Let $ p$ be a prime, $ n$ a natural number, and $ S$ a set of cardinality $ p^n$ . Let $ \textbf{P}$ be a family of partitions of $ S$ into nonempty parts of sizes divisible by $ p$ such that the intersection of any two parts that occur in any of the partitions has at most one element. How large can $ |\textbf{P}|$ be?

The company has $100$ people. For any $k$, we can find a group of $k$ people such that there are two (different from them) strangers, each of them knows all of these $k$ people. At what maximum $k$ is this possible?

For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$

The positive integers are partitioned into two infinite sets so that the sum of any $2023$ distinct integers in one set is also in that set. Prove that one set contains all the odd positive integers, and one set contains all the even positive integers. [i]Proposed by Evan Chang (squareman), USA[/i]

In a rectangle $ABCD$ two not intersecting circles $\omega_1$ and $\omega_2$ are drawn such that $\omega_1$ is tangent to $AB$ and $AD$ at points $P$ and $S$ respectively, and $\omega_2$ is tangent to $CB$ and $CD$ at $T$ and $Q$ respectively. It is known that $PQ=11, ST=10, BD=14$. Find the distance between centers of circles $\omega_1$ and $\omega_2$. [i]I. Voronovich[/i]

Two linear functions $f(x)$ and $g(x)$ satisfy the properties that for all $x$, $\bullet$ $f(x) + g(x) = 2$ $\bullet$ $f(f(x)) = g(g(x))$ and $f(0) = 2022$. Compute $f(1)$.

$ABCD$ is a convex quadrilateral. Take the $12$ points which are the feet of the altitudes in the triangles $ABC, BCD, CDA, DAB$. Show that at least one of these points must lie on the sides of $ABCD$.

On the blackboard are written the $400$ integers $1, 2, 3, \cdots , 399, 400$. Luis erases $100$ of these numbers, then Martin erases another $100$. Martin wins if the sum of the $200$ erased numbers equals the sum of those not deleted; otherwise, he wins Luis. Which of the two has a winning strategy? What if Luis deletes $101$ numbers and Martín deletes $99$? In each case, explain how the player with the winning strategy can ensure victory.

Which of the cones listed below can be formed from a $ 252^\circ$ sector of a circle of radius $ 10$ by aligning the two straight sides? [asy]import graph;unitsize(1.5cm);defaultpen(fontsize(8pt));draw(Arc((0,0),1,-72,180),linewidth(.8pt));draw(dir(288)--(0,0)--(-1,0),linewidth(.8pt));label("$10$",(-0.5,0),S);draw(Arc((0,0),0.1,-72,180));label("$252^{\circ}$",(0.05,0.05),NE);[/asy] [asy] import three; picture mainframe; defaultpen(fontsize(11pt)); picture conePic(picture pic, real r, real h, real sh) { size(pic, 3cm); triple eye = (11, 0, 5); currentprojection = perspective(eye); real R = 1, y = 2; triple center = (0, 0, 0); triple radPt = (0, R, 0); triple negRadPt = (0, -R, 0); triple heightPt = (0, 0, y); draw(pic, arc(center, radPt, negRadPt, heightPt, CW)); draw(pic, arc(center, radPt, negRadPt, heightPt, CCW), linetype("8 8")); draw(pic, center--radPt, linetype("8 8")); draw(pic, center--heightPt, linetype("8 8")); draw(pic, negRadPt--heightPt--radPt); label(pic, (string) r, center--radPt, dir(270)); if (h != 0) { label(pic, (string) h, heightPt--center, dir(0)); } if (sh != 0) { label(pic, (string) sh, heightPt--radPt, dir(0)); } return pic; } picture pic1; pic1 = conePic(pic1, 6, 0, 10); picture pic2; pic2 = conePic(pic2, 6, 10, 0); picture pic3; pic3 = conePic(pic3, 7, 0, 10); picture pic4; pic4 = conePic(pic4, 7, 10, 0); picture pic5; pic5 = conePic(pic5, 8, 0, 10); picture aux1; picture aux2; picture aux3; add(aux1, pic1.fit(), (0,0), W); label(aux1, "$\textbf{(A)}$", (0,0), 22W, linewidth(4)); label(aux1, "$\textbf{(B)}$", (0,0), 3E); add(aux1, pic2.fit(), (0,0), 35E); add(aux2, aux1.fit(), (0,0), W); label(aux2, "$\textbf{(C)}$", (0,0), 3E); add(aux2, pic3.fit(), (0,0), 35E); add(aux3, aux2.fit(), (0,0), W); label(aux3, "$\textbf{(D)}$", (0,0), 3E); add(aux3, pic4.fit(), (0,0), 35E); add(mainframe, aux3.fit(), (0,0), W); label(mainframe, "$\textbf{(E)}$", (0,0), 3E); add(mainframe, pic5.fit(), (0,0), 35E); add(mainframe.fit(), (0,0), N); [/asy]

On the sides of the triangle \(\triangle ABC\) lie the following points: \(K\) and \(L\) on \(AB\), \(M\) on \(BC\), and \(N\) on \(CA\). Let \[ P = AM\cap BN,\quad R = KM\cap LN,\quad S = KN\cap LM, \] and let the line \(CS\) meet \(AB\) at \(Q\). Prove that the points \(P\), \(Q\), and \(R\) are collinear.

Let $ABC$ be a triangle with circumcenter $O$, angle bisector $AD$ with $D \in BC$ and altitude $AE$ with $E \in BC$. The lines $AO$ and $BC$ meet at $I$. The circumcircle of $\triangle ADE$ meets $AB, AC$ at $F, G$ and $FG$ meets $BC$ at $H$. The circumcircles of triangles $AHI$ and $ABC$ meet at $J$. Show that $AJ$ is a symmedian in $\triangle ABC$

Let $ P(z)$ be a polynomial of degree $ n$ with complex coefficients, \[ P(0)\equal{}1, \;\textrm{and}\ \;|P(z)|\leq M\ \;\textrm{for}\ \;|z| \leq 1\ .\] Prove that every root of $ P(z)$ in the closed unit disc has multiplicity at most $ c\sqrt{n}$, where $ c\equal{}c(M) >0$ is a constant depending only on $ M$. [i]G. Halasz[/i]

There are $n$ people in a city, and each of them has exactly $1000$ friends (friendship is always symmetric). Prove that it is possible to select a group $S$ of people such that at least $\frac{n}{2017}$ persons in $S$ have exactly two friends in $S$.

A projectile, thrown with initial velocity $v_0$ in a direction making angle $\alpha$ with the horizontal, is acted on by no force except gravity. Find the lenght of its path until it strikes a horizontal plane through the starting point. Show that the flight is longest when $$\sin \alpha \log(\sec \alpha+ \tan \alpha)=1.$$

Given a geometric progression of five terms, each a positive integer less than $100$. The sum of the five terms is $211$. If $S$ is the sum of those terms in the progression which are squares of integers, then $S$ is: $\textbf{(A)}\ 0\qquad \textbf{(B)}\ 91\qquad \textbf{(C)}\ 133\qquad \textbf{(D)}\ 195\qquad \textbf{(E)}\ 211$

The set $A=\{1,2,3,\cdots, 10\}$ contains the numbers $1$ through $10$. A subset of $A$ of size $n$ is competent if it contains $n$ as an element. A subset of $A$ is minimally competent if it itself is competent, but none of its proper subsets are. Find the total number of minimally competent subsets of $A$.

Let $2561$ given points on a circle be colored either red or green. In each step, all points are recolored simultaneously in the following way: if both direct neighbors of a point $P$ have the same color as $P$, then the color of $P$ remains unchanged, otherwise $P$ obtains the other color. Starting with the initial coloring $F_1$, we obtain the colorings $F_2, F_3,\dots$ after several recoloring steps. Determine the smallest number $n$ such that, for any initial coloring $F_1$, we must have $F_n = F_{n+2}$.

For any positive integer $ n>1$, let $ P\left(n\right)$ denote the largest prime divisor of $ n$. Prove that there exist infinitely many positive integers $ n$ for which \[ P\left(n\right)<P\left(n\plus{}1\right)<P\left(n\plus{}2\right).\]

The graphs of a square trinomial and its derivative divide the coordinate plane into four parts. How many roots does this square trinomial has?

For a positive integer $n$ that is not a power of two, we define $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.

In Cartesian coordinates, two areas $M,N$ are defined below: $M:y\geq0,y\leq x,y\leq 2-x$; $N:t\leq x\leq t+1$. $t$ is a real number that $t\in[0,1]$. Then the area of $M\cap N$ is $\text{(A)}-t^2+t+\frac{1}{2}\qquad\text{(B)}-2t^2+2t\qquad\text{(C)}1-2t^2\qquad\text{(D)}\frac{1}{2}(t-2)^2$

A hyper-primitive root is a k-tuple $ (a_{1},a_{2},\dots,a_{k})$ and $ (m_{1},m_{2},\dots,m_{k})$ with the following property: For each $ a\in\mathbb N$, that $ (a,m) \equal{} 1$, has a unique representation in the following form: \[ a\equiv a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}\dots a_{k}^{\alpha_{k}}\pmod{m}\qquad 1\leq\alpha_{i}\leq m_{i}\] Prove that for each $ m$ we have a hyper-primitive root.

Let $a$, $b$, $c$, and $d$ be positive integers such that the following two inequalities hold: $a < 10^{20} \cdot c$ and $b > 10^{23} \cdot d$. Determine the minimum possible value of the total number of positive integer pairs $(n, m)$ for which $n \cdot m = 2^{2023}$ and $$ \frac {ab}{n} + \frac{cd}{m} < \frac{(a + c)(b + d)}{n + m}$$ [i]Proposed by Stijn Cambie, Belgium[/i]

Let $N_9$ be the answer to problem 9. In a rainforest, there is a row of nine rocks labeled from $1$ to $N_9$. A gecko is standing on rock $1$. The gecko jumps according to following rules: [list] [*] if it is on rock $1$, then it will jump with equal probability to any of the other rocks. [*] if it is on rock $R$ and $R$ is prime, then it will jump to rock $N_9$. [*] if it is on rock $4$, then it will jump with equal probability to rock $1$ or rock $6$. [*] if it is on rock $R$ and $R$ is composite with $4<R<N_9$, then it will jump with equal probability to rock $Q$ or $S$, where $Q$ is the greatest composite number less than $R$, and $S$ is the smallest composite number greater than $R$. [*] if it is on rock $N_9$, then it stops jumping. [/list] The expected number of jumps the gecko will take to reach rock $N_9$ is $\tfrac{p}{q}$, where $p$ are $q$ relatively prime positive integers. Compute $p+q$.

The Olcommunity consists of the next seven people: Christopher Took, Humberto Brandybuck, German son of Isildur, Leogolas, Argimli, Samzamora and Shago Baggins. This community needs to travel from the Olcomashire to Olcomordor to save the world. Each person can take with them a total of $4$ day-provisions, that can be transferred to other people that are on the same day of traveling, as long as nobody holds more than $4$ day-provisions at the time. If a person returns to Olcomashire, they will be too tired to go out again. What is the farthest away Olcomordor can be from Olcomashire, so that Shago Baggins can get to Olcomordor, and the rest of the Olcommunity can return save to Olcomashire? Note: All the members of the Olcommunity should eat exactly one day-provision while they are away from Olcomashire. The members only travel an integer number of days on each direction. Members of the Olcommunity may leave Olcomashire on different days.