Source: 2017 Canadian Open Math Challenge, Problem A1 The average of the numbers $2$, $5$, $x$, $14$, $15$ is $x$. Determine the value of $x$ .

Source: 2017 Canadian Open Math Challenge, Problem A2 An equilateral triangle has sides of length $4$cm. At each vertex, a circle with radius $2$cm is drawn, as shown in the figure below. The total area of the shaded regions of the three circles is $a\cdot \pi \text{cm}^2$. Determine $a$. [asy][asy] size(2.5cm); draw(circle((0,2sqrt(3)/3),1)); draw(circle((1,-sqrt(3)/3),1)); draw(circle((-1,-sqrt(3)/3),1)); draw((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle); fill(((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle),mediumgray); draw((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle); fill(((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle),mediumgray); draw((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle); fill(((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle),mediumgray); [/asy][/asy]

Source: 2017 Canadian Open Math Challenge, Problem A3 Two $1$ × $1$ squares are removed from a $5$ × $5$ grid as shown. [asy][asy] size(3cm); for(int i = 0; i < 6; ++i) { for(int j = 0; j < 6; ++j) { if(j < 5) { draw((i, j)--(i, j + 1)); } } } draw((0,1)--(5,1)); draw((0,2)--(5,2)); draw((0,3)--(5,3)); draw((0,4)--(5,4)); draw((0,5)--(1,5)); draw((2,5)--(5,5)); draw((0,0)--(2,0)); draw((3,0)--(5,0)); [/asy][/asy] Determine the total number of squares of various sizes on the grid.

Source: 2017 Canadian Open Math Challenge, Problem A4 Three positive integers $a$, $b$, $c$ satisfy $$4^a \cdot 5^b \cdot 6^c = 8^8 \cdot 9^9 \cdot 10^{10}.$$ Determine the sum of $a + b + c$.

Source: 2017 Canadian Open Math Challenge, Problem 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. If Andrew made exactly $1/3$ of his free throw attempts and Beatrice made exactly $3/5$ of her free throw attempts, what is the highest number of successful free throws they could have made between them?

Source: 2017 Canadian Open Math Challenge, Problem 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 woman. The total number of handshakes is $106$. Determine the value of $a \cdot b$.

Source: 2017 Canadian Open Math Challenge, Problem B3 Regular decagon (10-sided polygon) $ABCDEFGHIJ$ has area $2017$ square units. Determine the area (in square units) of the rectangle $CDHI$. [asy][asy] pair A,B,C,D,E,F,G,H,I,J; A = (0.809016994375, 0.587785252292); B = (0.309016994375, 0.951056516295); C = (-0.309016994375, 0.951056516295); D = (-0.809016994375, 0.587785252292); E = (-1, 0); F = (-0.809016994375, -0.587785252292); G = (-0.309016994375, -0.951056516295); H = (0.309016994375, -0.951056516295); I = (0.809016994375, -0.587785252292); J = (1, 0); label("$A$",A,NE); label("$B$",B,NE); label("$C$",C,NW); label("$D$",D,NW); label("$E$",E,E); label("$F$",F,E); label("$G$",G,SW); label("$H$",H,S); label("$I$",I,SE); label("$J$",J,2*dir(0)); fill(C--D--H--I--cycle,mediumgrey); draw(polygon(10)); [/asy][/asy]

Source: 2017 Canadian Open Math Challenge, Problem 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 $f(x) = ax2 + bx + c$. Two distinct real numbers $r$ and $s$ satisfy $f(r) = s$ and $f(s) = r$. If $rs = 2017$, determine the smallest possible value of $a$.

Source: 2017 Canadian Open Math Challenge, Problem 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 digit of $n$ reading from left to right, cannot be $0$). $\qquad$(a) Determine $P(2017)$. $\qquad$(b) Determine all numbers $n$ such that $P(n) = 4$. $\qquad$(c) Determine with an explanation whether there exists a number $n$ for which $P(n) - P(n + 1) > 50$.

Source: 2017 Canadian Open Math Challenge, Problem 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 $\cos(x)$ are periodic with least period $2\pi$. $\qquad$(a) Let a function $g(x)$ be periodic with the least period $T = \pi$. Determine the least period of $g(x/3)$. $\qquad$(b) Determine the least period of $H(x) = sin(8x) + cos(4x)$ $\qquad$(c) Determine the least periods of each of $G(x) = sin(cos(x))$ and $F(x) = cos(sin(x))$.

Source: 2017 Canadian Open Math Challenge, Problem C3 Let $XYZ$ be an acute-angled triangle. Let $s$ be the side-length of the square which has two adjacent vertices on side $YZ$, one vertex on side $XY$ and one vertex on side $XZ$. Let $h$ be the distance from $X$ to the side $YZ$ and let $b$ be the distance from $Y$ to $Z$. [asy][asy] pair S, D; D = 1.27; S = 2.55; draw((2, 4)--(0, 0)--(7, 0)--cycle); draw((1.27,0)--(1.27+2.55,0)--(1.27+2.55,2.55)--(1.27,2.55)--cycle); label("$X$",(2,4),N); label("$Y$",(0,0),W); label("$Z$",(7,0),E); [/asy][/asy] (a) If the vertices have coordinates $X = (2, 4)$, $Y = (0, 0)$ and $Z = (4, 0)$, find $b$, $h$ and $s$. (b) Given the height $h = 3$ and $s = 2$, find the base $b$. (c) If the area of the square is $2017$, determine the minimum area of triangle $XYZ$.

Source: 2017 Canadian Open Math Challenge, Problem C4 Let n be a positive integer and $S_n = \{1, 2, . . . , 2n - 1, 2n\}$. A perfect pairing of $S_n$ is defined to be a partitioning of the $2n$ numbers into $n$ pairs, such that the sum of the two numbers in each pair is a perfect square. For example, if $n = 4$, then a perfect pairing of $S_4$ is $(1, 8),(2, 7),(3, 6),(4, 5)$. It is not necessary for each pair to sum to the same perfect square. (a) Show that $S_8$ has at least one perfect pairing. (b) Show that $S_5$ does not have any perfect pairings. (c) Prove or disprove: there exists a positive integer $n$ for which $S_n$ has at least $2017$ different perfect pairings. (Two pairings that are comprised of the same pairs written in a different order are considered the same pairing.)