For how many integers $1\leq k\leq 2013$ does the decimal representation of $k^k$ end with a $1$?

Hexagon $ABCDEF$ has a circumscribed circle and an inscribed circle. If $AB = 9$, $BC = 6$, $CD = 2$, and $EF = 4$. Find $\{DE, FA\}$.

Let $S$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2009}$. What is $\log_2(S)$?

If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$, determine the product of all possible values of $ab$.

Call a polygon [i]normal[/i] if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer?

I have $8$ unit cubes of different colors, which I want to glue together into a $2\times 2\times 2$ cube. How many distinct $2\times 2\times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are.

Let $m,n$ be positive integers with $m \ge n$. Let $S$ be the set of pairs $(a,b)$ of relatively prime positive integers such that $a,b \le m$ and $a+b > m$. For each pair $(a,b)\in S$, consider the nonnegative integer solution $(u,v)$ to the equation $au - bv = n$ chosen with $v \ge 0$ minimal, and let $I(a,b)$ denote the (open) interval $(v/a, u/b)$. Prove that $I(a,b) \subseteq (0,1)$ for every $(a,b)\in S$, and that any fixed irrational number $\alpha\in(0,1)$ lies in $I(a,b)$ for exactly $n$ distinct pairs $(a,b)\in S$. [i]Victor Wang, inspired by 2013 ISL N7[/i]

Let $ABC$ be an acute triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\Omega$. Let $M$ be the midpoint of $AH$ and $N$ the midpoint of $BH$. Assume the points $M$, $N$, $O$, $H$ are distinct and lie on a circle $\omega$. Prove that the circles $\omega$ and $\Omega$ are internally tangent to each other. [i]Dhroova Aiylam and Evan Chen[/i]

22. Let the function $f : \mathbb{Z} \to \mathbb{Z}$ take only integer inputs and have integer outputs. For any integers $x$ and $y$, f satisfies $$f(x) + f(y) = f(x + 1) + f(y - 1)$$ If $f(2016) = 6102$ and $f(6102) = 2016$, what is $f(1)$? 23. Let $d$ be a randomly chosen divisor of $2016$. Find the expected value of $$\frac{d^2}{d^2 + 2016}$$ 24. Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjecent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$?

In a rectangular box $ABCDEFGH$ with edge lengths $AB = AD = 6$ and $AE = 49$, a plane slices through point $A$ and intersects edges $BF$, $FG$, $GH$, $HD$ at points $P$, $Q$, $R$, $S$ respectively. Given that $AP = AS$ and $PQ = QR = RS$, find the area of pentagon $APQRS$.

In triangle $ABC$, $AB = 13$, $BC = 14$, $CA = 15$. Squares $ABB_1A_2$, $BCC_1B_2$, $CAA_1B_2$ are constructed outside the triangle. Squares $A_1A_2A_3A_4$, $B_1B_2B_3B_4$ are constructed outside the hexagon $A_1A_2B_1B_2C_1C_2$. Squares $A_3B_4B_5A_6$, $B_3C_4C_5B_6$, $C_3A_4A_5C_6$ are constructed outside the hexagon $A_4A_3B_4B_3C_4C_3$. Find the area of the hexagon $A_5A_6B_5B_6C_5C_6$.

16. Create a cube $C_1$ with edge length $1$. Take the centers of the faces and connect them to form an octahedron $O_1$. Take the centers of the octahedron’s faces and connect them to form a new cube $C_2$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons. 17. Let $p(x) = x^2 - x + 1$. Let $\alpha$ be a root of $p(p(p(p(x)))$. Find the value of $$(p(\alpha) - 1)p(\alpha)p(p(\alpha))p(p(p(\alpha))$$ 18. An $8$ by $8$ grid of numbers obeys the following pattern: 1) The first row and first column consist of all $1$s. 2) The entry in the $i$th row and $j$th column equals the sum of the numbers in the $(i - 1)$ by $(j - 1)$ sub-grid with row less than i and column less than $j$. What is the number in the 8th row and 8th column?

13. How many functions $f : \{0, 1\}^3 \to \{0, 1\}$ satisfy the property that, for all ordered triples $(a_1, a_2, a_3)$ and $(b_1, b_2, b_3)$ such that $a_i \ge b_i$ for all $i$, $f(a_1, a_2, a_3) \ge f(b_1, b_2, b_3)$? 14. The very hungry caterpillar lives on the number line. For each non-zero integer $i$, a fruit sits on the point with coordinate $i$. The caterpillar moves back and forth; whenever he reaches a point with food, he eats the food, increasing his weight by one pound, and turns around. The caterpillar moves at a speed of $2^{-w}$ units per day, where $w$ is his weight. If the caterpillar starts off at the origin, weighing zero pounds, and initially moves in the positive $x$ direction, after how many days will he weigh $10$ pounds? 15. Let $ABCD$ be an isosceles trapezoid with parallel bases $AB = 1$ and $CD = 2$ and height $1$. Find the area of the region containing all points inside $ABCD$ whose projections onto the four sides of the trapezoid lie on the segments formed by $AB,BC,CD$ and $DA$.

Find the integer closest to \[\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}\]

10. Michael is playing basketball. He makes $10\%$ of his shots, and gets the ball back after $90\%$ of his missed shots. If he does not get the ball back he stops playing. What is the probability that Michael eventually makes a shot? 11. How many subsets $S$ of the set $\{1, 2, \ldots , 10\}$ satisfy the property that, for all $i \in [1, 9]$, either $i$ or $i + 1$ (or both) is in S? 12. A positive integer $\overline{ABC}$, where $A, B, C$ are digits, satisfies $$\overline{ABC} = B^C - A$$ Find $\overline{ABC}$.

4. A square can be divided into four congruent figures as shown: [asy] size(2cm); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((0,1)--(2,1)); [/asy] For how many $n$ with $1 \le n \le 100$ can a unit square be divided into $n$ congruent figures? 5. If $x + 2y - 3z = 7$ and $2x - y + 2z = 6$, determine $8x + y$. 6. Let $ABCD$ be a rectangle, and let $E$ and $F$ be points on segment $AB$ such that $AE = EF = FB$. If $CE$ intersects the line $AD$ at $P$, and $PF$ intersects $BC$ at $Q$, determine the ratio of $BQ$ to $CQ$.