A scalene triangle $ABC$ is inscribed within circle $\omega$. The tangent to the circle at point $C$ intersects line $AB$ at point $D$. Let $I$ be the center of the circle inscribed within $\triangle ABC$. Lines $AI$ and $BI$ intersect the bisector of $\angle CDB$ in points $Q$ and $P$, respectively. Let $M$ be the midpoint of $QP$. Prove that $MI$ passes through the middle of arc $ACB$ of circle $\omega$.

Given a regular $26$-gon. Prove that for any $9$ vertices of that regular $26$-gon, then there exists three vertices that forms an isosceles triangle.

Find all polynomials $f$ with non-negative integer coefficients such that for all primes $p$ and positive integers $n$ there exist a prime $q$ and a positive integer $m$ such that $f(p^n)=q^m$.

a) Show that for each function $f:\mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{R}$, there exists a function $g:\mathbb{Q}\rightarrow \mathbb{R}$ with $f(x,y) \leq g(x)+g(y) $ for all $x,y\in \mathbb{Q}$. b) Find a function $f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, for which there is no function $g:\mathbb{Q}\rightarrow \mathbb{R}$ such that $f(x,y) \leq g(x)+g(y) $ for all $x,y\in \mathbb{R}$.

Let $\Gamma$ be the circumscribed circle of a triangle $ABC$ and let $D$ be a point at line segment $BC$. The circle passing through $B$ and $D$ tangent to $\Gamma$ and the circle passing through $C $and $D$ tangent to $\Gamma$ intersect at a point $E \ne D$. The line $DE$ intersects $\Gamma$ at two points $X$ and $Y$ . Prove that $|EX| = |EY|$.

Several numbers are written in a row. In each move, Robert chooses any two adjacent numbers in which the one on the left is greater than the one on the right, doubles each of them and then switches them around. Prove that Robert can make only a finite number of moves.

Let $A,B\in\mathcal{M}_n(\mathbb{R})$ two real, symmetric matrices with nonnegative eigenvalues. Prove that $A^3+B^3=(A+B)^3$ if and only if $AB=O_n$.

Let $f$ and $g$ be functions from the set $A$ to the same set $A$. We define $f$ to be a functional $n$-th root of $g$ ($n$ is a positive integer) if $f^n(x) = g(x)$, where $f^n(x) = f^{n-1}(f(x)).$ (a) Prove that the function $g : \mathbb R \to \mathbb R, g(x) = 1/x$ has an infinite number of $n$-th functional roots for each positive integer $n.$ (b) Prove that there is a bijection from $\mathbb R$ onto $\mathbb R$ that has no nth functional root for each positive integer $n.$

Andy has $4$ coins $c_1, c_2, c_3, c_4$ such that the probability that coin $c_i$ with $1 \leq i \leq 4$ lands tails is $\frac{1}{2^i}$. Andy flips each coin exactly once. The probability that only one coin lands on heads can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Anthony Yang[/i]

Let $d$ be a positive integer and $1 < a \le (d+2)/(d+1)$. For given $x_0, x_1,\dots, x_d \in (0, a-1)$, let $x_{k+1} = x_k (a - x_{k-d})$, $k \ge d$. Prove that $\lim_{k \to \infty} x_k = a-1$.

Find the smallest number $n \geq 5$ for which there can exist a set of $n$ people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances. [i]Bulgaria[/i]

All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement “No school bigger than Euclid HS sold more T-shirts than Euclid HS”? $(\textbf{A})$ All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS. $(\textbf{B})$ No school that sold more T-shirts than Euclid HS is bigger than Euclid HS. $(\textbf{C})$ All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS. $(\textbf{D})$ All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS. $(\textbf{E})$ All schools smaller than Euclid HS sold more T-shirts than Euclid HS.

[b]p1. [/b](a) Show that for every positive integer $m > 1$, there are positive integers $x$ and $y$ such that $x^2 - y^2 = m^3$. (b) Find all pairs of positive integers $(x, y)$ such that $x^6 = y^2 + 127$. [b]p2.[/b] (a) Let $P(x)$ be a polynomial with integer coefficients. Suppose that $P(0)$ is an odd integer and that $P(1)$ is also an odd integer. Show that if $c$ is an integer then $P(c)$ is not equal to $0$. (b) Let P(x) be a polynomial with integer coefficients. Suppose that $P(1,000) = 1,000$ and $P(2,000) = 2,000.$ Explain why $P(3,000)$ cannot be equal to $1,000$. [b]p3.[/b] Triangle $\vartriangle ABC$ is created from points $A(0, 0)$, $B(1, 0)$ and $C(1/2, 2)$. Let $q, r$, and $s$ be numbers such that $0 < q < 1/2 < s < 1$, and $q < r < s$. Let D be the point on $AC$ which has $x$-coordinate $q$, $E$ be the point on AB which has $x$-coordinate $r$, and $F$ be the point on $BC$ that has $x$-coordinate $s$. (a) Find the area of triangle $\vartriangle DEF$ in terms of $q, r$, and $s$. (b) If $r = 1/2$, prove that at least one of the triangles $\vartriangle ADE$, $\vartriangle CDF$, or $\vartriangle BEF$ has an area of at least $1/4$. [b]p4.[/b] In the Gregorian calendar: (i) years not divisible by $4$ are common years, (ii) years divisible by $4$ but not by $100$ are leap years, (iii) years divisible by $100$ but not by $400$ are common years, (iv) years divisible by $400$ are leap years, (v) a leap year contains $366$ days, a common year $365$ days. From the information above: (a) Find the number of common years and leap years in $400$ consecutive Gregorian years. Show that $400$ consecutive Gregorian years consists of an integral number of weeks. (b) Prove that the probability that Christmas falls on a Wednesday is not equal to $1/7$. [b]p5.[/b] Each of the first $13$ letters of the alphabet is written on the back of a card and the $13$ cards are placed in a row in the order $$A,B,C,D,E, F, G,H, I, J,K, L,M$$ The cards are then turned over so that the letters are face down. The cards are rearranged and again placed in a row, but of course they may be in a different order. They are rearranged and placed in a row a second time and both rearrangements were performed exactly the same way. When the cards are turned over the letters are in the order $$B,M, A,H, G,C, F,E,D, L, I,K, J$$ What was the order of the letters after the cards were rearranged the first time? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number of cases in which one of the numbers 1, 2, 3, 4, and 5 is written at each vertex of an equilateral triangle so that the following conditions are satisfied. (However, the same number is counted as one when rotated, and the same number can be written multiple times.) $ \bigstar $ The product of the two numbers written at each end of the sides of an equilateral triangle is an even number.

There were made 7 golden, 7 silver and 7 bronze for a tournament. All the medals of the same material should weigh the same and the medals of different materials should have different weight. However, it so happened that exactly one medal had a wrong weight. If this medal is golden, it is lighter than a standard golden medal; if it is bronze, it is heavier than a standard bronze one; if it is silver, it may be lighter or heavier than a standard silver one. Is it possible to find the nonstandard one for sure, using three weighings on a balance scale with no weights?

Let $M$ be an arbitrary interior point of a tetrahedron $ABCD$, and let $S_A,S_B,S_C,S_D$ be the areas of the faces $BCD,ACD,ABD,ABC$, respectively. Prove that $$S_A\cdot MA+S_B\cdot MB+S_C\cdot MC+S_D\cdot MD\ge9V,$$where $V$ is the volume of $ABCD$. When does equality hold?

For a triangle $ \triangle ABC (\angle B > \angle C) $, $ D $ is a point on $ AC $ satisfying $ \angle ABD = \angle C $. Let $ I $ be the incenter of $ \triangle ABC $, and circumcircle of $ \triangle CDI $ meets $ AI $ at $ E ( \ne I )$. The line passing $ E $ and parallel to $ AB $ meets the line $ BD $ at $ P $. Let $ J $ be the incenter of $ \triangle ABD $, and $ A' $ be the point such that $ AI = IA' $. Let $ Q $ be the intersection point of $ JP $ and $ A'C $. Prove that $ QJ = QA' $.

Prove that for each positive integer $m$, one can find $m$ consecutive positive integers like $n$ such that the following phrase doesn't be a perfect power: $$\left(1^3+2018^3\right)\left(2^3+2018^3\right)\cdots \left(n^3+2018^3\right)$$ [i]Proposed by Navid Safaei[/i]

Mustafa bought a big rug. The seller measured the rug with a ruler that was supposed to measure one meter. As it turned out to be $30$ meters long by $20$ meters wide, he charged Rs $120.000$ Rs. When Mustafa arrived home, he measured the rug again and realized that the seller had overcharged him by $9.408$ Rs. How many centimeters long is the ruler used by the seller?

Isabella's house has $ 3$ bedrooms. Each bedroom is $ 12$ feet long, $ 10$ feet wide, and $ 8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $ 60$ square feet in each bedroom. How many square feet of walls must be painted? $ \textbf{(A)}\ 678 \qquad \textbf{(B)}\ 768 \qquad \textbf{(C)}\ 786 \qquad \textbf{(D)}\ 867 \qquad \textbf{(E)}\ 876$

Let $n$ be a positive integer $\geq 2$ . Consider a $n$ by $n$ grid with all entries $1$. Define an operation on a square to be changing the signs of all squares adjacent to it but not the sign of its own. Find all $n$ such that it is possible after a finite sequence of operations to reach a $n$ by $n$ grid with all entries $-1$

Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.

The zeroes of a fourth degree polynomial $f(x)$ form an arithmetic progression. Prove that the three zeroes of the polynomial $f'(x)$ also form an arithmetic progression.

Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on side $BC$. Points $X$ and $Y$ are chosen on lines $BI$ and $CI$ respectively such that $DXIY$ is a parallelogram. Points $E$ and $F$ are chosen on side $BC$ such that $AX$ and $AY$ are the angle bisectors of angles $\angle BAE$ and $\angle CAF$ respectively. Let $\omega$ be the circle tangent to segment $EF$, the extension of $AE$ past $E$, and the extension of $AF$ past $F$. Prove that $\omega$ is tangent to the circumcircle of triangle $ABC$.

In the triangle $ABC$ we have $30^o$ at the vertex $A$, and $50^o$ at the vertex $B$. Let $O$ be the center of inscribed circle. Show that $AC + OC = AB$.