumcenter of triangle AP Q lies on a fixed circle. 20. Issue 434 – August 2013 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 21 21. Issue 435 – September 2013 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 22. Issue 436 – October 2013 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 22 23. Issue 437 – November 2013 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 24. Issue 438 – December 2013 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 23 25. Issue 439 – January 2014 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 26. Issue 440 – February 2014 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 24 27. Issue 441 – March 2014 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 28. Issue 442 – April 2014 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 25 29. Issue 443 – May 2014 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 30. Issue 444 – June 2014 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 26 31. Issue 445 – July 2014 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 32. Issue 446 – August 2014 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 27 33. Issue 447 – September 2014 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 34. Issue 448 – October 2014 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 28 35. Issue 449 – November 2014 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 36. Issue 450 – December 2014 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 29 37. Issue 452 – January 2015 38. Issue 453 – February 2015 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 39. Issue 454 – March 2015 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 30 40. Issue 455 – April 2015 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 41. Issue 456 – May 2015 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 31 42. Issue 457 – June 2015 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 43. Issue 458 – July 2015 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 32 44. Issue 459 – August 2015 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 45. Issue 460 – September 2015 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 33 46. Issue 461 – October 2015 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 47. Issue 462 – November 2015 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 34 48. Issue 463 – December 2015 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 35 49. Issue 464 – January 2016 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 50. Issue 465 – February 2016 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 36 51. Issue 466 – March 2016 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 52. Issue 467 – April 2016 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 37 53. Issue 468 – May 2016 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 54. Issue 469 – June 2016 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 38 55. Issue 470 – July 2016 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 56. Issue 471 – August 2016 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 39 57. Issue 472 – September 2016 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 58. Issue 472 – October 2016 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 40 59. Issue 473 – November 2016 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 60. Issue 474 – December 2016 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 41 61. Issue 475 – January 2017 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 62. Issue 476 – February 2017 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 42 63. Issue 477 – March 2017 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 64. Issue 478 – April 2017 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 43 65. Issue 479 – May 2017 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 66. Issue 480 – June 2017 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 44 67. Issue 481 – July 2017 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 68. Issue 482 – August 2017 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 45 69. Issue 483 – September 2017 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 70. Issue 484 – October 2017 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 46 71. Issue 485 – November 2017 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. 72. Issue 486 – December 2017 Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem 10. Problem 11. Problem 12. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 47 73. Issue 487 – January 2018 Problem 1. Find all pairs of prime numbers (p, q) satisfying pq − q p = 79. Problem 2. Find integers a, b, c, d satisfying √ √ 3 a2 + b2 + c2 = a + b + c = d. Problem 3. Let a, b, c be positive numbers such that a + b + c = 1. Prove that 1 1 1 21 + + ≥ . a b c 1 + 36abc Problem 4. Given a triangle ABC circumscribing a circle (O). The sides AB, BC and CA are tangent to (O) at D, E and F respectively and assume furthermore that EC = 2EB. Suppose that EI is a diameter of (O). Through D draw a line which is parallel to BC. This line intersects the line segment EF at K. Prove that A, I,K are collinear. Problem 5. collinear. Find the maximal number M such that the inequality x2 ≥ M [x]{x} holds for every x (where [x], {x} respectively are the integral part and the fractional part of x). Problem 6. Solve the equation √ x3 + x + 6 = 2(x + 1) 3 + 2x − x2 . Problem 7. Solve the system of equations 1 1 1 √x + √y + √z = 3 r r r 2 2 2 4 4 4 4 4 4 . y z x + y y + z x 4 4 4 z + x + + = + + y z x 2 2 2 Problem 8. Given a triangle ABC with the exradii ra , rb , rc , the medians ma , mb , mc and the area S. Prove that √ ra2 + rb2 + rc2 ≥ 3 3S + (ma − mb )2 + (mb − mc )2 + (mc − ma )2 . Problem 9. Given a positive integer n and positive numbers a1 , a2 , . . . an . Find a real number λ such that ax1 + ax2 + . . . + ax2 ≥ n + λx, ∀x ∈ R. Problem 10. a) Prove that for every positive integer n the equation 2012x (x2 − n2 ) = 1 has aunique solution (denoted by xn ). b) Find lim (xn+1 − xn ). n→∞ Problem 11. Let T be the set of all positive factors of n = 20042010 . Suppose that S be an arbitrary nonempty subset of T satisfying the fact that for all a, b belong to S and a > b then a is not divisible by b. Find the maximal number of elements of such subset S. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 48 Problem 12. Given a triangle ABC whose the incircle (I) is tangent to BC at D. Let H be the perpendicular projection of A on BC. Let N be the midpoint pf AH. The line through D and N intersects CA, AB respectively at J and S. Assume that BJ intersects CS at P . Suppose that DA, DP intersect (I) respectively at G, L. Prove that B, C, G, L lie on some circle. 74. Issue 488 – February 2018 Problem 1. Give a pentagon ABCDE. Assume that BC is parallel to AD, CD is parallel to BE, DE is parallel to AC, and AE is parallel to BD. Show that AB is parallel to CE. Problem 2. Prove that 1 + 13 + 15 + . . . + 1 + 12 + 13 + . . . + 1 4035 1 2018 > 2019 . 4036 Problem 3. Given a triangle ABC with AB 6= AC. A circle (O) passing through B, C intersects the line segments AB and AC at M and N respectively. Let P be the intersection of BN and CM . Let Q be the midpoint of the arc BC which does not contian M , N . Let K be the incenter of P BC. Show that KQ always goes through a fixed point when (O) varies. Problem 4. Given two triples (a, b, c); (x, y, z) none of them contains all 0’s, such that a + b + c = x + y + z = ax + by + cz = 0. Prove that the expression P = (b + c)2 (y + z)2 + is a constant. ab + bc + ca xy + yz + zx \= Problem 5. Outside a triangle ABC, draw triangles ABD, BCE, CAF such that ADB 0 \ = CF [ \ = CBE \ = CAF [ = α. Prove that DF = AE. BEC A = 90 , ABD Problem 6. Show that the following sum is a positive integer 2 1 1 1 1 1 1 S =1 + + + . . . + + 1 + + + ... + + 2 3 2017 2 3 2017 2 2 1 1 1 1 + + + ... + + ... + . 2 3 2017 2017 Problem 7. Solve the system of equations ( 2x5 − 2x3 y − x2 y + 10x3 + y 2 − 5y √ (x + 1) y − 5 − y + 3x2 − x + 2 Problem 8. Prove that x0 = cos =0 . =0 π 8π 10π + cos + cos is a solution of the equation 21 21 21 4x3 + 2x2 − 7x − 5 = 0. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 49 Problem 9. In any triangle ABC, show that 1 cos(A − B) + cos(B − C) + cos(C − A) ≤ 2 a+b b+c c+a + + c a b . Problem 10. Given 6 positive numbers a, b, c, x, y, z and assume that x + y + z = 1. Show that ax + by + cz ≥ ax by cz . Problem 11. Given an infinite sequence of positive integers a1 < a2 < . . . an < . . . such that ai+1 − ai ≥ 8 for all i = 1, 2, 3, . . .. For each n, let sn = a1 + a2 + . . . an . Show that for each n, there are at lease two square numbers inside the half-open interval [sn , sn+1 ). Problem 12. Given two positive sequences (an )n≥0 and (bn )n≥0 which are determined as follows √ 1 + an+1 a0 = 3, b0 = 2, a2n + 1 = b2n , an + bn = , ∀n ∈ N. 1 − an+1 Show that they converges and find the limits. 75. Issue 489 – March 2018 Problem 1. Find all pairs of integers (x, y) satisfying x2 + x = 32018y + 1. Problem 2. Given a triangle ABC with ∠B = 450 , ∠C = 300 . Let BM be one of the \ medians of ABC. Find the angle AM B. Problem 3. Given real numbers x, y satisfying 0 < x, y < 1. Find the minimum value of the expression 2xy − x − y + 1 F = x2 + y 2 + . 4xy Problem 4. Given a circle (O) with a diameter AB. On (O) pick a point C (C is different from A and B). Draw CH perpendicular to AB at H. Choose M and N on the line segments CH and BC respectively such that M N is parallel to AB. Through N draw a line perpendicular to BC. This line intersects the ray AM at D. On the line DO choose two points F and K such that O is the midpoint of F K. The lines AF and AK respectively intersect (O) at P and Q. Prove that D, P , Q are colinear. Problem 5. Suppose that the polynomial f (x) = x3 + ax2 + bx + c has 3 non-negative real solutions. Find the maximal real number α so that f (x) ≥ α(x − a)3 , ∀x ≥ 0. Problem 6. Solve the equation √ 1 (1 − 2 sin x)(cos 2x + sin 2x) = . 2 MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 50 Problem 7. Given the following system of equations yz(y + z − x) =a x+y+z zx(z + x − y) =b x+y+z xy(x + y − z) = c x+y+z where a, b, c are positive parameters. a) Show that the system always has a positive solution. b) Solve the system when a = 2, b = 5, c = 10. Problem 8. Suppose that a, b, c are the lengths of three sides of a triangle. Prove that ab bc ca 1 2r + 2 + 2 ≥ + 2 2 2 2 a +b b +c c +a 2 R with R, r respectively are the inradius and the circumradius of the triangle. Problem 9. For any integer n which is greater than 3, let √ 120 √ 3 √ 60 P = 3· 4 . . . n −n n − 1. Show that √ 24n2 +24n 3n2 +n−12 ≤ P < √ 8 3. Problem 10. Find natural numbers n so that 4m + 2n + 29 cannot be a perfect square for any natural number m. Problem 11. The sequence (an ) is given as follows 1 a1 = , 2 an+1 = (an − 1)2 , n ∈ N∗ . 2 − an a) Find lim an . √ 2 a1 + a2 + . . . + an ≥1− for all n ∈ N∗ . b) Show that n 2 Problem 12. Given a triangle ABC inscribed in a circle (O). A point P varies on (O) but is different from A, B and C. Choose M , N respectively on P B, P C so that AM P N is a parallelogram. a) Prove that there exists a fixed point which is equidistant from M and N . b) Prove that the Euler line of AM N always goes through a fixed point. n→∞ 76. Issue 490 – April 2018 Problem 1. The natural number a is coprime with 210. Dividing a by 210 we get the remainder r satisfying 1 < r < 120. Prove that r is prime. Problem 2. Find all natural solutions of the equation 3xyz − 5yz + 3x + 3z = 5. MATHEMATICAL OLYMPIAD CONTESTS COLLECTION – WWW.MOLYMPIAD.BLOGSPOT.COM 51 Problem 3. Given a half circle with the center O, the diameter AB, and the radius OD perpendicular to AB. A point C is moving on the arc BD. The line AC intersects OD at M . Prove that the circumcenter I of the triangle DM C always belongs to a fixed line. Problem 4. Let x, y be real numbers such that x3 + y 3 = 2. Find the minimum value of the expression 9 P = x2 + y 2 + . x+y Problem 5. Given positive numbers a, b, c satisfying abc = 1. Prove that 1 1 1 + + ≤ 1. a5 + b 5 + c 2 b 5 + c 5 + a2 c 5 + a5 + b 2 Problem 6. Find all positive integers a, b such that r q √ π 8 + 32 + 768 = a cos . b Problem 7. Given non-zero numbers a, b, c, d satisfying b2 = ac, c2 = bd, b3 + 27c3 + 8d3 6= 0. Show that a3 + 27b3 + 8c3 a = 3 . d b + 27c3 + 8d3 Problem 8. Given a triangle ABC. Let (K) be the circle passing through A, C and is tangent to AB and let (L) be the circle passing through A, B and is tangent to AC. Assume that (K) intersects (L) at another point D which is different from A. Assume that AK, AL respectively intersect DB, DC at E and F . Let M , N respectively be the midpoints of BE, CF . Prove that A, M , N are colinear. Problem 9. Given real numbers a, b, c such that 2(a2 b2 + b2 c2 + c2 a2 ) ≥ a4 + b4 + c4 . Prove that |b + c − a| + |c + a − b| + |a + b − c| + |a + b + c| = 2(|a| + |b| + |c|). Problem 10. Find all triples of positive integers (a, b, c) such that 2a + 5b = 7c . Problem 11. The sequence (un ) is determined as follows u1 = 14, u2 = 20, u3 = 32, un+2 = 4un+1 − 8un + 8un−1 , ∀n ≥ 2. Show that u2018 = 5 · 22018 . Problem 12. Given a triangle ABC with (O) is the circumcircle and I is the incenter. Let D be the second intersection of AI and (O). Let E be the intersection between BC and the line pasing through I and perpecdicular to AI. Assume that K, L respectively are the intersections between BC, DE and the line passing through I and perpendicular to OI. Prove that KI = KL. The MOlympiad is a project to collect and assemble the problems and official solutions of mathematical olympiad competitions from around the world. It includes International Contests (IMO, APMO,...), National Olympiads from all country around the world, Team Selection Tests to IMO, National and Regional Contests, Undergraduate Contests (IMC, IMS,...), and some other competitions. This site contains information, problems, and preparation materials for math olympiads. We will update scripts every year once we will have new script. Any help is most welcome. This may include problems, solutions and feedback about certain competitions. If you want to contribute problems and solutions, we are particularly interested in those on a dark background. We are generally very happy to receive solutions to the contests that are not yet offered on the site, although it may be a good idea to contact us first, please send to e-mail address bbt.molympiad@gmail.com. The materials should be in one of the following forms (in the order of preference): TEX, PDF, Word. Your help will be duly acknowledged.