Given an acute triangle $ABC$ with $O$ as its circumcenter. Line $AO$ intersects $BC$ at $D$. Points $E$, $F$ are on $AB$, $AC$ respectively such that $A$, $E$, $D$, $F$ are concyclic. Prove that the length of the projection of line segment $EF$ on side $BC$ does not depend on the positions of $E$ and $F$.

Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

[b]p1.[/b] There are $16$ students in a class. Each month the teacher divides the class into two groups. What is the minimum number of months that must pass for any two students to be in different groups in at least one of the months? [b]p2.[/b] Find all functions $f(x)$ defined for all real $x$ that satisfy the equation $2f(x) + f(1 - x) = x^2$. [b]p3.[/b] Arrange the digits from $1$ to $9$ in a row (each digit only once) so that every two consecutive digits form a two-digit number that is divisible by $7$ or $13$. [b]p4.[/b] Prove that $\cos 1^o$ is irrational. [b]p5.[/b] Consider $2n$ distinct positive Integers $a_1,a_2,...,a_{2n}$ not exceeding $n^2$ ($n>2$). Prove that some three of the differences $a_i- a_j$ are equal . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\omega_1$ and $\omega_2$ be two circles intersecting at points $P$ and $Q$. The tangent line closer to $Q$ touches $\omega_1$ and $\omega_2$ at $M$ and $N$ respectively. If $P Q = 3$, $QN = 2$, and $MN = P N$, what is $QM^2$?

A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A=(-3, 2)$ and $B=(3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths? $ \textbf{(A)}\ 161 \qquad \textbf{(B)}\ 185 \qquad \textbf{(C)}\ 195 \qquad \textbf{(D)}\ 227 \qquad \textbf{(E)}\ 255 $

A man born in the first half of the nineteenth century was $ x$ years old in the year $ x^2$. He was born in: $ \textbf{(A)}\ 1849 \qquad \textbf{(B)}\ 1825 \qquad \textbf{(C)}\ 1812 \qquad \textbf{(D)}\ 1836 \qquad \textbf{(E)}\ 1806$

Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$. After he graded Payton's test, the class average became $81$. What was Payton's score on the test? $\textbf{(A) }81\qquad\textbf{(B) }85\qquad\textbf{(C) }91\qquad\textbf{(D) }94\qquad\textbf{(E) }95$

Let a triangle $ABC$ be given. On a ruler three segment congruent to the sides of this triangle are marked. Using this ruler construct the orthocenter of the triangle formed by the tangency points of the sides of $ABC$ with its incircle.

We say that $ n$ squares in a $ n\times n$ board are scattered if no two of them are in the same row or column.In every square of this board is witten a natural number so that the sum of numbrs in $ n$ scattered squares is always the same and no row or no column contains two equal numbers .It turned out that the numbers on the main diagonal are arranged in the increasing order ,and that their product is the smallest among all products of $ n$ scattered numbers .Prove that scattered numbers with the greatest product are exactly those on the other diagonal.

If $a-1=b+2=c-3=d+4$, which of the four quantities $a,b,c,d$ is the largest? $ \textbf{(A)}\ a \qquad\textbf{(B)}\ b \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ d \qquad\textbf{(E)}\ \text{no one is always largest} $

Let $\mathbb{Q}^+$ be the set of all positive rational numbers. Find all functions $f:\mathbb{Q}^+\rightarrow \mathbb{Q}^+$ satisfying $f(1)=1$ and \[ f(x+n)=f(x)+nf(\frac{1}{x}) \forall n\in\mathbb{N},x\in\mathbb{Q}^+\]

Find the sum of the values of $x$ such that $\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x$, where $x$ is measured in degrees and $100< x< 200$.

In fields of a $19 \times 19$ table are written integers so that any two lying on neighboring fields differ at most by $2$ (two fields are neighboring if they share a side). Find the greatest possible number of mutually different integers in such a table.

Let $P(x)$ be a polynomial with integer coefficients. Prove that if there is an integer $k$ such that none of the integers $P(1),P(2), ..., P(k)$ is divisible by $k$, then $P(x)$ does not have integer roots.

There are 100 boxers, each of them having different strengths, who participate in a tournament. Any of them fights each other only once. Several boxers form a plot. In one of their matches, they hide in their glove a horse shoe. If in a fight, only one of the boxers has a horse shoe hidden, he wins the fight; otherwise, the stronger boxer wins. It is known that there are three boxers who obtained (strictly) more wins than the strongest three boxers. What is the minimum number of plotters ? [i]Proposed by N. Kalinin[/i]

Twelve places have been arranged at a round table for members of the Jury, with a name tag at each place . Professor K. being absent-minded instead of occupying his place, sits down at the next place (clockwise) . Each of the other Jury members in turn either occupies the place assigned to this member or, if it has been already occupied, sits down at the first free place in the clockwise order. The resulting seating arrangement depends on the order in which the Jury members come to the table. How many different seating arrangements of this kind are possible? (A Shapovalov)

Let $f(x)=x^4-4x^3-3x^2-4x+1$. Compute the sum of the real roots of $f(x)$.

In triangle $ABC$, $D$ is the middle point of side $BC$ and $M$ is a point on segment $AD$ such that $AM=3MD$. The barycenter of $ABC$ and $M$ are on the inscribed circumference of $ABC$. Prove that $AB+AC>3BC$.

A birdbath is designed to overflow so that it will be self-cleaning. Water flows in at the rate of 20 milliliters per minute and drains at the rate of 18 milliliters per minute. One of these graphs shows the volume of water in the birdbath during the filling time and continuing into the overflow time. Which one is it? [asy] size(450); defaultpen(linewidth(0.8)); path[] p={origin--(8,8)--(14,8), (0,10)--(4,10)--(14,0), origin--(14,14), (0,14)--(14,14), origin--(7,7)--(14,0)}; int i; for(i=0; i<5; i=i+1) { draw(shift(21i,0)*((0,16)--origin--(14,0))); draw(shift(21i,0)*(p[i])); label("Time", (7+21i,0), S); label(rotate(90)*"Volume", (21i,8), W); } label("$A$", (0*21 + 7,-5), S); label("$B$", (1*21 + 7,-5), S); label("$C$", (2*21 + 7,-5), S); label("$D$", (3*21 + 7,-5), S); label("$E$", (4*21 + 7,-5), S); [/asy] $\text{(A)}\ \text{A} \qquad \text{(B)}\ \text{B} \qquad \text{(C)}\ \text{C} \qquad \text{(D)}\ \text{D} \qquad \text{(E)}\ \text{E}$

There is located real number $f(A)$ in any point A on the plane. It's known that if $M$ will be centroid of triangle $ABC$ then $f(M)=f(A)+f(B)+f(C)$. Prove that $f(A)=0$ for all points A.

Let $\mathcal{A}_n$ be the set of $n$-tuples $x = (x_1, ..., x_n)$ with $x_i \in \{0, 1, 2\}$. A triple $x, y, z$ of distinct elements of $\mathcal{A}_n$ is called [i]good[/i] if there is some $i$ such that $\{x_i, y_i, z_i\} = \{0, 1, 2\}$. A subset $A$ of $\mathcal{A}_n$ is called [i]good[/i] if every three distinct elements of $A$ form a good triple. Prove that every good subset of $\mathcal{A}_n$ has at most $2(\frac{3}{2})^n$ elements.

Find the real numbers $ x>1 $ having the property that $ \sqrt[n]{\lfloor x^n \rfloor } $ is an integer for any natural number $ n\ge 2. $ [i]Mihai Piticari[/i] and [i]Dan Popescu[/i]

Let $n(n\geq2)$ be a natural number and $a_1,a_2,...,a_n$ natural positive real numbers. Determine the least possible value of the expression $$E_n=\frac{(1+a_1)\cdot(a_1+a_2)\cdot(a_2+a_3)\cdot...\cdot(a_{n-1}+a_n)\cdot(a_n+3^{n+1})} {a_1\cdot a_2\cdot a_3\cdot...\cdot a_n}$$

A positive integer $n$ is [i]magical[/i] if $\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor=\lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil$. Find the number of magical integers between $1$ and $10,000$ inclusive.

Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold: $$d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1$$ [i]Proposed by Mykyta Kharin[/i]