A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.

Points $M$ and $N$ lie on the side $BC$ of the regular triangle $ABC$ ($M$ is between $B$ and $N$), and $\angle MAN=30^\circ.$ The circumcircles of triangles $AMC$ and $ANB$ meet at a point $K.$ Prove that the line $AK$ passes through the circumcenter of triangle $AMN.$

Sides of a triangle are equal to $3$, $4$ and $5$. Each side is extended until it intersects the bisector of the external angle to the angle opposite to it. Three such points are obtained in all. Prove that one of the three points we get is the midpoint of the segment joining the other two points. (V. Prasolov)

Let $n$ be an integer, and let $X$ be a set of $n+2$ integers each of absolute value at most $n$. Show that there exist three distinct numbers $a, b, c \in X$ such that $c=a+b$.

Consider an unbiased coin which is tossed infinitely many times. Let $A_n$ be the event that no two successive heads occur in the first $n$ tosses of this experiment. Then which of the following is incorrect : (A) $\lim_{n \to \infty} P(A_n)=0$ (B) $\lim_{n \to \infty}3^n P(A_n)=0$ (C) $2^nP(A_n) +2^{n+1}P(A_{n+1})=2^{n+2}P(A_{n+2}$ (D) $\lim_{n \to \infty} \frac{P(A_n)}{P(A_{n+1})}$ is lesser than $1.2$

Find an acutangle triangle such that its sides and altitudes have integer length.

Let $A{}$ be a set with $n{}$ $(n\geq3)$ elements. Iterations $f^2,f^2,\ldots$ of the function $f:A\rightarrow A$ are defined as $f^2(x)=f(f(x)), f^{i+1}=f(f^i(x)), \forall i\geq2$. Find the number of functions $f:A\rightarrow A$ with the property: the function $f^{n-2}$ is constant, but $f^{n-3}$ is not.

Consider the polynomials $P(x)=x^{6}-x^{5}-x^{3}-x^{2}-x$ and $Q(x)=x^{4}-x^{3}-x^{2}-1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x)=0,$ find $P(z_{1})+P(z_{2})+P(z_{3})+P(z_{4}).$

A polynomial $P$ with integer coefficients is [i]square-free[/i] if it is not expressible in the form $P = Q^2R$, where $Q$ and $R$ are polynomials with integer coefficients and $Q$ is not constant. For a positive integer $n$, let $P_n$ be the set of polynomials of the form $$1 + a_1x + a_2x^2 + \cdots + a_nx^n$$ with $a_1,a_2,\ldots, a_n \in \{0,1\}$. Prove that there exists an integer $N$ such that for all integers $n \geq N$, more than $99\%$ of the polynomials in $P_n$ are square-free. [i]Navid Safaei, Iran[/i]

Prove that in any triangle $ABC$ the following inequality holds: \[ \frac{a^{2}}{b+c-a}+\frac{b^{2}}{a+c-b}+\frac{c^{2}}{a+b-c}\geq 3\sqrt{3}R. \] [i]Laurentiu Panaitopol[/i]

All the cells in a $8* 8$ board are colored white. Omar and Asaad play the following game: in the beginning Omar colors $n$ cells red, then Asaad chooses $4$ rows and $4$ columns and colors them black. Omar wins if there is at least one red cell. Find the least possible value for n such that Omar can always win regardless of Asaad's move.

Let $x,y,z\in \mathbb{R}^*$ such that $xy,yz,zx\in \mathbb{Q}$. a) Prove that $x^2+y^2+z^2$ is rational; b) If $x^3+y^3+z^3$ is rational, prove that $x,y,z$ are rational. [i]Marius Ghergu[/i]

In a right triangle, the altitude from a vertex to the hypotenuse splits the hypotenuse into two segments of lengths $a$ and $b$. If the right triangle has area $T$ and is inscribed in a circle of area $C$, find $ab$ in terms of $T$ and $C$.

[b]p1.[/b] The number $2020$ is very special: the sum of its digits is equal to the product of its nonzero digits. How many such four digit numbers are there? (Numbers with only one nonzero digit, like $3000$, also count) [b]p2.[/b] A locker has a combination which is a sequence of three integers between $ 0$ and $49$, inclusive. It is known that all of the numbers in the combination are even. Let the total of a lock combination be the sum of the three numbers. Given that the product of the numbers in the combination is $12160$, what is the sum of all possible totals of the locker combination? [b]p3.[/b] Given points $A = (0, 0)$ and $B = (0, 1)$ in the plane, the set of all points P in the plane such that triangle $ABP$ is isosceles partitions the plane into $k$ regions. The sum of the areas of those regions that are bounded is $s$. Find $ks$. [b]p4.[/b] Three families sit down around a circular table, each person choosing their seat at random. One family has two members, while the other two families have three members. What is the probability that every person sits next to at least one person from a different family? [b]p5.[/b] Jacob and Alexander are walking up an escalator in the airport. Jacob walks twice as fast as Alexander, who takes $18$ steps to arrive at the top. Jacob, however, takes $27$ steps to arrive at the top. How many of the upward moving escalator steps are visible at any point in time? [b]p6.[/b] Points $A, B, C, D, E$ lie in that order on a circle such that $AB = BC = 5$, $CD = DE = 8$, and $\angle BCD = 150^o$ . Let $AD$ and $BE$ intersect at $P$. Find the area of quadrilateral $PBCD$. [b]p7.[/b] Ivan has a triangle of integers with one number in the first row, two numbers in the second row, and continues up to eight numbers in the eighth row. He starts with the first $8$ primes, $2$ through $19$, in the bottom row. Each subsequent row is filled in by writing the least common multiple of two adjacent numbers in the row directly below. For example, the second last row starts with$ 6, 15, 35$, etc. Let P be the product of all the numbers in this triangle. Suppose that P is a multiple of $a/b$, where $a$ and $b$ are positive integers and $a > 1$. Given that $b$ is maximized, and for this value of $b, a$ is also maximized, find $a + b$. [b]p8.[/b] Let $ABCD$ be a cyclic quadrilateral. Given that triangle $ABD$ is equilateral, $\angle CBD = 15^o$, and $AC = 1$, what is the area of $ABCD$? [b]p9.[/b] Let $S$ be the set of all integers greater than $ 1$. The function f is defined on $S$ and each value of $f$ is in $S$. Given that $f$ is nondecreasing and $f(f(x)) = 2x$ for all $x$ in $S$, find $f(100)$. [b]p10.[/b] An [i]origin-symmetric[/i] parallelogram $P$ (that is, if $(x, y)$ is in $P$, then so is $(-x, -y)$) lies in the coordinate plane. It is given that P has two horizontal sides, with a distance of $2020$ between them, and that there is no point with integer coordinates except the origin inside $P$. Also, $P$ has the maximum possible area satisfying the above conditions. The coordinates of the four vertices of P are $(a, 1010)$, $(b, 1010)$, $(-a, -1010)$, $(-b, -1010)$, where a, b are positive real numbers with $a < b$. What is $b$? [b]p11.[/b] What is the remainder when $5^{200} + 5^{50} + 2$ is divided by $(5 + 1)(5^2 + 1)(5^4 + 1)$? [b]p12.[/b] Let $f(n) = n^2 - 4096n - 2045$. What is the remainder when $f(f(f(... f(2046)...)))$ is divided by $2047$, where the function $f$ is applied $47$ times? [b]p13.[/b] What is the largest possible area of a triangle that lies completely within a $97$-dimensional hypercube of side length $1$, where its vertices are three of the vertices of the hypercube? [b]p14.[/b] Let $N = \left \lfloor \frac{1}{61} \right \rfloor + \left \lfloor\frac{3}{61} \right \rfloor+\left \lfloor \frac{3^2}{61} \right \rfloor+... +\left \lfloor\frac{3^{2019}}{61} \right \rfloor$. Given that $122N$ can be expressed as $3^a - b$, where $a, b$ are positive integers and $a$ is as large as possible, find $a + b$. Note: $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$. [b]p15.[/b] Among all ordered triples of integers $(x, y, z)$ that satisfy $x + y + z = 8$ and $x^3 + y^3 + z^3 = 134$, what is the maximum possible value of $|x| + |y| + |z|$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that a positive integer $m$ is a sum of two triangular numbers if and only if $4m+1$ is a sum of two squares.

An ellipse with foci $B_0,B_1$ intersects $AB_i$ at $C_i$ $(i=0,1)$. Let $P_0$ be a point on ray $AB_0$. $Q_0$ is a point on ray $C_1B_0$ such that $B_0P_0=B_0Q_0$; $P_1$ is on ray $B_1A$ such that $C_1Q_0=C_1P_1$; $Q_1$ is on ray $B_1C_0$ such that $B_1P_1=B_1Q_1$; $P_2$ is on ray $AB_0$ such that $C_0Q_1=C_0Q_2$. Prove that $P_0=P_2$ and that the four points $P_0,Q_0,Q_1,P_1$ are concyclic.

$BB_1$ and $CC_1$ are altitudes in $\Delta ABC$. Let $B_1 C_1$ intersect the circumscribed circle of $\Delta ABC$ in points $E$ and $F$. Let $k$ be a circle passing through $E$ and $F$ in such way that the center of $k$ lies on the arc $\widehat{BAC}$. We denote with $M$ the middle point of $BC$. $X$ and $Y$ are the points on $k$ for which $MX$ and $MY$ are tangent to $k$. Let $EX\cap FY=S_1,EY\cap FX=S_2,BX\cap CY=U,$ and $BY\cap CX=V$. Prove that $S_1 S_2$ and $UV$ intersect in the orthocenter of $\Delta ABC$.

Let $a, b, c, d, e \in \mathbb{R}^+$ and $f:\{(x, y) \in (\mathbb{R}^+)^2|c-dx-ey > 0\}\to \mathbb{R}^+$ be given by $f(x, y) = (ax)(by)(c- dx- ey)$. Find the maximum value of $f$.

Rio likes fruit, and one day she decides to pick persimmons. She picks a total of $12$ persimmons from the first $5$ trees she sees. Rio has $5$ more trees to pick persimmons from. If she wants to pick an average of $4$ persimmons per tree overall, what is the average number of persimmons that she must pick from each of the last $5$ trees for her goal?

When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$? $\textbf{(A)}~6\qquad\textbf{(B)}~8\qquad\textbf{(C)}~9\qquad\textbf{(D)}~10\qquad\textbf{(E)}~11$

An equilateral triangle has side length $ 6$. What is the area of the region containing all points that are outside the triangle and not more than $ 3$ units from a point of the triangle? $ \textbf{(A)}\ 36\plus{}24\sqrt{3} \qquad \textbf{(B)}\ 54\plus{}9\pi \qquad \textbf{(C)}\ 54\plus{}18\sqrt{3}\plus{}6\pi \qquad \textbf{(D)}\ \left(2\sqrt{3}\plus{}3\right)^2\pi \\ \textbf{(E)}\ 9\left(\sqrt{3}\plus{}1\right)^2\pi$

Let $a, b, m, n$ be positive integers such that $gcd(a, b) = 1$ and $a > 1$. Prove that if $$a^m+b^m \mid a^n+b^n$$then $m \mid n$.

Consider the figure below, not drawn to scale. In this figure, assume that$AB \perp BE$ and $AD \perp DE$. Also, let $AB = \sqrt6$ and $\angle BED =\frac{\pi}{6}$ . Find $AC$. [img]https://cdn.artofproblemsolving.com/attachments/2/d/f87ac9f111f02e261a0b5376c766a615e8d1d8.png[/img]

A six dimensional "cube" (a $6$-cube) has $64$ vertices at the points $(\pm 3,\pm 3,\pm 3,\pm 3,\pm 3,\pm 3).$ This $6$-cube has $192\text{ 1-D}$ edges and $240\text{ 2-D}$ edges. This $6$-cube gets cut into $6^6=46656$ smaller congruent "unit" $6$-cubes that are kept together in the tightly packaged form of the original $6$-cube so that the $46656$ smaller $6$-cubes share 2-D square faces with neighbors ($\textit{one}$ 2-D square face shared by $\textit{several}$ unit $6$-cube neighbors). How many 2-D squares are faces of one or more of the unit $6$-cubes?

As usual, let $\sigma (N)$ denote the sum of all the (positive integral) divisors of $N.$ (Included among these divisors are $1$ and $N$ itself.) For example, if $p$ is a prime, then $\sigma (p)=p+1.$ Motivated by the notion of a "perfect" number, a positive integer $N$ is called "quasiperfect" if $\sigma (N) =2N+1.$ Prove that every quasiperfect number is the square of an odd integer.