# Mathematical and Youth Magazine Problems ```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
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
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 &lt; 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.
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
+
+
&lt;
.
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.
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 &gt; 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 &lt; ri ≤ ,
ri = 1
2 i=1
1
(n &gt; 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
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
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 &lt; 1
n+k+1
k=0
n
1
where e = lim 1 +
.
n→∞
n
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
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.
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.
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.
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.
13. Issue 427 – January 2013
9a
Problem 1. Let a = 123456789. Which number is greater 20129
a9
or 2013a ?.
Problem 2. Let ABC (AB &lt; AC) be a triangle, with two altitudes BD, CE and AB = c,
AC = b, BD = hb , CE = hc . Prove that
cn + hcn &lt; 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 &middot; 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
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 &lt; 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 &times; 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 &lt; 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)
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 &gt; 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 &gt; 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 &middot; OE &middot; OF ≤ 27AB &middot; BC &middot; 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.
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).
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 &gt; 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
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
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?.
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 .
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&middot;e
a) R = 2
,
p − e2
8SR
= 2p2 , where 2p = a + b + c + d.
b) a2 + b2 + c2 + d2 +
e
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 &gt; b then a is not divisible by b. Find the maximal number of elements of such subset S.
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
&gt;
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.
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 &lt; a2 &lt; . . . an &lt; . . . 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 &lt; x, y &lt; 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
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&middot;
4 . . . n −n n − 1.
Show that
√
24n2 +24n
3n2 +n−12 ≤ P &lt;
√
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 &lt; r &lt; 120. Prove that r is prime.
Problem 2. Find all natural solutions of the equation
3xyz − 5yz + 3x + 3z = 5.
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 &middot; 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
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