143 0
【COMC 2016】Two lines intersect at a point Q at an angle θ◦, where 0 < θ < 180.
c4.two lines intersect at a point q at an angle θ◦, where 0 &lt; θ &lt; 180. a frog is originally at a point other than q on the angle bisector of this angle. the frog alternately ..

130 0
【COMC 2016】Let A = (0, a), O = (0, 0), C = (c, 0), B = (c, b), where a, b, c are positive integers.
c3.let a = (0, a), o = (0, 0), c = (c, 0), b = (c, b), where a, b, c are positive integers. let p = (p, 0) be the point on line segment oc that minimizes the distance ap + pb, over.. 93 0
【COMC 2016】Alice and Bob play a game, taking turns, playing on a row of n seats.
c2.alice and bob play a game, taking turns, playing on a row of n seats. on a player’s turn, he or she places a coin on any seat provided there is no coin on that seat or on an adj..

101 0
【COMC 2016】A sequence of three numbers a, b, c form an arithmetic sequence if the difference between
c1.a sequence of three numbers a, b, c form an arithmetic sequence if the difference between successive terms in the sequence is the same. that is, when b − a = c − b. (a) the se..

115 0
【COMC 2016】Let n be a positive integer.
b4.let n be a positive integer. given a real number x, let [图片] be the greatest integer less than or equal to x. for example,[图片] and [图片]. define a sequence a1, a2, a3, . . . .. 82 0
【COMC 2016】A hexagon ABCDEF has AB = 18cm, BC = 8cm, CD = 10cm, DE = 15cm, EF = 20cm, FA = 1cm
b3.a hexagon abcdef has ab = 18cm, bc = 8cm, cd = 10cm, de = 15cm, ef = 20cm, fa = 1cm, ∠fab = 90°,∠cde = 90° and bc is parallel to ef. determine the area of this hexagon, in cm2...

98 0
【COMC 2016】The squares of a 6 × 6 square grid are each labelled with a point value.
b2.the squares of a 6 × 6 square grid are each labelled with a point value. as shown in the diagram below, the point value of the square in row i and column j is i × j. [图片] .. 122 0
【COMC 2016】 If the cubic equation x3 ? 10x2 + Px ? 30 = 0 has three positive integer roots
b1.if the cubic equation x3 ? 10x2 + px ? 30 = 0 has three positive integer roots, determine the value of p...

83 0
【COMC 2016】Three males and two females write their names on sheets of paper
a4.three males and two females write their names on sheets of paper, and randomly arrange them in order, from left to right. what is the probability that all of the female names ..

104 0
【COMC 2016】Determine the pair of real numbers x, y which satisfy the system of equations:
a3.determine the pair of real numbers x, y which satisfy the system of equations:..

88 0
【COMC 2016】A square is inscribed in a circle, as shown in the figure.
a2.a square is inscribed in a circle, as shown in the figure. if the area of the circle is 16π cm2 and the area of the square is s cm2, what is the value of s?..

84 0
【COMC 2016】Pat has ten tests to write in the school year.
a1.pat has ten tests to write in the school year. he can obtain a maximum score of 100 on each test. the average score of pat’s first eight tests is 80 and the average score of all..

137 0
【COMC 2017】Let n be a positive integer and Sn = {1, 2, . . . , 2n - 1, 2n}.
c4.let n be a positive integer and sn = {1, 2, . . . , 2n - 1, 2n}. a perfect pairing of sn is defined to be a partitioning of the 2n numbers into n pairs, such that the sum of the..

91 0
【COMC 2017】Let XYZ be an acute-angled triangle.
c3.let xyz be an acute-angled triangle. let s be the side-length of the square which has two adjacent vertices on side y z, one vertex on side xy and one vertex on side xz. let h b.. 104 0
【COMC 2017】A function f(x) is periodic with period T > 0 if f(x + T) = f(x) for all x.
c2.a function f(x) is periodic with period t &gt; 0 if f(x + t) = f(x) for all x. the smallest such number t is called the least period. for example, the functions sin(x) and cos(x..

112 0
【COMC 2017】For a positive integer n, we define function P(n) to be the sum of the digits of n plus
c1.for a positive integer n, we define function p(n) to be the sum of the digits of n plus the number of digits of n. for example, p(45) = 4 + 5 + 2 = 11. (note that the first digi..

79 0
【COMC 2017】Numbers a, b and c form an arithmetic sequence if b - a = c - b.
b4.numbers a, b and c form an arithmetic sequence if b - a = c - b. let a, b, c be positive integers forming an arithmetic sequence with a &lt; b &lt; c. let f(x) = ax2 + bx + c. t..

91 0
【COMC 2017】Regular decagon (10-sided polygon) ABCDEFGHIJ has area 2017 square units
b3.regular decagon (10-sided polygon) abcdefghij has area 2017 square units. determine the area (in square units) of the rectangle cdhi. [图片].. 96 0
【COMC 2017】There are twenty people in a room, with a men and b women.
b2.there are twenty people in a room, with a men and b women. each pair of men shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a wo..

99 0
【COMC 2017】Andrew and Beatrice practice their free throws in basketball.
b1.andrew and beatrice practice their free throws in basketball. one day, they attempted a total of 105 free throws between them, with each person taking at least one free throw. i..

96 0
【COMC 2017】Three positive integers a, b, c satisfy 4a × 5b × 6c = 88 × 99 × 1010.
a4.three positive integers a, b, c satisfy 4a × 5b × 6c = 88 × 99 × 1010 . determine the sum of a + b + c...

76 0
【COMC 2017】Two 1 × 1 squares are removed from a 5 × 5 grid as shown.
a3.two 1 × 1 squares are removed from a 5 × 5 grid as shown. [图片] determine the total number of squares of various sizes on the grid... 87 0
【COMC 2017】An equilateral triangle has sides of length 4cm.
a2.an equilateral triangle has sides of length 4cm. at each vertex, a circle with radius 2cm is drawn, as shown in the figure below. the total area of the shaded regions of the thr.. 86 0
【COMC 2017】The average of the numbers 2, 5, x, 14, 15 is x. Determine the value of x.
a1.the average of the numbers 2, 5, x, 14, 15 is x. determine the value of x...

149 0
【COMC2018】Given a positive integer N, Matt writes N in decimal on a blackboard
c4.given a positive integer n, matt writes n in decimal on a blackboard, without writing any of the leading 0s. every minute he takes two consecutive digits, erases them, and repla..

102 0
Consider a convex quadrilateral ABCD. 【COMC2018】
c3.consider a convex quadrilateral abcd. let rays ba and cd intersect at e, rays da and cb intersect at f, and the diagonals ac and bd intersect at g. it is given that the triangle..

127 0
【COMC2018】Alice has two boxes A and B. Initially box A contains n coins and box B is empty.
c2.alice has two boxes a and b. initially box a contains n coins and box b is empty. on each turn, she may either move a coin from box a to box b, or remove k coins from box a, whe..

91 0
【COMC2018】At Math-ee -Mart, cans of cat food arearranged in an pentagonal pyramid of 15 layers high
c1.at math-ee -mart, cans of cat food arearranged in an pentagonal pyramid of 15 layers high, with 1 can in the top layer, 5 cans in the second layer, 12 cans in the third layer, 2.. 90 0
【COMC2018】Determine the number of 5-tuples of integers (x1, x2, x3, x4, x5) such that
b4.determine the number of 5-tuples of integers (x1, x2, x3, x4, x5) such that (a) xi ≥ i for 1 ≤ i ≤ 5; (b)[图片].. 117 0
【COMC2018】The doubling sum function is defined by
b3.the doubling sum function is defined by[图片] for example, we have d(5, 3) = 5 + 10 + 20 = 35 and d(11, 5) = 11 + 22 + 44 + 88 + 176 = 341. determine the smallest positive .. 