Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF.$ The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4} {5}$ the area of $ABC.$ The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$? [i]Author: Ray Li[/i]

For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$

For the real number $a$ find the number of solutions $(x, y, z)$ of a system of the equations: $\left\{\begin{array}{lll} x+y^2+z^2=a \\ x^2+y+z^2=a \\ x^2+y^2+z=a\end{array}\right.$

How many three-digit perfect squares are there such that if each digit is increased by one, the resulting number is also a perfect square? (A): $1$, (B): $2$, (C): $4$, (D): $8$, (E) None of the above.

Let $S$ be the set $\{-1, 1\}^n$, that is, $n$-tuples such that each coordinate is either $-1$ or $1$. For \[s = (s_1, s_2, \ldots, s_n), t = (t_1, t_2, \ldots, t_n) \in \{-1, 1\}^n,\] define $s \odot t = (s_1t_1, s_2t_2, \ldots, s_nt_n)$. Let $c$ be a positive constant, let $f : S \to \{-1, 1\}$ be a function such that there are at least $(1-c) \cdot 2^{2n}$ pairs $(s, t)$ with $s, t \in S$ such that $f(s \odot t) = f(s)f(t)$. Show that there exists a function $f'$ such that $f'(s \odot t) = f'(s)f'(t)$ for all $s, t \in S$ and $f(s) = f'(s)$ for at least $(1-10c) \cdot 2^n$ values of $s \in S$.

A regular hexagon has positive integer side length. A laser is emitted from one of the hexagon’s corners, and is reflected off the edges of the hexagon until it hits another corner. Let $a$ be the distance that the laser travels. What is the smallest possible value of $a^2$ such that $a > 2019$? You need not simplify/compute exponents.

How many nonnegative integers with at most $40$ digits consisting of entirely zeroes and ones are divisible by $11$?

Triangle $ ABC$ has incentre $ I$ and the incircle touches $ BC, CA$ at $ D, E$ respectively. Let $ BI$ meet $ DE$ at $ G$. Show that $ AG$ is perpendicular to $ BG$.

Consider a board having 2 rows and $n$ columns. Thus there are $2n$ cells in the board. Each cell is to be filled in by $0$ or $1$ . [list=a] [*] In how many ways can this be done such that each row sum and each column sum is even? [*] In how many ways can this be done such that each row sum and each column sum is odd? [/list]

For what values of $a$ does the system of equations \[x^2 = y^2,(x-a)^2 +y^2 = 1\]have exactly 2 solutions?

The integer $202020$ is a multiple of $91$. For each positive integer $n$, show how $n$ additional $2$'s may be inserted into the digits of $202020$ such that the resulting $(n+6)$-digit number is also a multiple of $91$. For example, a possible way to do this when $n=5$ is [u]2[/u]2020[u]2[/u]20[u]222[/u] (the inserted $2$'s are underlined).

Let $ABCD$ be a cyclic quadrilateral. Let $DA$ and $BC$ intersect at $E$ and let $AB$ and $CD$ intersect at $F$. Assume that $A, E, F$ all lie on the same side of $BD$. Let $P$ be on segment $DA$ such that $\angle CPD = \angle CBP$, and let $Q$ be on segment $CD$ such that $\angle DQA = \angle QBA$. Let $AC$ and $PQ$ meet at $X$. Prove that, if $EX = EP$, then $EF$ is perpendicular to $AC$.

A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white? $ \textbf{(A)}\ \dfrac{1}{9} \qquad \textbf{(B)}\ \dfrac{1}{4} \qquad \textbf{(C)}\ \dfrac{4}{9} \qquad \textbf{(D)}\ \dfrac{5}{9} \qquad \textbf{(E)}\ \dfrac{19}{27}$

Let $a,b,c > 0$ be real numbers with the sum equal to $3$. Show that: $$\frac{a+3}{3a+bc}+\frac{b+3}{3b+ca}+\frac{c+3}{3c+ab} \ge 3$$

Points $X$, $Y$, and $Z$ lie on a circle with center $O$ such that $XY=12$. Points $A$ and $B$ lie on segment $XY$ such that $OA=AZ=ZB=BO=5$. Compute $AB$.

Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. Prove that at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle. [i](Dan Schwarz)[/i]

Points $M$ and $N$ are taken on the hypotenuse $AB$ of a right triangle $ABC$ so that $BC = BM$ and $AC = AN$. Prove that the angle $MCN$ is equal to $45$ degrees. (Folklore)

Find the minimal $N$ such that any $N$-element subset of $\{1, 2, 3, 4,...,7\}$ has a subset $S$ such that the sum of elements of $S$ is divisible by $7$.

Let $BC=6$, $BX=3$, $CX=5$, and let $F$ be the midpoint of $\overline{BC}$. Let $\overline{AX}\perp\overline{BC}$ and $AF=\sqrt{247}$. If $AC$ is of the form $\sqrt{b}$ and $AB$ is of the form $\sqrt{c}$ where $b$ and $c$ are nonnegative integers, find $2c+3b$.

When the fraction $\dfrac{49}{84}$ is expressed in simplest form, then the sum of the numerator and the denominator will be $\text{(A)}\ 11 \qquad \text{(B)}\ 17 \qquad \text{(C)}\ 19 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 133$

In a unit square one hundred segments are drawn from the centre to the sides dividing the square into one hundred parts (triangles and possibly quadruilaterals). If all parts have equal perimetr $p$, show that $\dfrac{14}{10} < p < \dfrac{15}{10}$.

Let $f(x) = 3(|x|+|x-1|-|x+1|)$ and let $x_{n+1}=f(x_n)$ $\forall n \ge 0$. How many real number $x_0$ are there, that satisfy $x_0=x_{2007}$ and $x_0,x_1,x_2,...,x_{2006}$ are distinct?

Find all three digit numbers $n$ which are equal to the number formed by three last digit of $n^2$.

Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$. [i]Calvin Deng.[/i]

Chad and Chad2 run competing rare candy stores at Princeton. Chad has a large supply of boxes of candy, each box containing three candies and costing him \$ $3$ to purchase from his supplier. He charges \$ $1.50$ per candy per student. However, any rare candy in an opened box must be discarded at the end of the day at no profit. Chad knows that at each of $8$am, $10$am, noon, $2$pm, $4$pm, and $6$pm, there will be one person who wants to buy one candy, and that they choose between Chad and Chad2 at random. (He knows that those are the only times when he might have a customer.) Chad may refuse sales to any student who asks for candy. If Chad acts optimally, his expected daily profit can be written in simplest form as $\frac{m}{n}$. Find $m + n$. (Chad’s profit is \$ $1.50$ times the number of candies he sells, minus $3 per box he opens.)