Determine the number of odd integers $1 \le n \le 100$ with the property that \[ \sum_{\substack{1 \le k \le n \\ \gcd(k,n) = 1}} \cos\left(\frac{2\pi k}{n} \right) = 1 \quad\text{and}\quad \sum_{\substack{1 \le k \le n \\ \gcd(k,n) = 1}} \sin\left(\frac{2\pi k}{n} \right) = 0. \] [i]Based on a proposal by Mayank Pandey[/i]

Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card. Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$. [i]Proposed by Carl Schildkraut and Colin Tang[/i]

Let $E$ be intersection of the diagonals of convex quadrilateral $ABCD$. It is given that $m(\widehat{EDC}) = m(\widehat{DEC})=m(\widehat{BAD})$. If $F$ is a point on $[BC]$ such that $m(\widehat{BAF}) + m(\widehat{EBF})=m(\widehat{BFE})$, show that $A$, $B$, $F$, $D$ are concyclic.

Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that \[{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}\] is divisible by $p^{2}$.

What is the number of ways the numbers from $1$ to $14$ can be split into $7$ pairs such that for each pair, the greater number is at least $2$ times the smaller number? $\textbf{(A) }108\qquad\textbf{(B) }120\qquad\textbf{(C) }126\qquad\textbf{(D) }132\qquad\textbf{(E) }144$

An odd natural number $k$ is given. Consider a composite number $n$. We define $d(n)$ the set of proper divisors of number $n$. If for some number $m$, $d(m)$ is equal to $d(n) \cup \{ k \}$, we call $n$ a good number. prove that there exist only finitely many good numbers. (A proper divisor of a number is any divisor other than one and the number itself.)

Let the sequence ($a_n$) be defined by $a_n = n^6 +5n^4 -12n^2 -36, n \ge 2$. (a) Prove that any prime number divides some term in this sequence. (b) Prove that there is a positive integer not dividing any term in the sequence. (c) Determine the least $n \ge 2$ for which $1989 | a_n$.

Let $S$ be an infinite set of positive integers and define: $T=\{ x+y|x,y \in S , x \neq y \} $ Suppose that there are only finite primes $p$ so that: 1.$p \equiv 1 \pmod 4$ 2.There exists a positive integer $s$ so that $p|s,s \in T$. Prove that there are infinity many primes that divide at least one term of $S$.

100 couples are invited to a traditional Modolvan dance. The $200$ people stand in a line, and then in a $\textit{step}$, (not necessarily adjacent) many swap positions. Find the least $C$ such that whatever the initial order, they can arrive at an ordering where everyone is dancing next to their partner in at most $C$ steps.

Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that: $$a^3 + b^3 + c^3 = 2016$$

What is the number of $4$ digit natural numbers such that the sum of digits is even? (A)$4999$ (B)$5000$ (C)$5050$ (D)$4500$

Given the point $O$ in the space and $1979$ straight lines $l_1, l_2, ... , l_{1979}$ containing it. Not a pair of lines is orthogonal. Given a point $A_1$ on $l_1$ that doesn't coincide with $O$. Prove that it is possible to choose the points $A_i$ on $l_i$ ($i = 2, 3, ... , 1979$) in so that $1979$ pairs will be orthogonal: $A_1A_3$ and $l_2$, $A_2A_4$ and $l_3$,$ ...$ , $A_{i-1}A_{i+1}$ and $l_i$,$ ...$ , $A_{1977}A_{1979}$ and $l_{1978}$, $A_{1978}A_1$ and $l_{1979}$, $A_{1979}A_2$ and $l_1$

Nanna and Sofie move in the same direction along two parallel paths, which are $200$ meters apart. Nanna's speed is $3$ meters per second, Sofie's only $1$ meter per second. A tall, cylindrical building with a diameter of $100$ meters is located in the middle between the two paths. Since the building first once the line of sight breaks between the girls, their distance between them is $200$ metres. How long will it be before the two girls see each other again?

Let $x_1,x_2,\ldots,x_n\ge0$, $\alpha,\beta>0$, $\beta\ge\alpha$, $t\in\mathbb R$, such that $x_1^{x_2^t}\cdot x_2^{x_3^t}\cdots x_n^{x_1^t}=1$. Then prove that $$x_1^\beta x_2^t+x_2^\beta x_3^t+\ldots+x_n^\beta x_1^t\ge x_1^\alpha x_2^t+x_2^\alpha x_3^t+\ldots+x_n^\alpha x_1^t.$$ [i]Proposed by Marius Drăgan[/i]

Prove that for $a, b, c \in [0;1]$, $$(1-a)(1+ab)(1+ac)(1-abc) \leq (1+a)(1-ab)(1-ac)(1+abc).$$

Let $P$ be a right prism whose two bases are equilateral triangles with side length 2. The height of $P$ is $2\sqrt{3}$. Let $l$ be the line connecting the centroids of the bases. Remove the solid, keeping only the bases. Rotate one of the bases $180^\circ$ about $l$. Let $T$ be the convex hull of the two current triangles. What is the volume of $T$?

Let $ABC$ be an acute triangle such that $AH = HD$, where $H$ is the orthocenter of $ABC$ and $D \in BC$ is the foot of the altitude from the vertex $A$. Let $\ell$ denote the line through $H$ which is tangent to the circumcircle of the triangle $BHC$. Let $S$ and $T$ be the intersection points of $\ell$ with $AB$ and $AC$, respectively. Denote the midpoints of $BH$ and $CH$ by $M$ and $N$, respectively. Prove that the lines $SM$ and $TN$ are parallel.

Let $0!!=1!!=1$ and $n!!=n\cdot (n-2)!!$ for all integers $n\geq 2$. Find all positive integers $n$ such that \[\dfrac{(2^n+1)!!-1}{2^{n+1}}\] is an integer.

In a right-angled triangle, $a$ and $b$ denote the lengths of the two catheti. A circle with radius $r$ has the center on the hypotenuse and touches both catheti. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$.

Define $a*b=\frac{a-b}{1-ab}$. What is $(1*(2*(3*\cdots (n*(n+1))\cdots )))$?

Does there exist a field such that its multiplicative group is isomorphism to its additive group? [i]Proposed by Alexandre Chapovalov, New York University, Abu Dhabi[/i]

Right triangle ABC with a right angle at A has AB = 20 and AC = 15. Point D is on AB with BD = 2. Points E and F are placed on ray CA and ray CB, respectively, such that CD is a median of $\triangle$ CEF. Find the area of $\triangle$CEF.

We call a polynomial $P(x)=a_dx^d+...+a_0$ of degree $d$ [i]nice[/i] if $$\frac{2021(|a_d|+...+|a_0|)}{2022}<max_{0 \le i \le d}|a_i|$$ Initially Shayan has a sequence of $d$ distinct real numbers; $r_1,...,r_d \neq \pm 1$. At each step he choose a positive integer $N>1$ and raises the $d$ numbers he has to the exponent of $N$, then delete the previous $d$ numbers and constructs a monic polynomial of degree $d$ with these number as roots, then examine whether it is nice or not. Prove that after some steps, all the polynomials that shayan produces would be nice polynomials Proposed by [i]Navid Safaei[/i]

Let $a_1,a_2,...,a_{15}$ be positive integers for which the number $a_k^{k+1} - a_k$ is not divisible by $17$ for any $k = 1,...,15$. Show that there are integers $b_1,b_2,...,b_{15}$ such that: (i) $b_m - b_n$ is not divisible by $17$ for $1 \le m < n \le 15$, and (ii) each $b_i$ is a product of one or more terms of $(a_i)$.

Given a sequence $(x_1,x_2,\ldots, x_n)$ of 0's and 1's, let $A$ be the number of triples $(x_i,x_j,x_k)$ with $i<j<k$ such that $(x_i,x_j,x_k)$ equals $(0,1,0)$ or $(1,0,1)$. For $1\leq i \leq n$, let $d_i$ denote the number of $j$ for which either $j < i$ and $x_j = x_i$ or else $j > i$ and $x_j\neq x_i$. (a) Prove that \[A = \binom n3 - \sum_{i=1}^n\binom{d_i}2.\] (Of course, $\textstyle\binom ab = \tfrac{a!}{b!(a-b)!}$.) [5 points] (b) Given an odd number $n$, what is the maximum possible value of $A$? [15 points]