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*发表时间：*2022-02-15 14:50:36

169
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c4.two lines intersect at a point q at an angle θ◦, where 0 < θ < 180. a frog is originally at a point other than q on the angle bisector of this angle. the frog alternately ..

*发表时间：*2022-02-15 14:47:01

163
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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..

*发表时间：*2022-02-15 14:45:17

117
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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..

*发表时间：*2022-02-15 14:36:27

136
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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..

*发表时间：*2022-02-15 14:23:06

161
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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, . . . ..

*发表时间：*2022-02-15 14:18:48

109
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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...

*发表时间：*2022-02-15 14:12:26

125
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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] ..

*发表时间：*2022-02-15 14:09:57

160
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b1.if the cubic equation x3 ? 10x2 + px ? 30 = 0 has three positive integer roots, determine the value of p...

*发表时间：*2022-02-15 14:04:36

113
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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 ..

*发表时间：*2022-02-15 13:57:58

130
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a3.determine the pair of real numbers x, y which satisfy the system of equations:..

*发表时间：*2022-02-15 13:52:51

120
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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?..

*发表时间：*2022-02-15 13:47:46

115
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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..

*发表时间：*2022-02-15 11:51:45

180
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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..

*发表时间：*2022-02-15 11:49:40

125
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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..

*发表时间：*2022-02-15 11:47:27

129
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c2.a function f(x) is periodic with period t > 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..

*发表时间：*2022-02-15 11:46:19

148
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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..

*发表时间：*2022-02-15 11:44:32

108
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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 < b < c. let f(x) = ax2 + bx + c. t..

*发表时间：*2022-02-15 11:43:09

117
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b3.regular decagon (10-sided polygon) abcdefghij has area 2017 square units. determine the area (in square units) of the rectangle cdhi. [图片]..

*发表时间：*2022-02-15 11:40:56

125
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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..

*发表时间：*2022-02-15 11:38:54

131
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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..

*发表时间：*2022-02-15 11:36:41

128
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a4.three positive integers a, b, c satisfy 4a × 5b × 6c = 88 × 99 × 1010 . determine the sum of a + b + c...

*发表时间：*2022-02-15 11:34:41

104
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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...

*发表时间：*2022-02-15 11:31:55

117
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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..

*发表时间：*2022-02-15 11:28:52

112
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a1.the average of the numbers 2, 5, x, 14, 15 is x. determine the value of x...

*发表时间：*2022-02-13 18:13:13

187
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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..

*发表时间：*2022-02-13 18:11:09

138
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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..

*发表时间：*2022-02-13 18:09:23

171
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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..

*发表时间：*2022-02-13 18:06:53

125
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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..

*发表时间：*2022-02-13 17:55:58

117
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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]..

*发表时间：*2022-02-13 17:52:48

155
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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 ..