紫色彗星数学竞赛真题（2019年高中组）

PURPLE COMET! MATH MEET April 2019

HIGH SCHOOL - PROBLEMS

Copyright Titu Andreescu and Jonathan Kane

Problem 1

Ivan, Stefan, and Katia divided 150 pieces of candy among themselves so that Stefan and Katia each got twice as many pieces as Ivan received. Find the number of pieces of candy Ivan received.

 

Problem 2

The large square in the diagram below with sides of length 8 is divided into 16 congruent squares. Find the

area of the shaded region.

 

Problem 3

The mean of diffffers from the mean of where m and n are relatively prime positive integers. Find m + n.

 

Problem 4

The diagram below shows a sequence of equally spaced parallel lines with a triangle whose vertices lie on these lines. The segment  is 6 units longer than the segment Find the length of segment

Problem 5

Evaluate

 

Problem 6

A pentagon has four interior angles each equal to 110° . Find the degree measure of the fififth interior angle.

 

Problem 7

Find the number of real numbers x that satisfy the equation

(3^x)^x+2 + (4^x)^x+2 − (6^x)^x+2 = 1

 

Problem 8

The diagram below shows a 12 by 20 rectangle split into four strips of equal widths all surrounding an isosceles triangle. Find the area of the shaded region.

Problem 9

Find the positive integer n such that 32 is the product of the real number solutions of

x ^log2(x3)−n = 13.

 

Problem 10

Find the number of positive integers less than 2019 that are neither multiples of 3 nor have any digits that are multiples of 3.

 

Problem 11

Let m > n be positive integers such that 3(3mn − 2)² − 2(3m − 3n)² = 2019. Find 3m + n.

 

Problem 12

The following diagram shows four adjacent 2 × 2 squares labeled 1, 2, 3, and 4. A line passing through the lower left vertex of square 1 divides the combined areas of squares 1, 3, and 4 in half so that the shaded region has area 6. The difffference between the areas of the shaded region within square 4 and the shaded region within square 1 is where p and q are relatively prime positive integers. Find p + q.

 

Problem 13

There are relatively prime positive integers m and n so that the parabola with equation y = 4x² is tangent to the parabola with equation x = y² +  . Find m + n.

 

Problem 14

The circle centered at point A with radius 19 and the circle centered at point B with radius 32 are both internally tangent to a circle centered at point C with radius 100 such that point C lies on segment  Point M is on the circle centered at A and point N is on the circle centered at B such that line MN is a common internal tangent of those two circles. Find the distance MN.

Problem 15

Suppose a is a real number such that sin(π · cos a) = cos(π · sin a). Evaluate 35 sin2 (2a) + 84 cos2 (4a).

 

Problem 16

Find the number of ordered triples of sets (T₁, T₂, T₃) such that

1. each of T₁, T₂, and T₃ is a subset of {1, 2, 3, 4},
2. T₁ ⊆ T₂ ∪ T₃,
3. T₂ ⊆ T₁ ∪ T₃, and
4. T₃ ⊆ T₁ ∪ T₂.

Problem 17

The following diagram shows equilateral triangle △ABC and three other triangles congruent to it. The other three triangles are obtained by sliding copies of △ABC a distance AB along a side of △ABC in the directions from A to B, from B to C, and from C to A. The shaded region inside all four of the triangles has area 300. Find the area of △ABC.

 

Problem 18

A container contains fifive red balls. On each turn, one of the balls is selected at random, painted blue, and returned to the container. The expected number of turns it will take before all fifive balls are colored blue is  where m and n are relatively prime positive integers. Find m + n.

 

Problem 19

Find the remainder when is divided by 2019.

 

Problem 20

In the diagram below, points D, E, and F are on the inside of equilateral △ABC such that D is on , E  is on and the triangles △AEC, △BDA, and △CFB are congruent. Given that AB = 10 and DE = 6, the perimeter of 4 BDA is  where a, b, c, and d are positive integers, b and d are relatively prime, and c is not divisible by the square of any prime. Find a + b + c + d.

 

Problem 21

Each of the 48 faces of eight 1 × 1 × 1 cubes is randomly painted either blue or green. The probability that these eight cubes can then be assembled into a 2 × 2 × 2 cube in a way so that its surface is solid green can be written where p and q are prime numbers and m and n are positive integers. Find p + q + m + n.

 

Problem 22

Let a and b positive real numbers such that (65a² + 2ab + b² )(a² + 8ab + 65b² ) = (8a² + 39ab + 7b²)² . Then

one possible value of  satisfifies where m and n are positive integers. Find m + n.

 

Problem 23

Find the number of ordered pairs of integers (x, y) such that

 

Problem 24

A 12-sided polygon is inscribed in a circle with radius r. The polygon has six sides of length that alternate with six sides of length 2. Find r².

 

Problem 25

The letters AAABBCC are arranged in random order. The probability no two adjacent letters will be the

same is , where m and n are relatively prime positive integers. Find m + n.

 

Problem 26

Let D be a regular dodecahedron, which is a polyhedron with 20 vertices, 30 edges, and 12 regular pentagon faces. A tetrahedron is a polyhedron with 4 vertices, 6 edges, and 4 triangular faces. Find the number of tetrahedra with positive volume whose vertices are vertices of D.

 

Problem 27

Binhao has a fair coin. He writes the number +1 on a blackboard. Then he flflips the coin. If it comes up

heads (H), he writes +, and otherwise, if he flflips tails (T), he writes −. Then he flflips the coin again. If it comes up heads, he writes +14 , and otherwise he writes −14 . Binhao continues to flflip the coin, and on the

nth flflip, if he flflips heads, he writes +, and otherwise he writes −. For example, if Binhao flflips HHTHTHT, he writes The probability that Binhao will generate a

series whose sum is greater than where p and q are relatively prime positive integers. Find p + 10q.

 

Problem 28

There are positive integers m and n such that m² − n = 32 and is a real root of the polynomial x⁵ − 10x³ + 20x − 40. Find m + n.

 

Problem 29

In a right circular cone, A is the vertex, B is the center of the base, and C is a point on the circumference of the base with BC = 1 and AB = 4. There is a trapezoid ABCD with  A right circular cylinder whose surface contains the points A, C, and D intersects the cone such that its axis of symmetry is perpendicular to the plane of the trapezoid, and  is a diameter of the cylinder. A sphere radius r lies inside the cone and inside the cylinder. The greatest possible value of r is , where a, b, c, and d are positive integers, a and d are relatively prime, and b is not divisible by the square of any prime. Find a + b + c + d.

 

Problem 30

A derangement of the letters ABCDEF is a permutation of these letters so that no letter ends up in the position it began such as BDECFA. An inversion in a permutation is a pair of letters xy where x appears before y in the original order of the letters, but y appears before x in the permutation. For example, the derangement BDECFA has seven inversions: AB, AC, AD, AE, AF, CD, and CE. Find the total number of inversions that appear in all the derangements of ABCDEF.

 

扫描下方二维码获取本年度学术活动答案及解析

还可免费领取最新年度真题及答案解析