Twenty players participated in a chess tournament. Each player faced every other player exactly once and each match ended with either player winning or a draw. In this tournament, it was noticed that for every match that ended in a draw, each of the other $18$ players won at least one of the two players involved in it. We also know that at least two games ended in a draw. Show that it is possible to name the players as $P_1, P_2, ...., P_{20}$ so that player $P_k$ beat player $P_{k+1}$, to each $k \in \{ 1, 2, 3,... , 19\}$.

Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$.

Consider a partition of $\{1,2,\ldots,900\}$ into $30$ subsets $S_1,S_2,\ldots,S_{30}$ each with $30$ elements. In each $S_k$, we paint the fifth largest number blue. Is it possible that, for $k=1,2,\ldots,30$, the sum of the elements of $S_k$ exceeds the sum of the blue numbers?

Given a $2025 \times 2025$ board and $k$ chips lying on the table. Initially, the board is empty. It is allowed to place a chip from the table on any free square $A$ if two conditions are met simultaneously: – the cell next to $A$ from below has a chip (or $A$ is on the bottom edge of the board); – the cell next to $A$ on the left has a chip (or $A$ is on the left edge of the board). In addition, it is allowed to remove any chip from the board and put it on the table. At what minimum $k$ can a chip appear in the upper-right corner of the board?

Kevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?

Points $D$, $E$, $F$ lie on circle $O$ such that the line tangent to $O$ at $D$ intersects ray $\overrightarrow{EF}$ at $P$. Given that $PD = 4$, $PF = 2$, and $\angle FPD = 60^o$, determine the area of circle $O$.

A teacher suggests four possible books for students to read. Each of six students selects one of the four books. How many ways can these selections be made if each of the books is read by at least one student?

Find the greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$.

Given positive integers$ m,n$ such that $ m < n$. Integers $ 1,2,...,n^2$ are arranged in $ n \times n$ board. In each row, $ m$ largest number colored red. In each column $ m$ largest number colored blue. Find the minimum number of cells such that colored both red and blue.

There are several bears living on the $2025$ islands of the Arctic Ocean. Every bear sometimes swims from one island to another. It turned out that every bear made at least one swim in a year, but no two bears made equal swams. At the same time, exactly one swim was made between each two islands $A$ and $B$: either from $A$ to $B$ or from $B$ to $A$. Prove that there were no bears on some island at the beginning and at the end of the year. [i]A. Kuznetsov[/i]

Find all $x$ such that $ 6^x + 27^{x-1} = 8^x - 1 $.

Let $v_1, \ldots, v_{12}$ be unit vectors in $\mathbb{R}^3$ from the origin to the vertices of a regular icosahedron. Show that for every vector $v \in \mathbb{R}^3$ and every $\varepsilon>0$, there exist integers $a_1, \ldots, a_{12}$ such that $\left\|a_1 v_1+\cdots+a_{12} v_{12}-v\right\|<\varepsilon$.

Find all functions $f : N \to N$ such that $f(p)$ divides $f(n)^p -n$ by any natural number $n$ and prime number $p$.

Each vertex of an $m\times n$ grid is colored blue, green or red in such a way that all the boundary vertices are red. We say that a unit square of the grid is properly colored if: $(i)$ all the three colors occur at the vertices of the square, and $(ii)$ one side of the square has the endpoints of the same color. Show that the number of properly colored squares is even.

How many rectangles are there on an $8 \times 8$ checkerboard? [img]https://cdn.artofproblemsolving.com/attachments/9/e/7719117ae393d81a3e926acb567f850cc1efa9.png[/img]

Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 14$

How to arrange three right circular cylinders of diameter $a/2$ and height $a$ into an empty cube with side $a$ so that the cylinders could not change position inside the cube? Each cylinder can, however, rotate about its axis of symmetry.

Trapezoid $ABCD$ has properties $AB \parallel CD$, $AB = 15$, $CD = 27$, and $BC = AD = 10$. A smaller trapezoid $EF GH$ is drawn within$ ABCD$ with $AB\parallel EF$, $BC\parallel F G$, $CD\parallel GH$, and $DA\parallel HE$ such that each edge in $ABCD$ is a distance $2$ away from the corresponding edge in $EF GH$. Compute the area of $EF GH$.

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

A $4\times4$ grid has its 16 cells colored arbitrarily in three colors. A [i]swap[/i] is an exchange between the colors of two cells. Prove or disprove that it always takes at most three swaps to produce a line of symmetry, regardless of the grid's initial coloring. [i]Proposed by Matthew Babbitt[/i]

Viswam walks half a mile to get to school each day. His route consists of $10$ city blocks of equal length and he takes one minute to walk each block. Today, after walking $5$ blocks, Viswam discovers he has to take a detour, walking $3$ blocks of equal length instead of one block to reach the next corner. From the time he starts his detour what speed, in mph, must he walk in order to get to school at his usual time? $\textbf{(A)}~4.0\qquad\textbf{(B)}~4.2\qquad\textbf{(C)}~4.5\qquad\textbf{(D)}~4.8\qquad\textbf{(E)}~5.0\qquad$

In the triangle $ABC$, one of the angles of which is equal to $48^o$, side lengths satisfy $(a-c)(a+c)^2+bc(a+c)=ab^2$. Express in degrees the measures of the other two angles of this triangle.

[u]Set 7[/u] [b]p19.[/b] The polynomial $x^4 + ax^3 + bx^2 - 32x$, where$ a$ and $b$ are real numbers, has roots that form a square in the complex plane. Compute the area of this square. [b]p20.[/b] Tetrahedron $ABCD$ has equilateral triangle base $ABC$ and apex $D$ such that the altitude from $D$ to $ABC$ intersects the midpoint of $\overline{BC}$. Let $M$ be the midpoint of $\overline{AC}$. If the measure of $\angle DBA$ is $67^o$, find the measure of $\angle MDC$ in degrees. [b]p21.[/b] Last year’s high school graduates started high school in year $n- 4 = 2017$, a prime year. They graduated high school and started college in year $n = 2021$, a product of two consecutive primes. They will graduate college in year $n + 4 = 2025$, a square number. Find the sum of all $n < 2021$ for which these three properties hold. That is, find the sum of those $n < 2021$ such that $n -4$ is prime, n is a product of two consecutive primes, and $n + 4$ is a square. PS. You should use hide for answers.

A point $D$ is taken on the arc $BC$ of the circumcircle of triangle $ABC$ which does not contain $A$. A point $E$ is taken at the intersection of the interior region of the triangles $ABC$ and $ADC$ such that $m(\widehat{ABE})=m(\widehat{BCE})$. Let the circumcircle of the triangle $ADE$ meets the line $AB$ for the second time at $K$. Let $L$ be the intersection of the lines $EK$ and $BC$, $M$ be the intersection of the lines $EC$ and $AD$, $N$ be the intersection of the lines $BM$ and $DL$. Prove that $$m(\widehat{NEL})=m(\widehat{NDE})$$

Consider the set $L$ of binary strings of length less than or equal to $9$, and for a string $w$ define $w^{+}$ to be the set $\{w,w^2,w^3,\ldots\}$ where $w^k$ represents $w$ concatenated to itself $k$ times. How many ways are there to pick an ordered pair of (not necessarily distinct) elements $x,y\in L$ such that $x^{+}\cap y^{+}\neq \varnothing$?