AMC
Pascal/Cayley/Fermat
Lynx
COMC
??
100
Three positive integers are equally spaced on a number line. The middle number is 15, and the largest number is 4 times the smallest number. What is the smallest of these three numbers?
6
100
The integers a, b and c satisfy the equations a + 5 = b and 5 + b = c and b + c = a. The value of b is
-10
100
Let x, y, and z be positive integers for which x + 2y = 60; y + 2z = 70; z + 2x = 110 What is the value of x + y + z?
80
100
John had a box of candies. On the first day he ate exactly half of the candies and gave one to his little sister. On the second day he ate exactly half of the remaining candies and gave one to his little sister. On the third day he ate exactly half of the remaining candies and gave one to his little sister, at which point no candies remained. How many candies were in the box at the start?
14
200
A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?
1
200
Carl and Andr´e are running a race. Carl runs at a constant speed of x m/s. Andr´e runs at a constant speed of y m/s. Carl starts running, and then Andr´e starts running 20 s later. After Andr´e has been running for 10 s, he catches up to Carl. The ratio y : x is equivalent to
3:1
200
Let T be the area of an equilateral triangle with side length 12 units, and let H be the area of a regular hexagon with side length 4 units. What is the value of the ratio T/H ?
3/2
200
Initially, there are four red balls, seven green balls, eight blue balls, ten white balls, and eleven black balls on a table. Every minute, we may repaint one of the balls into any of the other four colours. What is the minimum number of minutes after which the number of balls of each of the five colours is the same?
5
300
Isosceles triangle 
has 
, and a circle with radius 
is tangent to line 
at 
and to line 
at 
. What is the area of the circle that passes through vertices 
, , and 
26pi
300
A sequence has 101 terms, each of which is a positive integer. If a term, n, is even, the next term is equal to 1 2 n + 1. If a term, n, is odd, the next term is equal to 1 2 (n + 1). For example, if the first term is 7, then the second term is 4 and the third term is 3. If the first term is 16, the 101st term is
2
300
Consider the sequence t1, t2, t3, . . . , t15, t16, t17. This sequence has the following three properties. (i) Each of the 17 integers from 1 to 17 appears exactly once in the sequence. (ii) The sum of each pair of consecutive terms is a perfect square. (For example, t1+t2 is a perfect square, t2+t3 is a perfect square, and so on.) (iii) t1 = 17. What is the value of t5?
10
300
The floor function of any real number a is the integer number denoted by ⌊a⌋ such that ⌊a⌋ ≤ a and ⌊a⌋ > a − 1. For example, ⌊5⌋ = 5, ⌊π⌋ = 3 and ⌊−1.5⌋ = −2. Find the difference between the largest integer solution of the equation ⌊x/3⌋ = 102 and the smallest integer solution of the equation ⌊x/3⌋ = −102.
614
300
How many math contests have there been in total this year?
Name them all
(AMC 8,10,12 counts as 1)
13
400
Square pyramid ABCDE has base ABCD, which measures 3 cm on a side, and altitude AE perpendicular to the base, which measures 6 cm. Point P lies on BE, one third of the way from B to E; point Q lies on DE, one third of the way from D to E; and point R lies on CE, two thirds of the way from C to E. What is the area, in square centimeters, of 
?
2√2
400
Quadrilateral ABCD has ∠BCD = ∠DAB = 90◦ . The perimeter of ABCD is 224 and its area is 2205. One side of ABCD has length 7. The remaining three sides have integer lengths. The sum of the squares of the side lengths of ABCD is S. What is the integer formed by the rightmost two digits of S?
60
400
Let A, B, C, D, E, F be six points equally spread out around a circle. Draw all 15 edges connecting two of these six points. Aponi picks 8 of these 15 edges and colours them red; the remaining 7 edges are then coloured blue. Consider all triangles that can be formed from three of these six points. For each triangle that has only red edges, Aponi scores 1 point. For each triangle that has only blue edges, Aponi scores 2 points. For all other triangles (e.g. a triangle with two red edges and one blue edge), Aponi scores 0 points. Let X be the maximum score that Aponi can obtain, and let Y be the minimum score that Aponi can obtain. What is the value of X − Y ?
9
400
Determine all integers a for which a/(1011 − a) is an even integer.
1010, 1012, 1008, 1014, 674, 1348, 0, 2022
400
What is the date of the first math club meeting this school year?
Oct. 3