Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$ [i]Note: A primitive root of $r$ of a prime $p$ is an integer $1 \le r < p$ such that the smallest positive integer $k$ such that $r^k \equiv 1 \pmod{p}$ is $k = p-1$.[/i]

Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$, there is a circle passing through all four or a line passing through some three. [i]Carl Lian.[/i]

Prove that $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \ge \dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+2(a+b+c)$$ for the all $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2+2abc \le 1$.

A frog is located on a unit square of an infinite grid oriented according to the cardinal directions. The frog makes moves consisting of jumping either one or two squares in the direction it is facing, and then turning according to the following rules: i) If the frog jumps one square, it then turns $90^\circ$ to the right; ii) If the frog jumps two squares, it then turns $90^\circ$ to the left. Is it possible for the frog to reach the square exactly $2024$ squares north of the initial square after some finite number of moves if it is initially facing: a) North; b) East?

Consider a finite set of points $T\in\mathbb{R}^n$ contained in the $n$-dimensional unit ball centered at the origin, and let $X$ be the convex hull of $T$. Prove that for all positive integers $k$ and all points $x\in X$, there exist points $t_1, t_2, \dots, t_k\in T$, not necessarily distinct, such that their centroid \[\frac{t_1+t_2+\dots+t_k}{k}\]has Euclidean distance at most $\frac{1}{\sqrt{k}}$ from $x$. (The $n$-dimensional unit ball centered at the origin is the set of points in $\mathbb{R}^n$ with Euclidean distance at most $1$ from the origin. The convex hull of a set of points $T\in\mathbb{R}^n$ is the smallest set of points $X$ containing $T$ such that each line segment between two points in $X$ lies completely inside $X$.)

$N $ different numbers are written on blackboard and one of these numbers is equal to $0$.One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $-2016$ and $2016$ were written on blackboard(and some other numbers maybe). Find the smallest possible value of $N $.

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

$ ABCD$ is a convex quadrilateral for which $ AB$ is the longest side. Points $ M$ and $ N$ are located on sides $ AB$ and $ BC$ respectively, so that each of the segments $ AN$ and $ CM$ divides the quadrilateral into two parts of equal area. Prove that the segment $ MN$ bisects the diagonal $ BD$.

In the usual coordinate system $ xOy$ a line $ d$ intersect the circles $ C_1:$ $ (x\plus{}1)^2\plus{}y^2\equal{}1$ and $ C_2:$ $ (x\minus{}2)^2\plus{}y^2\equal{}4$ in the points $ A,B,C$ and $ D$ (in this order). It is known that $ A\left(\minus{}\frac32,\frac{\sqrt3}2\right)$ and $ \angle{BOC}\equal{}60^{\circ}$. All the $ Oy$ coordinates of these $ 4$ points are positive. Find the slope of $ d$.

Let $f$ be a function satisfying $f(x+y+z)=f(x)+f(y)+f(z)$ for all integers $x$, $y$, $z$. Suppose $f(1)=1$, $f(2)=2$. Then $\lim \limits_{n\to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r)$ equals [list=1] [*] 4 [*] 6 [*] 12 [*] 24 [/list]

Let S be a non-empty set with an associative operation that is left and right cancellative (xy=xz implies y=z, and yx = zx implies y = z). Assume that for every a in S the set {a^n : n = 0,1,2...} is finite. Must S be a group? I haven't had much group theory at this point...

Let K be a bounded, d-dimensional convex polyhedron that is not simplex and P is a point on K. Show that if vertices $P_1 , ..., P_k$ are not all on the same face of K, then one of them can be omitted so that the convex hull of the remaining vertices of K still contains P. [hide=note]caratheodory's theorem might be useful. [/hide]

From the set of numbers $1 , 2, 3, . . . , 1985$ choose the largest subset such that the difference between any two numbers in the subset is not a prime number (the prime numbers are $2, 3 , 5 , 7,... , 1$ is not a prime number) .

Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin $O$ that cuts them both, then these polygons are called "properly placed". Find the least $m \in \mathbb{N}$, such that for any group of properly placed polygons, $m$ lines can drawn through $O$ and every polygon is cut by at least one of these $m$ lines.

A sequence of moves is performed starting on the three letter string "$BAD.$'' A move consists of inserting an additional instance of the three letter string "$BAD$'' between any two consecutive letters of the current string, to achieve an elongated string. After $n$ moves, how many distinct possible strings of length $3n+3$ can be achieved? For example, after one move the strings "$BBADAD$'' and "$BABADD$'' are achievable. [i]Proposed by squareman (Evan Chang), USA[/i]

Positive integers $m, n$ are such that $mn$ is divisible by $9$ but not divisible by $27$. Rectangle $m \times n$ is cut into corners, each consisting of three cells. There are four types of such corners, depending on their orientation; you can see them on the figure below. Could it happen that the number of corners of each type was the exact square of some positive integer? [i]Proposed by Oleksiy Masalitin[/i] [img]https://i.ibb.co/Y8QSHyf/Kyiv-MO-2023-10-4.png[/img]

Given a triangle $ABC$. Let $O$ be the circumcenter of this triangle $ABC$. Let $H$, $K$, $L$ be the feet of the altitudes of triangle $ABC$ from the vertices $A$, $B$, $C$, respectively. Denote by $A_{0}$, $B_{0}$, $C_{0}$ the midpoints of these altitudes $AH$, $BK$, $CL$, respectively. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$, respectively. Prove that the four lines $A_{0}D$, $B_{0}E$, $C_{0}F$ and $OI$ are concurrent. (When the point $O$ concides with $I$, we consider the line $OI$ as an arbitrary line passing through $O$.)

Let $ a,b,c$ be real numbers such that $ ab,bc,ca$ are rational numbers. Prove that there are integers $ x,y,z$, not all of them are $ 0$, such that $ ax\plus{}by\plus{}cz\equal{}0$.

Suppose that $a, b, x$ are positive integers such that \[x^{a+b}=a^bb\] Prove that $a=x$ and $b=x^x$.

If the statement "All shirts in this store are on sale." is false, then which of the following statements must be true? I. All shirts in this store are at non-sale prices. II. There is some shirt in this store not on sale. III. No shirt in this store is on sale. IV. Not all shirts in this store are on sale. $ \textbf{(A)}\ \text{II only} \qquad \textbf{(B)}\ \text{IV only} \qquad \textbf{(C)}\ \text{I and III only} \qquad$ $ \textbf{(D)}\ \text{II and IV only} \qquad \textbf{(E)}\ \text{I, II and IV only}$

In triangle $ABC,$ the medians to the sides $\overline{AB}$ and $\overline{AC}$ are perpendicular. Prove that $\cot B+\cot C\ge \frac23.$

$A, B$ and $C$ are points on the circumference of a circle with centre $O$. Tangents are drawn to the circumcircles of triangles $OAB$ and $OAC$ at $P$ and $Q$ respectively, where $P$ and $Q$ are diametrically opposite $O$. The two tangents intersect at $K$. The line $CA$ meets the circumcircle of $\triangle OAB$ at $A$ and $X$. Prove that $X$ lies on the line $KO$.

Unless of isomorphisms, how many simple four-vertex graphs are there?

Suppose $ 2015= a_1 <a_2 < a_3<\cdots <a_k $ be a finite sequence of positive integers, and for all $ m, n \in \mathbb{N} $ and $1\le m,n \le k $, $$ a_m+a_n\ge a_{m+n}+|m-n| $$ Determine the largest possible value $ k $ can obtain.

Let $a,b,c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that $P(a)=b+c$, $P(b) = c+a$, $P(c) = a+b$, and $P(a+b+c) = -16$, find $P(0)$. [i]Author: Alex Zhu[/i]