Let $n\ge 3$ be a positive integer. Find the largest real number $t_n$ as a function of $n$ such that the inequality \[\max\left(|a_1+a_2|, |a_2+a_3|, \dots ,|a_{n-1}+a_{n}| , |a_n+a_1|\right) \ge t_n \cdot \max(|a_1|,|a_2|, \dots ,|a_n|)\] holds for all real numbers $a_1, a_2, \dots , a_n$ . [i]Proposed by Rohan Goyal and Rijul Saini[/i]

Determine all real-valued functions $f$ on the set of real numbers satisfying \[2f(x)=f(x+y)+f(x+2y)\] for all real numbers $x$ and all non-negative real numbers $y$.

Let $H, K, L$ be the feet from the altitudes from vertices $A, B, C$ of the triangle $ABC$, respectively. Prove that $| AK | \cdot | BL | \cdot| CH | = | HK | \cdot | KL | \cdot | LH | = | AL | \cdot | BH | \cdot | CK | $.

A $k \times k$ array contains each of the numbers $1, 2, \dots, m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k = 3^n$ ($n \in \mathbb{N}^+$)? [i]Attila Sztranyák and Péter Erben[/i]

Twenty-five points are given on the plane. Among any three of them, one can choose two less than one inch apart. Prove that there are 13 points among them which lie in a circle of radius 1.

A unit square is cut by $n$ straight lines . Prove that in at least one of these parts one can completely fit a square with side $\frac{1}{n+1}$ [hide=original wording]Одиничний квадрат розрізано $n$ прямими на частини. Доведіть, що хоча б в одній з цих частин можна повністю розмістити квадрат зі стороною $\frac{1}{n+1}$[/hide] [hide=notes] The selection panel jury made a mistake because the solution known to it turned out to be incorrect. As it turned out, the assertion of the problem is still correct, although it cannot be proved by simple methods, see. article: Keith Ball. Тhe plank problem for symmetric bodies // Іпѵепііопез МаіЬешаІіеае. — 1991. — Ѵоі. 104, по. 1. — Р. 535-543. [url]https://arxiv.org/abs/math/9201218[/url][/hide]

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

The figure on the right shows the map of Squareville, where each city block is of the same length. Two friends, Alexandra and Brianna, live at the corners marked by $A$ and $B$, respectively. They start walking toward each other's house, leaving at the same time, walking with the same speed, and independently choosing a path to the other's house with uniform distribution out of all possible minimum-distance paths [that is, all minimum-distance paths are equally likely]. What is the probability they will meet? [asy] size(200); defaultpen(linewidth(0.8)); for(int i=0;i<=2;++i) { for(int j=0;j<=4;++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle); } } for(int i=3;i<=4;++i) { for(int j=3;j<=6;++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle); } } label("$A$",origin,SW); label("$B$",(5,7),SE); [/asy]

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

Find all pairs of nonnegative integers $(x, p)$, where $p$ is prime, that verify $$x(x+1)(x+2)(x+3)=1679^{p-1}+1680^{p-1}+1681^{p-1}.$$

In the figure below, choose point $D$ on $\overline{BC}$ so that $\triangle ACD$ and $\triangle ABD$ have equal perimeters. What is the area of $\triangle ABD$? [asy]draw((0,0)--(4,0)--(0,3)--(0,0)); label("$A$", (0,0), SW); label("$B$", (4,0), ESE); label("$C$", (0, 3), N); label("$3$", (0, 1.5), W); label("$4$", (2, 0), S); label("$5$", (2, 1.5), NE);[/asy] $\textbf{(A) }\frac{3}{4}\qquad\textbf{(B) }\frac{3}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\frac{12}{5}\qquad\textbf{(E) }\frac{5}{2}$

Triangle $\vartriangle ABC$ has $m\angle C = 135^o$, and $D$ is the foot of the altitude from $C$ to $AB$. We are told that $CD = 2$ and that $AD$ and $BD$ are finite positive integers. What is the sum of all distinct possible values of $AB$?

Let $ m,n$ be integers such that $ 0\le m\le 2n$. Then prove that the number $ 2^{2n \plus{} 2} \plus{} 2^{m \plus{} 2} \plus{} 1$ is perfect square iff $ m \equal{} n$.

Solve in nonnegative integers the equation $5^t + 3^x4^y = z^2$.

Each of the subsets $ A_1$, $ A_2$, $ \dots,$ $ A_n$ of a 2009-element set $ X$ contains at least 4 elements. The intersection of every two of these subsets contains at most 2 elements. Prove that in $ X$ there is a 24-element subset $ B$ containing neither of the sets $ A_1$, $ A_2$, $ \dots,$ $ A_n$.

A cube has one of its vertices and all edges connected to that vertex deleted. How many ways can the letters from the word "$AMONGUS$" be placed on the remaining vertices of the cube so that one can walk along the edges to spell out "$AMONGUS$"? Note that each vertex will have at most $1$ letter, and one vertex is deleted and not included in the walk

Given integers $a$ and $b$ such that $a^2+b^2$ is divisible by $11$. Prove that $a$ and $b$ are both divisible by $11$.

Some bishops and knights are placed on an infinite chessboard, where each square has side length $1$ unit. Suppose that the following conditions hold: [list] [*] For each bishop, there exists a knight on the same diagonal as that bishop (there may be another piece between the bishop and the knight). [*] For each knight, there exists a bishop that is exactly $\sqrt{5}$ units away from it. [*] If any piece is removed from the board, then at least one of the above conditions is no longer satisfied. [/list] If $n$ is the total number of pieces on the board, find all possible values of $n$. [i]Sheldon Kieren Tan[/i]

A parallelogram $ABCD$ is split by the diagonal $BD$ into two equal triangles. A regular hexagon is inscribed into the triangle $ABD$ so that two of its consecutive sides lie on $AB$ and $AD$ and one of its vertices lies on $BD$. Another regular hexagon is inscribed into the triangle $CBD{}$ so that two of its consecutive vertices lie on $CB$ and $CD$ and one of its sides lies on $BD$. Which of the hexagons is bigger? [i]Konstantin Knop[/i]

$p,q$ satisfies $px+q\geq \ln x$ at $a\leq x\leq b\ (0<a<b)$. Find the value of $p,q$ for which the following definite integral is minimized and then the minimum value. \[\int_a^b (px+q-\ln x)dx\]

Given $n$ sets $A_i$, with $| A_i | = n$, prove they may be indexed $A_i = \{a_{i,j} \mid j=1,2,\ldots,n \}$, in such way that the sets $B_j = \{a_{i,j} \mid i=1,2,\ldots,n \}$, $1\leq j\leq n$, also have $| B_j | = n$. (Anonymous)

[b]p19.[/b] $A_1A_2...A_{12}$ is a regular dodecagon with side length $1$ and center at point $O$. What is the area of the region covered by circles $(A_1A_2O)$, $(A_3A_4O)$, $(A_5A_6O)$, $(A_7A_8O)$, $(A_9A_{10}O)$, and $(A_{11}A_{12}O)$? $(ABC)$ denotes the circle passing through points $A,B$, and $C$. [b]p20.[/b] Let $N = 2000... 0x0 ... 00023$ be a $2023$-digit number where the $x$ is the $23$rd digit from the right. If$ N$ is divisible by $13$, compute $x$. [b]p21.[/b] Alice and Bob each visit the dining hall to get a grilled cheese at a uniformly random time between $12$ PM and $1$ PM (their arrival times are independent) and, after arrival, will wait there for a uniformly random amount of time between $0$ and $30$ minutes. What is the probability that they will meet? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose we list the decimal representations of the positive even numbers from left to right. Determine the $2015^{th}$ digit in the list.

Two truth tellers and two liars are positioned in a line, where every person is distinguishable. How many ways are there to position these four people such that everyone claims that all people directly adjacent to them are liars? $\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }12\qquad\textbf{(E) }16$

An acute triangle $ABC$ is given. Let $D$ be an arbitrary inner point of the side $AB$. Let $M$ and $N$ be the feet of the perpendiculars from $D$ to $BC$ and $AC$, respectively. Let $H_1$ and $H_2$ be the orthocentres of triangles $MNC$ and $MND$, respectively. Prove that the area of the quadrilateral $AH_1BH_2$ does not depend on the position of $D$ on $AB$.