Let $p,q,r$ be prime numbers such that $2p>q$, $q > 2r$ and $q>p+r$. Prove that $p+q+r\ge 20$.

Find all integers $n$ and $m$, $n > m > 2$, and such that a regular $n$-sided polygon can be inscribed in a regular $m$-sided polygon so that all the vertices of the $n$-gon lie on the sides of the $m$-gon.

Let $\vartriangle ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP,BP,CP$ meet the sides $BC,CA,AB$ in the points $X,Y,Z$ respectively. Prove that $XY \cdot YZ\cdot ZX \ge XB\cdot YC\cdot ZA$.

For the integer $n>1$, define $D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}$. Prove that for any integer $k>1$, there exists pairwise distinct positive integers $n_1,n_2,\ldots,n_k$ such that $n_1,\ldots,n_k>1$ and $|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2$.

Let $k$ be a positive integer. The little one and the magician on the skywalk play a game. Initially, there are $N = 2^k$ distinct balls line up in a row, with each of the ball covered by a cup. On each turn, the little one chooses two cups, then the magician can either swap the balls in the two cups, or do a fake move so that the balls in the two cups stay the same. The little one cannot distinguish whether the magician fakes a move on not, nor can she observe the balls inside the cups. After $M = k \times 2^{k-1}$ turns, the magician opens all cups so the little one can check the ball in each of the cups. If the little one can identify whether the magician fakes a move or not for each of the $M$ turns, then the little one win. Prove that the little one has a winning strategy. [i] Proposed by usjl[/i]

When making a batch of $N \ge 5$ coins, a worker mistakenly made two coins from a different material (all coins look the same). The boss knows that there are exactly two such coins, that they weigh the same, but differ in weight from the others. The employee knows what coins these are and that they are lighter than others. He needs, after carrying out two weighings on cup scales without weights, to convince his boss that the coins are counterfeit easier than real ones, and in which coins are counterfeit. Can he do it?

$X$ has $n$ elements. $F$ is a family of subsets of $X$ each with three elements, such that any two of the subsets have at most one element in common. Show that there is a subset of $X$ with at least $\sqrt{2n}$ members which does not contain any members of $F$.

There are $n$ matches on the table ($n > 1$). Two players take turns shooting them from the table. On the first move, the player removes any number of matches from the table from $1$ to $n - 1$, and then each time you can take no more matches from the table, than the partner took with the previous move. The one who took the last match wins.. Find all $n$ for which the first player can provide win for yourself.

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.