Let $n \ge 1$ and $x_1, \ldots, x_n \ge 0$. Prove that $$ (x_1 + \frac{x_2}{2} + \ldots + \frac{x_n}{n}) (x_1 + 2x_2 + \ldots + nx_n) \le \frac{(n+1)^2}{4n} (x_1 + x_2 + \ldots + x_n)^2 .$$

In the triangle $ABC$ let $AD$ be the interior angle bisector of $\angle ACB$, where $D\in AB$. The circumcenter of the triangle $ABC$ coincides with the incenter of the triangle $BCD$. Prove that $AC^2 = AD\cdot AB$.

One side of a square sheet of paper is colored red, the other - in blue. On both sides, the sheet is divided into $n^2$ identical square cells. In each of these $2n^2$ cells is written a number from $1$ to $k$. Find the smallest $k$,for which the following properties hold simultaneously: (i) on the red side, any two numbers in different rows are distinct; (ii) on the blue side, any two numbers in different columns are different; (iii) for each of the $n^2$ squares of the partition, the number on the blue side is not equal to the number on the red side.

The positive integers $x_1$, $x_2$, $\ldots$, $x_5$, $x_6 = 144$ and $x_7$ are such that $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n=1,2,3,4$. Determine the value of $x_7$.

For how many integer value of $ m$ does the lines $ 13x\plus{}11y \equal{} 700$ and $ y \equal{} mx\minus{}1$ intersect in a point with integer valued coordinats? A. None B. 1 C. 2 D. 3 E. Infinitely many

Given positive integer $n$ and positive number $M$. For all arithmetic squence $a_1,a_2,\cdots,$ that $a_1^2+a_{n+1}^2\leq M$, find the maximum value of $S=a_{n+1}+a_{n+2}+\cdots,a_{2n+1}$.

[b]p1.[/b] (a) Show that for every set of three integers, we can find two of them whose average is also an integer. (b) Show that for every set of $5$ integers, there is a subset of three of them whose average is an integer. [b]p2.[/b] Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two different quadratic polynomials such that $f(7) + f(11) = g(7) + g(11)$. (a) Show that $f(9) = g(9)$. (b) Show that $x = 9$ is the only value of $x$ where $f(x) = g(x)$. [b]p3.[/b] Consider a rectangle $ABCD$ and points $E$ and $F$ on the sides $BC$ and $CD$, respectively, such that the areas of the triangles $ABE$, $ECF$, and $ADF$ are $11$, $3$, and $40$, respectively. Compute the area of rectangle $ABCD$. [img]https://cdn.artofproblemsolving.com/attachments/f/0/2b0bb188a4157894b85deb32d73ab0077cd0b7.png[/img] [b]p4.[/b] How many ways are there to put markers on a $8 \times 8$ checkerboard, with at most one marker per square, such that each of the $8$ rows and each of the $8$ columns contain an odd number of markers? [b]p5.[/b] A robot places a red hat or a blue hat on each person in a room. Each person can see the colors of the hats of everyone in the room except for his own. Each person tries to guess the color of his hat. No communication is allowed between people and each person guesses at the same time (so no timing information can be used, for example). The only information a person has is the color of each other person’s hat. Before receiving the hats, the people are allowed to get together and decide on their strategies. One way to think of this is that each of the $n$ people makes a list of all the possible combinations he could see (there are $2^{n-1}$ such combinations). Next to each combination, he writes what his guess will be for the color of his own hat. When the hats are placed, he looks for the combination on his list and makes the guess that is listed there. (a) Prove that if there are exactly two people in the room, then there is a strategy that guarantees that always at least one person gets the right answer for his hat color. (b) Prove that if you have a group of $2008$ people, then there is a strategy that guarantees that always at least $1004$ people will make a correct guess. (c) Prove that if there are $2009$ people, then there is no strategy that guarantees that always at least $1005$ people will make a correct guess. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two equally oriented regular $2n$-gons $A_1A_2\ldots A_{2n}$ and $B_1B_2\ldots B_{2n}$ are given. The perpendicular bisectors $\ell_i$ of the segments $A_iB_i$ are drawn. Let the lines $\ell_i$ and $\ell_{i+1}$ intersect at the point $K_i$ (hereafter we reduce indices modulo $2n$). Denote by $m_i$ the line $K_iK_{i+n}$. Prove that $n{}$ lines $m_i$ intersect at one point and at that the angles between the lines $m_i$ and $m_{i+1}$ are equal. [i]Proposed by Chan Quang Hung (Vietnam)[/i]

Andrew has $10$ balls in a bag, each a different color. He randomly picks a ball from the bag $4$ times, with replacement. The expected number of distinct colors among the balls he picks is $\tfrac{p}{q}$, where $\gcd(p, q) = 1$ and $p, q > 0$. What is $p + q$?

Let \( A \) and \( B \) be positive integers, and let \( S \) be a set of positive integers with the following properties: (1) For every non-negative integer $k$, $\text{ } A^k \in S$; (2) If a positive integer $ n \in S$, then every positive divisor of $ n$ is in $S$; (3) If $m ,n \in S$ and $m,n$ are coprime, then $mn \in S$; (4) If $n \in S$, then $An + B \in S$. Prove that all positive integers coprime to \( B \) are in \( S \).

Let $I$ be the incenter of a triangle $ABC$ with $\angle BAC=60^\circ$. A line through $I$ parallel to $AC$ intersects $AB$ at $F$. Let $P$ be the point on the side $BC$ such that $3BP=BC$. Prove that $\angle BFP=\frac12\angle ABC$.

In a television series about incidents in a conspicuous town there are $n$ citizens staging in it, where $n$ is an integer greater than $3$. Each two citizens plan together a conspiracy against one of the other citizens. Prove that there exists a citizen, against whom at least $\sqrt{n}$ other citizens are involved in the conspiracy.

A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two. How many three-element subsets of the set of integers $\left\{z\in\mathbb{Z}\mid -2011<z<2011\right\}$ are arithmetic and harmonic? (Remark: The arithmetic mean $A(a,b)$ and the harmonic mean $H(a,b)$ are defined as \[A(a,b)=\frac{a+b}{2}\quad\mbox{and}\quad H(a,b)=\frac{2ab}{a+b}=\frac{2}{\frac{1}{a}+\frac{1}{b}}\mbox{,}\] respectively, where $H(a,b)$ is not defined for some $a$, $b$.)

$AB$ and $AC$ are tangents to a circle $\omega$ with center $O$ at $B,C$ respectively. Point $P$ is a variable point on minor arc $BC$. The tangent at $P$ to $\omega$ meets $AB,AC$ at $D,E$ respectively. $AO$ meets $BP,CP$ at $U,V$ respectively. The line through $P$ perpendicular to $AB$ intersects $DV$ at $M$, and the line through $P$ perpendicular to $AC$ intersects $EU$ at $N$. Prove that as $P$ varies, $MN$ passes through a fixed point.

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of integers. Define $a_n^{(0)} = a_n$ for all $n \in \mathbb{N}$. For all $M \geq 0$, we define $(a_n^{(M + 1)})_{n \in \mathbb{N}}:\, a_n^{(M + 1)} = a_{n + 1}^{(M)} - a_n^{(M)}, \forall n \in \mathbb{N}$. We say that $(a_n)_{n \in \mathbb{N}}$ is $\textrm{(M + 1)-self-referencing}$ if there exists $k_1$ and $k_2$ fixed positive integers such that $a_{n + k_1} = a_{n + k_2}^{(M + 1)}, \forall n \in \mathbb{N}$. (a) Does there exist a sequence of integers such that the smallest $M$ such that it is $\textrm{M-self-referencing}$ is $M = 2022$? (a) Does there exist a stricly positive sequence of integers such that the smallest $M$ such that it is $\textrm{M-self-referencing}$ is $M = 2022$?

Let $p(x)$ be a polynomial with integer coefficients and let $n$ be an integer. Suppose that there is a positive integer $k$ for which $f^{(k)}(n) = n$, where $f^{(k)}(x)$ is the polynomial obtained as the composition of $k$ polynomials $f$. Prove that $p(p(n)) = n$.

A quadrilateral $ABCD$ has an incircle $\Gamma$. The points $X, Y$ are chosen so that $AX-CX=AB-BC$, $BX-DX=BC-CD$, $CY-AY=AD-DC$ and $DY-BY=AB-AD$. Given that the center of $\Gamma$ lies on $XY$, show that $AC, BD, XY$ are concurrent.

In $\triangle ABC$, $AB = 360$, $BC = 507$, and $CA = 780$. Let $M$ be the midpoint of $\overline{CA}$, and let $D$ be the point on $\overline{CA}$ such that $\overline{BD}$ bisects angle $ABC$. Let $F$ be the point on $\overline{BC}$ such that $\overline{DF} \perp \overline{BD}$. Suppose that $\overline{DF}$ meets $\overline{BM}$ at $E$. The ratio $DE: EF$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Evaluate $\frac{3^4 + 3^7}{84}$. $ \mathrm{(A) \ } 27 \qquad \mathrm{(B) \ } 29 \qquad \mathrm {(C) \ } 33 \qquad \mathrm{(D) \ } 37 \qquad \mathrm{(E) \ } 39$

Suppose that $x, y$, and $z$ are positive real numbers satisfying $$\begin{cases} x^2 + xy + y^2 = 64 \\ y^2 + yz + z^2 = 49 \\ z^2 + zx + x^2 = 57 \end{cases}$$ Then $\sqrt[3]{xyz}$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Find all values of $n$ such that there exists a rectangle with integer side lengths, perimeter $n$, and area $2n$.

Lyuba, Tanya, Lena and Ira ran across a flat field. At some point it turned out that among the pairwise distances between them there are distances of $1, 2, 3, 4$ and $5$ meters, and there are no other distances. Give an example of how this could be.

Prove that some (or one) of any $100$ integers can always be chosen so that the sum of the chosen integers is divisible by $100$.

Let $ a_0 $ be a real number. The sequence $ \{a_n \} $ is given by $ a_ {n + 1} = 3 ^ n-5a_n $, $ n = 0,1,2, \ldots $. a) Express the general member $ a_n $ through $ a_0 $ and $ n. $ b) Find such $ a_0, $ that $ a_ {n + 1}> a_n, $ for every $ n. $

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[yf(x+1)=f(x+y-f(x))+f(x)f(f(y))\] for all $x,y \in \mathbb{R}$.