Given are real numbers $a<b<c<d$. Determine all permutations $p,q,r,s$ of the numbers $a,b,c,d$ for which the value of the sum $$(p-q)^2+(q-r)^2+(r-s)^2+(s-p)^2$$is minimal.

Prove that for every integer $n \ge 1$ there exists a real number $a$ such that for any integer $m \ge 1$ the number $[a^m] + 1$ is divisible by $n$ ($[x]$ denotes the largest integer that does not exceed $x$).

Find all positive integers $m$ and $n$ such that the inequality: \[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \] is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.

Let $B$ be the set of all binary integers that can be written using exactly 5 zeros and 8 ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer 1 is obtained.

Suppose that $S$ is a finite set of points in the plane such that the area of triangle $\triangle ABC$ is at most $1$ whenever $A,B,$ and $C$ are in $S.$ Show that there exists a triangle of area $4$ that (together with its interior) covers the set $S.$

An infinite sequence of positive real numbers $x_0,x_1,x_2,...$ is called $vasco$ if it satisfies the following properties: (a) $x_0=1,x_1=3$; and (b) $x_0+x_1+...+x_{n-1}\ge3x_{n}-x_{n+1}$, for every $n\ge1$. Find the greatest real number $M$ such that, for every $vasco$ sequence, the inequality $\frac{x_{n+1}}{x_{n}}>M$ is true for every $n\ge0$.

There are $11$ empty boxes and a pile of stones. Two players play the following game by alternating moves: In one move a player takes $10$ stones from the pile and places them into boxes, taking care to place no more than one stone in any box. The winner is the player after whose move there appear $21$ stones in one of the boxes for the first time. If a player wants to guarantee that they win the game, should they go first or second? Explain your reasoning.

Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$. She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by $2017$ and add $2$ to the result when it is her turn. Pat will always add $2019$ to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by $2018$. Mary wants to prevent Pat from winning the game. Determine, with proof, the smallest initial integer Mary could choose in order to achieve this.

Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$. [i]Evan O'Dorney.[/i]

Is the series $$\sum_{n=0}^{\infty} \frac{n!}{(n+1)^{n}}\cdot \left(\frac{19}{7}\right)^{n}$$ convergent or divergent?

Prove that there are infinitely many quadruplets of positive integers $(a,b,c,d)$, such that\\ $ab+1$, $bc+16$, $cd+4$, $ad+9$\\ are perfect squares

For a positive integer $n$, we will say that a sequence $a_1, a_2, \dots a_n$ where $a_i \in \{1, 2, \dots , n\}$ for all $i$ is $n$[i]-highly divisible[/i] if, for every positive integer $d$ that divides $n$ and every nonnegative integer $k$ less than $\frac{n}{d}$ we have that \[ d\;\Bigg\vert \sum_{i=kd+1}^{(k+1)d} a_i. \] Let $\chi(n)$ be the probability that a sequence $a_1, a_2, \dots, a_n$ where $a_i$ is chosen randomly from $\{1, 2, \dots n\}$ independently for all $i$ is $n$-highly divisible. Suppose that $n$ is a positive integer such that there exists a positive integer $m$ not divisible by 3 such that $3^{40}\chi(n)=\frac{1}{m}$. Compute the sum of all possible values of $n$. [i]Proposed by Jaedon Whyte[/i]

The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.

Find all functions $f:\mathbb{R}_{\ge 0} \to \mathbb{R}$ with \[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\] for all $x,y,z \in \mathbb{R}_{\ge 0}$.

Given several matrices of the same size. Given a positive integer $N$, let's say that a matrix is $N$-balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$. (i) Show that every $2N$-balanced matrix can be written as a sum of two $N$-balanced matrices. (ii) Show that every $3N$-balanced matrix can be written as a sum of three $N$-balanced matrices.

Let $ABCD$ be an isosceles trapezoid ($AD\parallel BC$), $\angle BAD = 80^o$, $\angle BDA = 60^o$. Point $P$ lies on $CD$ and $\angle PAD = 50^o$. Find $\angle PBC$

Prove that $9$-digit number, that contains all the decimal digits except zero and does not ends with $5$ can not be exact square.

Let $ABCD$ be a parallelogram in the plane. We draw two circles of radius $R$, one through the points $A$ and $B$, the other through $B$ and $C$. Let $E$ be the other intersection point of the circles. We assume that $E$ is not a vertex of the parallelogram. Show that the circle passing through $A, D$, and $E$ also has radius $R$.

In a quadrilateral $ABCD$ we have $AB||CD$ and $AB=2CD$. A line $\ell$ is perpendicular to $CD$ and contains the point $C$. The circle with centre $D$ and radius $DA$ intersects the line $\ell$ at points $P$ and $Q$. Prove that $AP\perp BQ$.

For how many primes $p$, $|p^4-86|$ is also prime? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

Complex numbers $a,b,w,x,y,z,p$ satisfy \begin{align*} \frac{(x-w)\lvert a-w \rvert}{(a-w)\lvert x-w \rvert}&=\text{(cyclic variants)};\\ \frac{(z-w)\lvert b-w \rvert}{(b-w)\lvert z-w \rvert}&=\text{(cyclic variants)};\\ p &= \frac{\sum_{\text{cyc}} \frac w{\lvert p-w \rvert}}{\sum_{\text{cyc}}\frac1{\lvert p-w \rvert}}; \end{align*} where cyclic sums, equations, etc. are wrt $w,x,y,z$. Prove that there exists a real $k$ such that \[\sum_{\text{cyc}} \frac{(x-w)(a-w)}{\lvert x-w\rvert (p-w)} =k\sum_{\text{cyc}} \frac{(z-w)(b-w)}{\lvert z-w\rvert(p-w)}.\] [i]Neal Yan[/i]

Let $ABC$ be a triangle with incenter $I$. Let $D$ be the point of intersection between the incircle and the side $BC$, the points $P$ and $Q$ are in the rays $IB$ and $IC$, respectively, such that $\angle IAP=\angle CAD$ and $\angle IAQ=\angle BAD$. Prove that $AP=AQ$.

Let $S = \{1, 2, \dots , 2021\}$, and let $\mathcal{F}$ denote the set of functions $f : S \rightarrow S$. For a function $f \in \mathcal{F},$ let \[T_f =\{f^{2021}(s) : s \in S\},\] where $f^{2021}(s)$ denotes $f(f(\cdots(f(s))\cdots))$ with $2021$ copies of $f$. Compute the remainder when \[\sum_{f \in \mathcal{F}} |T_f|\] is divided by the prime $2017$, where the sum is over all functions $f$ in $\mathcal{F}$.

Let $ABC$ be an acute-angled triangle. Construct points $A', B', C'$ on its sides $BC, CA, AB$ such that: - $A'B' \parallel AB$, - $C'C$ is the bisector of angle $A'C'B'$, - $A'C' + B'C'= AB$.

There are $30$ contestants and each contestant has $6$ friends each. $3$ people is selected from these $30$ contestants, and it is called $good~triple$, if either all three are mutual friends, or none of them are friends with each other. How many $good~triples$ are there? (Note: If contestant $A$ is friends with $B$, then $B$ is friends with $A$. Similarly, if $A$ is not friends with $B$, then $B$ is not friends with $A$)