All Star Quiz 2017
Geometry and Trigonometry
Time Alotted: 3 days
Part I: Demigod Questions. Each correct answer is worth 2 point.
√
1. 4ABC has AC = 18cm and BC = 18 3 cm. The medians from A and B intersect at right
angles. What is the length of AB?
2. A rectangle has dimensions 30by35 units. What is the least number of squares with integral
side lengths into which the rectangle ca be cut?
3. What is/are the equation/s of the tangent line/s to the circle x2 + y 2 = 16 that passes through
(1, 1)? Answers in slope-intercept form.
4. A cube and a sphere have equal surface areas. What is the ratio of the volume of the sphere
to the volume of the cube?
5. A point is selected inside a rectangle where it is 5, 12, and 13 cm from three consecutive
vertices. What is its distance from the fourth vertex?
6. The lengths of the medians of a triangle are 7, 11, 14, all in cm. What is the area of the
triangle?
7. Simplify
cos 5x + cos 3x
.
sin 5x − sin 3x
8. Five points O,A,B,C,D are taken in order on a horizontal line with distances OA = a, OB =
b, OC = c, OD = d. Let P be a point on BC such that AP : P D = BP : P C. Assuming that
a + c 6= b + d, express OP in terms of a, b, c and d.
9. What is the equation of the locus of a point so that the sum of the squares of its distances
from the points (0, 0), (1, 9), (3, 4) is always 107?
√
√
10. For what acute degree angles x does the equation sin x + 3 cos x = 2 satisfy?
Part II: Deity Questions. Each correct answer is worth 3 points.
1. The vertices of a polygon are located at the points (0, 0), (7, 1), (8, 4), (3, 6), (−4, 5), and(−7, 2).
What is he area of the polygon?
2. Two chords of a circle intersect at right angles. If the chords are of length 40cm and 18cm
√ and
the distance from the center of the circle to the point of intersection of the chords is 39 cm,
what is the area of the circle?
3. A point P is an external point to a circle O. Let AP and BP be tangent lines to the circle
and 6 AP O = 30◦ . LEt C and D be poitns on AP and BP , respectively and Q is the point of
tangency when CD is connected. What is the perimeter of triangle PCD if the radius of the
circle is 9 units?
4. Given that A, B, C are angles of a triangle and sin A : sin B : sin C = 7 : 8 : 9, what is the
ratio cos A : cos B : cos C?
5. A circle O has a radius 10 cm. Let AB be a minor arc where AB = 12cm. Let C be a point
on the interior of the circle where it is 5 cm from B and 12 cm from A. What is the area of
the region enclosed by the arc AB and the lines AC and BC? Answers in 3 decimal places.
6. A point is inside a square where it is equidistant from two consecutive vertices and an opposite
side. What is the ratio of that common distance d to the length of a side of the square s ?
7. The angles of a certain polygon are in arithmetic sequence. The smallest interior angle
measures 80◦ and the largest interior angle measures 160◦ . How many sides does the polygon
have?
8. How many circles of radius 5 cm can be cut from a paper with dimensions 1.2m by 1m?
9. In trapezoid ABCD with AB||CD, M,N are the midpoints of AD, BC respectively. Diagonals
BD and AC meet MN at points P and Q respectively. If AB = 37, CD = 63, find P Q.
10. Given that tan a, tan b, tan c are the roots of the equation x3 − 4x2 + 7x − 2 = 0. What is the
value of tan(a + b + c).
Part III: Olympian Questions. Each correct answer is worth 5 points.
1. What is the surface area of the sphere that is tangent to the plane 3x + 4y + 12z = 19 and is
centered at (4, −8, 0) ?
2. Triangle XY Z has points A on XY , B on Y Z and C on XZ. Given that XC : CZ = 1 : 2, ZB :
BY = 2 : 3, Y A : AX = 3 : 4 and QRS is the triangle that is formed when XB, Y C, AZ
intersect at Q, R, S, what is the ratios of the area of ABC and QRS?
3. Daniel and James are in a room with floors designed like a coordinate system. They want
to play a game where Daniel must start at a point where the coordinates are (20, 17) and he
must go to James at (−20, −17) the shortest possible path but Daniel must go to a point P
at the y-axis first and get a laser shield at (17, −20), fight minions until you reach a point Q
at the x-axis, and finally ran towards James. How long is P Q in exact form?
4. For what value/s of x in the interval (0◦ , 360◦ ) does the equation cos 2x + sin 2x = 3 cos x +
sin x − 2 satisfy?
5. What is the distance between the circumcenter and the incenter of a triangle of side lengths
18, 48, and 50?
6. What are the coordinates of the orthocenter of the triangle whose vertices are at (20, 17), (−17, 20),
and (0, 18)?
7. An electric cable of uniform linear density that is simply supported by two posts of the same
height has a horizontal span of 80 meters and the vertical distance from the top of the post
to the bottom most portion of the cable is 10 meters. If the height of the posts are 20 meters,
how high from the ground is a point on the cable that is 25 meters, along the horizontal, at
midspan?
8. Express the sum of cos 1◦ + cos 2◦ + · · · + cos 90◦ in terms of a single trigonometric function.
9. A ball is dropped from a height of 123 m. For every bounce, it rises two-thirds of its previous
height. What is the total distance the ball travelled until it come up to rest?
10. y = ax2 + bx + c is a parabola that passes through (−2, 1) and (2, 9). Given that it does not
intersect the x-axis, what is the range of the values for the abscissa of its vertex?
Part IV: Titan Questions. Each correct answer is worth 7 points.
1. Find all possible solutions of x in the equation sin x + sin 2x + sin 3x + sin 4x + sin 5x =
cos x + cos 2x + cos 3x + cos 4x + cos 5x in the interval (0, π).
2. A triangular lot ABC for which AB = 20m, BC = 15m, AC = 32m. Martin wants to divide
his lot into two equal areas where one endpoint of the dividing line divides AB in the ratio
2 : 5 where the point is nearer to A. What is the length of the dividing line? Answers in 4
decimal places.
3. A cake has a height of 40cm whose base is elliptical in shape where the major axis is 50cm
long while its minor axis is 30cm long. If every cross-section of the cake that is perpendicular
to the minor axis is a semi-ellipse, what is the total volume of the cake?
4. 4ABC has a, b and c as the respective opposite sides of its interior angle. If the given angles
and sides satisfy the condition a cos A + b cos B = c cos C, what type of triangle is 4ABC? If
it is a right-angled triangle, state which angle/s is/may be 90◦ .
5. Find two points M and N on the line 2x + y = 7 and on the circle x2 + y 2 + 2x − 2y = 167
such that the midpoint of M N is (2, 10)

sin x + sin y = 85/217
6. Solve for tan(x + y) if
204
cos x + cos y =
217
7. Three congruent circles are externally tangent to each other. Another circle isformed by
joining the three centers of the first three circles. What is the area of the region common to
the big circle and the three circles? Answers in 4 decimal places.
8. What is the equation of the ellipse √
that is centered at the origin, has latus rectum length of
12.5 units, and foci located at (0, ± 21 )?
9. A conical tank, has its radius on top, is filled with water at a height of 0.5 m. If the tanke is
2 m high, what will be the height of the water in the cone when it is placed in a the upside
down position?
10. Find the positive integer n such that arctan
1
1
1
1
π
+ arctan + arctan + arctan = .
3
4
5
n
4
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