At a two-day team competition in chess, three schools with $15$ pupils each attend. Each student plays one game against each player on the other two teams, ie a total of $30$ chess games per student. a) Is it possible for each student to play exactly $15$ games after the first day? b) Show that it is possible for each student to play exactly $16$ games after the first day. c) Assume that each student has played exactly $16$ games after the first day. Show that there are three students, one from each school, who have played their three parties

From among $2^{1/2},$ $3^{1/3},$ $8^{1/8},$ $9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are \[ \begin{array}{rlrlrlrl} \hbox {(A)}& 3^{1/3},\ 2^{1/2} \quad & \hbox {(B)}& 3^{1/3},\ 8^{1/8} \quad & \hbox {(C)}& 3^{1/3},\ 9^{1/9} \quad & \hbox {(D)}& 8^{1/8},\ 9^{1/9} \\ \hbox {(E)}& \multicolumn{3}{l}{\hbox{None of these}} \end{array} \]

Each day, Jenny ate $ 20\%$ of the jellybeans that were in her jar at the beginning of the day. At the end of the second day, $ 32$ remained. How many jellybeans were in the jar originally? $ \textbf{(A)}\ 40\qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 60\qquad \textbf{(E)}\ 75$

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

Let $\triangle ABC$ be an acute scalene triangle with incenter $I$ and incircle $\omega$. Two points $X$ and $Y$ are chosen on minor arcs $AB$ and $AC$, respectively, of the circumcircle of triangle $\triangle ABC$ such that $XY$ is tangent to $\omega$ at $P$ and $\overline{XY}\perp \overline{AI}$. Let $\omega$ be tangent to sides $AC$ and $AB$ at $E$ and $F$, respectively. Denote the intersection of lines $XF$ and $YE$ as $T$. Prove that if the circumcircles of triangles $\triangle TEF$ and $\triangle ABC$ are tangent at some point $Q$, then lines $PQ$, $XE$, and $YF$ are concurrent. [i]Proposed by Andrew Wen[/i]

Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$? $ \textbf{(A)}\ 180 \qquad\textbf{(B)}\ 184 \qquad\textbf{(C)}\ 188 \qquad\textbf{(D)}\ 192\qquad\textbf{(E)}\ 196 $

Seven classmates are comparing their end-of-year grades in $ 12$ subjects. They observe that for any two of them, there is some subject out of the $ 12$ where the two students got different grades. It is possible to choose n subjects out of the $ 12$ such that if the seven students only compare their grades in these $n$ subjects, it will still be true that for any two, there is some subject out of the n where they got different grades. What is the smallest value of $n$ for which such a selection is surely possible? Note: In Hungarian high schools, students receive an integer grade from $ 1$ to $5$ in each subject at the end of the year.

Determine all triples $(a,b,c)$ of real numbers such that $$ (2a+1)^2 - 4b = (2b+1)^2 - 4c = (2c+1)^2 - 4a = 5. $$

Let $P=[a_{ij}]_{3\times 9}$ be a $3\times 9$ matrix where $a_{ij}\ge 0$ for all $i,j$. The following conditions are given: [list][*]Every row consists of distinct numbers; [*]$\sum_{i=1}^{3}x_{ij}=1$ for $1\le j\le 6$; [*]$x_{17}=x_{28}=x_{39}=0$; [*]$x_{ij}>1$ for all $1\le i\le 3$ and $7\le j\le 9$ such that $j-i\not= 6$. [*]The first three columns of $P$ satisfy the following property $(R)$: for an arbitrary column $[x_{1k},x_{2k},x_{3k}]^T$, $1\le k\le 9$, there exists an $i\in\{1,2,3\}$ such that $x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})$.[/list] Prove that: a) the elements $u_1,u_2,u_3$ come from three different columns; b) if a column $[x_{1l},x_{2l},x_{3l}]^T$ of $P$, where $l\ge 4$, satisfies the condition that after replacing the third column of $P$ by it, the first three columns of the newly obtained matrix $P'$ still have property $(R)$, then this column uniquely exists.

Let $ABC$ be a triangle with right angle $ACB$. Denote by $F$ the projection of $C$ on $AB$. A circle $\omega$ touches $FB$ at point $P$, touches $CF$ at point $Q$, and the circumcircle of $ABC$ at point $R$. Prove that the points $A, Q, R$ all lie on the same line and $AP=AC$.

In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line. [i]Proposed by Theoklitos Parayiou, Cyprus [/i]

Suppose $X$ is a set with $|X| = 56$. Find the minimum value of $n$, so that for any 15 subsets of $X$, if the cardinality of the union of any 7 of them is greater or equal to $n$, then there exists 3 of them whose intersection is nonempty.

Define $\phi^!(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when \[ \sum_{\substack{2 \le n \le 50 \\ \gcd(n,50)=1}} \phi^!(n) \] is divided by $50$.

Let $x_i\ge -1$ and $\sum^n_{i=1}x_i^3=0$. Prove $\sum^n_{i=1}x_i \le \frac{n}{3}$.

Let $p$ be an odd prime and $S = \{1,2,3,\dots, p\}$ Assume that $U: S \rightarrow S$ is a bijection and $B$ is an integer such that $$B\cdot U(U(a)) - a \: \text{ is a multiple of} \: p \: \text{for all} \: a \in S$$ Show that $B^{\frac{p-1}{2}} -1$ is a multiple of $p$.

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\sin ^ 2 x}{\cos ^ 3 x}\ dx$.

Baron Munchhausen has a collection of stones, such that they are of $1000$ distinct whole weights, $2^{1000}$ stones of every weight. Baron states that if one takes exactly one stone of every weight, then the weight of all these $1000$ stones chosen will be less than $2^{1010}$, and there is no other way to obtain this weight by picking another set of stones of the collection. Can this statement happen to be true? [i](М. Антипов)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

Suppose $ABCD$ is a parallelogram. Consider circles $w_1$ and $w_2$ such that $w_1$ is tangent to segments $AB$ and $AD$ and $w_2$ is tangent to segments $BC$ and $CD$. Suppose that there exists a circle which is tangent to lines $AD$ and $DC$ and externally tangent to $w_1$ and $w_2$. Prove that there exists a circle which is tangent to lines $AB$ and $BC$ and also externally tangent to circles $w_1$ and $w_2$. [i]Proposed by Ali Khezeli[/i]

Prove that if $x, y$ and $n$ are positive integers such that $$x^{2024} + y^{2024} = 2^n,$$ then $x=y$.

There are $256$ lumps of metal that have different weights in pairs. With the help of a beam balance , one may now compare every two lumps. Find the smallest number $m$ such that you can be sure to find the heaviest as well as the lightest lump with the weighing process.

The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is: $\textbf{(A)}\ -\dfrac{2}{3} \qquad \textbf{(B)}\ -\dfrac{1}{3} \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{3}{8} $

Find the point in the closed unit disc $D=\{ (x,y) | x^2+y^2\le 1 \}$ at which the function $f(x,y)=x+y$ attains its maximum .

In the non-isosceles triangle $ABC$ consider the points $X$ on $[AB]$ and $Y$ on $[AC]$ such that $[BX]=[CY]$, $M$ and $N$ are the midpoints of the segments $[BC]$, respectively $[XY]$, and the straight lines $XY$ and $BC$ meet in $K$. Prove that the circumcircle of triangle $KMN$ contains a point, different from $M$ , which is independent of the position of the points $X$ and $Y$.

Compute: $ \frac{777^2 \minus{} 66^2}{777\plus{}66}$

Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.