Laurie plays a game called $\textit{bash}$ where she picks two distinct numbers between $1$ and $10$, inclusive, at random. She then finds their sum, product, and non-negative difference. At random, she picks two of these three numbers and tells them to Ali. If the probability that Ali is able to logically deduce the original numbers can be written as $\frac{m}{n}$, with $m$ and $n$ relatively prime, find $m+n$. [i]Proposed by [b] atmchallenge [/b][/i]

A knight is placed at the origin of the Cartesian plane. Each turn, the knight moves in an chess $\text{L}$-shape ($2$ units parallel to one axis and $1$ unit parallel to the other) to one of eight possible location, chosen at random. After $2016$ such turns, what is the expected value of the square of the distance of the knight from the origin?

The sides of a triangle are of lengths $13$, $14$, and $15$. The altitudes of the triangle meet at point $H$. If $AD$ is the altitude to the side length $14$, what is the ratio $HD:HA$? $\textbf{(A) } 3 : 11\qquad \textbf{(B) } 5 : 11\qquad \textbf{(C) } 1 : 2\qquad \textbf{(D) }2 : 3\qquad \textbf{(E) }25 : 33$

$ABCD$ is a convex quadrilateral where $\angle A = 45^\circ$ and $\angle C = 135^\circ$. $P$ is a point strictly inside $\triangle ABC$ such that $\angle BAP = \angle CAD$ and $\angle BCP = \angle ACD$. Prove that $PB \perp PD$ if and only if $AC \perp BD$.

Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. How many minutes did it take Joe to get from home to school? $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 7.3\qquad\textbf{(C)}\ 7.7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 8.3 $

A real number with absolute value less than $1$ is written in each cell of an $n\times n$ array, so that the sum of the numbers in each $2\times 2$ square is zero. Show that for odd $n$ the sum of all the numbers is less than $n$.

Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system \[ x_5+x_2=yx_1 \] \[ x_1+x_3=yx_2 \] \[ x_2+x_4=yx_3 \] \[ x_3+x_5=yx_4 \] \[ x_4+x_1=yx_5 \] where $y$ is a parameter.

We call a pair of polygons, $p$ and $q$, [i]nesting[/i] if we can draw one inside the other, possibly after rotation and/or reflection; otherwise we call them [i]non-nesting[/i]. Let $p$ and $q$ be polygons. Prove that if we can find a polygon $r$, which is similar to $q$, such that $r$ and $p$ are non-nesting if and only if $p$ and $q$ are not similar.

For a positive integer $k>1$ with $\gcd(k,2020)=1,$ we say a positive integer $N$ is [i]$k$-bad[/i] if there do not exist nonnegative integers $x$ and $y$ with $N=2020x+ky$. Suppose $k$ is a positive integer with $k>1$ and $\gcd(k,2020)=1$ such that the following property holds: if $m$ and $n$ are positive integers with $m+n=2019(k-1)$ and $m \geq n$ and $m$ is $k$-bad, then $n$ is $k$-bad. Compute the sum of all possible values of $k$. [i]Proposed by Jaedon Whyte[/i]

Given a circle $\Gamma$ with center $I$, and a triangle $\triangle ABC$ with all its sides tangent to $\Gamma$. A line $\ell$ is drawn such that it bisects both the area and the perimeter of $\triangle ABC$. Prove that line $\ell$ passes through $I$.

The triangle $ABC$ has $\angle ACB = 30^o$, $BC = 4$ cm and $AB = 3$ cm . Compute the altitudes of the triangle.

a) Let $ax^3 + bx^2 + cx + d$ be divisible by $5$ for given positive integers $a, b, c, d$ and any integer $x$. Prove that $a, b, c$ and $d$ are all divisible by $5$. b) Let $ax^4 + bx^3 + cx^2 + dx + e$ be divisible by $7$ for given positive integers $a, b, c, d, e$ and all integers $x$. Prove that $a, b, c, d$ and $e$ are all divisible by $7$.

$9$ horizontal and $9$ vertical lines are drawn through a square. Prove that it is possible to select $20$ rectangles so that the sides of each rectangle is a segment of one of the given lines (including the sides of the square), and for any two of the $20$ rectangles, it is possible to cover one of them with the other (rotations are allowed).

Find all $n \in \mathbb{N}$ such that $ 2^{n-1}$ divides $n!$.

For some integers $a$ and $b$ the function $f(x)=ax+b$ has the properties that $f(f(0))=0$ and $f(f(f(4)))=9$. Find $f(f(f(f(10))))$.

We denote by $S(k)$ the sum of digits of a positive integer number $k$. We say that the positive integer $a$ is $n$-good, if there is a sequence of positive integers $a_0$, $a_1, \dots , a_n$, so that $a_n = a$ and $a_{i + 1} = a_i -S (a_i)$ for all $i = 0, 1,. . . , n-1$. Is it true that for any positive integer $n$ there exists a positive integer $b$, which is $n$-good, but not $(n + 1)$-good? A. Antropov

Compute the sum of all positive integers $x$ such that $(x-17)\sqrt{x-1}+(x-1)\sqrt{x+15}$ is an integer.

There are $12$ acrobats who are assigned a distinct number ($1, 2, \cdots , 12$) respectively. Half of them stand around forming a circle (called circle A); the rest form another circle (called circle B) by standing on the shoulders of every two adjacent acrobats in circle A respectively. Then circle A and circle B make up a formation. We call a formation a “[i]tower[/i]” if the number of any acrobat in circle B is equal to the sum of the numbers of the two acrobats whom he stands on. How many heterogeneous [i]towers[/i] are there? (Note: two [i]towers[/i] are homogeneous if either they are symmetrical or one may become the other one by rotation. We present an example of $8$ acrobats (see attachment). Numbers inside the circle represent the circle A; numbers outside the circle represent the circle B. All these three formations are “[i]towers[/i]”, however they are homogeneous [i]towers[/i].)

Find the smallest real number $p$ such that the inequality $\sqrt{1^2+1}+\sqrt{2^2+1}+...+\sqrt{n^2+1} \le \frac{1}{2}n(n+p)$ holds for all natural numbers $n$.

Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ?

Which of the following sets of whole numbers has the largest average? \[ \textbf{(A)}\ \text{multiples of 2 between 1 and 101} \qquad \textbf{(B)}\ \text{multiples of 3 between 1 and 101} \\ \textbf{(C)}\ \text{multiples of 4 between 1 and 101} \qquad \textbf{(D)}\ \text{multiples of 5 between 1 and 101} \\ \textbf{(E)}\ \text{multiples of 6 between 1 and 101} \]

There is a complex number $K$ such that the quadratic polynomial $7x^2 +Kx + 12 - 5i$ has exactly one root, where $i =\sqrt{-1}$. Find $|K|^2$.

Let $B$ be a set of more than $\tfrac{2^{n+1}}{n}$ distinct points with coordinates of the form $(\pm 1, \pm 1, \cdots, \pm 1)$ in $n$-dimensional space with $n \ge 3$. Show that there are three distinct points in $B$ which are the vertices of an equilateral triangle.

Let $n\geq 4$ be an integer, and let $a_1,a_2,\ldots,a_n$ be positive real numbers such that \[ a_1^2+a_2^2+\cdots +a_n^2=1 . \] Prove that the following inequality takes place \[ \frac{a_1}{a_2^2+1}+\cdots +\frac{a_n}{a_1^2+1} \geq \frac{4}{5}\left( a_1 \sqrt{a_1}+\cdots +a_n \sqrt{a_n} \right)^2 . \] [i]Bogdan Enescu, Mircea Becheanu[/i]

On $\vartriangle ABC$ let $D$ be a point on side $\overline{AB}$, $F$ be a point on side $\overline{AC}$, and $E$ be a point inside the triangle so that $\overline{DE}\parallel \overline{AC}$ and $\overline{EF} \parallel \overline{AB}$. Given that $AF = 6, AC = 33, AD = 7, AB = 26$, and the area of quadrilateral $ADEF$ is $14$, nd the area of $\vartriangle ABC$.