Let $ABC$ be an isosceles triangle with apex at $C.$ Let $D$ and $E$ be two points on the sides $AC$ and $BC$ such that the angle bisectors $\angle DEB$ and $\angle ADE$ meet at $F,$ which lies on segment $AB.$ Prove that $F$ is the midpoint of $AB.$

Determine all polynomials $f$ with integer coefficients with the property that for any two distinct primes $p$ and $q$, $f(p)$ and $f(q)$ are relatively prime.

Given an equilateral triangle lattice composed of $\frac{n(n+1)}{2}$ points (as shown in the figure), record the number of equilateral triangles with three points in the lattice as vertices as $f(n)$. Find an expression for $f(n)$. [img]https://cdn.artofproblemsolving.com/attachments/7/f/1de27231e8ef9c1c6a3dfd590a7c71adc508d6.png[/img]

For natural numbers $a, b$, define $Z(a,b)=\frac{(3a)!\cdot (4b)!}{a!^4 \cdot b!^3}$. [b](a)[/b] Prove that $Z(a, b)$ is an integer for $a \leq b$. [b](b)[/b] Prove that for each natural number $b$ there are infinitely many natural numbers a such that $Z(a, b)$ is not an integer.[/list]

Let $x_0,x_1,x_2,\dots$ be a infinite sequence of real numbers, such that the following three equalities are true: I- $x_{2k}=(4x_{2k-1}-x_{2k-2})^2$, for $k\geq 1$ II- $x_{2k+1}=|\frac{x_{2k}}{4}-k^2|$, for $k\geq 0$ III- $x_0=1$ a) Determine the value of $x_{2022}$ b) Prove that there are infinite many positive integers $k$, such that $2021|x_{2k+1}$

If $x > y > 0$, then $\frac{x^{y}y^{x}}{y^{y}x^{x}} = $ $ \textbf{(A)}\ (x - y)^{y/x}\qquad\textbf{(B)}\ \left(\frac{x}{y}\right)^{x-y}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \left(\frac{x}{y}\right)^{y-x}\qquad\textbf{(E)}\ (x - y)^{x/y} $

Suppose that $a,b,c$ are positive numbers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is an integer. Show that $abc$ is a perfect cube.

If $ x < 0$, then which of the following must be positive? $ \textbf{(A)}\ \frac{x}{|x|}\qquad \textbf{(B)}\ \minus{}x^2\qquad \textbf{(C)}\ \minus{}2^x\qquad \textbf{(D)}\ \minus{}x^{\minus{}1}\qquad \textbf{(E)}\ \sqrt[3]{x}$

Let $ABCD$ be a cyclic quadrilateral with center $O$. Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$, while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$. Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$. Define $\omega_2$ analogously as the circle passing through $H$ and the feet of the perpendiculars from $H$ to $BC$ and $DA$. Show that the midpoint of $GH$ lies on the radical axis of $\omega_1$ and $\omega_2$. [i]Proposed by Yang Liu[/i]

How many tangents to the curve $y = x^3-3x\:\: (y = x^3 + px)$ can be drawn from different points in the plane?

For any positive numbers $a,b,c,x,y, z$, prove the inequality $ \frac{a^3}{x}+ \frac{b^3}{y}+ \frac{c^3}{z} \ge \frac{(a+b+c)^3}{3(x+y+z)}$

Let $n$ be a positive integer, where $n \geq 3$ and let $a_1, a_2, ..., a_n$ be the lengths of sides of some $n$-gon. Prove that $$a_1 + a_2 + ... + a_n \geq \sqrt{2 \cdot (a_1^2 + a_2^2 + ... + a_n^2)} $$

The terms of the sequence $ (a_i)$ defined by $ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} 2009} {1 \plus{} a_{n \plus{} 1}}$ for $ n \ge 1$ are positive integers. Find the minimum possible value of $ a_1 \plus{} a_2$.

It is possible to perform three operations $f, g$, and $h$ for positive integers: $f(n) = 10n, g(n) = 10n + 4$, and $h(2n) = n$; in other words, one may write $0$ or $4$ in the end of the number and one may divide an even number by $2$. Prove: every positive integer can be constructed starting from $4$ and performing a finite number of the operations $f, g,$ and $h$ in some order.

$x,y,z$ positive real numbers such that $xyz=1$ Prove that: $\frac{1}{x+y^{20}+z^{11}}+\frac{1}{y+z^{20}+x^{11}}+\frac{1}{z+x^{20}+y^{11}}\leq1$

Can we find positive reals $a_1, a_2, \dots, a_{2002}$ such that for any positive integer $k$, with $1 \leq k \leq 2002$, every complex root $z$ of the following polynomial $f(x)$ satisfies the condition $|\text{Im } z| \leq |\text{Re } z|$, \[f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,\] where $a_{2002+i}=a_i$, for $i=1,2, \dots, 2001$.

Three circles, $\omega_1$, $\omega_2$, $\omega_3$ are drawn, with $\omega_3$ externally tangent to $\omega_1$ at $C$ and internally tangent to $\omega_2$ at $D$. Say also that $\omega_1$, $\omega_2$ intersect at points $A, B$. Suppose the radius of $\omega_1$ is $20$, the radius of $\omega_2$ is $15$, and the radius of $\omega_3$ is $6$. Draw line $CD$, and suppose it meets $AB$ at point $X$. If $AB = 24$, then $CX$ can be written in the form $\frac{a \sqrt{b}}{c}$, where$ a, b, c$ are positive integers where $b$ is square-free, and $a, c$ are relatively prime. Find $a + b + c$.

An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\tfrac23$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle? [asy] size(5cm); fill((0,0)--(2/3,1.155/3)--(4-(4-2)/3,1.155/3)--(4,0)--cycle,lightgray*0.5+mediumgray*0.5); draw((0,0)--(4,0)--(2,2*sqrt(3))--cycle); //center: 2,1.155 draw((2/3,1.155/3)--(4-(4-2)/3,1.155/3)--(2,2*sqrt(3)-0.7697)--cycle); dot((0,0)^^(4,0)^^(2,2*sqrt(3))^^(2/3,1.155/3)^^(4-(4-2)/3,1.155/3)^^(2,2*sqrt(3)-0.7697)); draw((0,0)--(2/3,1.155/3)); draw((4,0)--(4-(4-2)/3,1.155/3)); draw((2,2*sqrt(3))--(2,2*sqrt(3)-0.7697)); [/asy] $\textbf{(A) } 1:3\qquad\textbf{(B) } 3:8\qquad\textbf{(C) } 5:12\qquad\textbf{(D) } 7:16\qquad\textbf{(E) } 4:9$

Let $p$ be a polynomial with integer coefficients such that $p(15)=6$, $p(22)=1196$, and $p(35)=26$. Find an integer $n$ such that $p(n)=n+82$.

Let $C_1$, $C_2$ and $C_3$ be three parallel chords of a circle on the same side of the center. The distance between $C_1$ and $C_2$ is the same as the distance between $C_2$ and $C_3$. The lengths of the chords are 20, 16, and 8. The radius of the circle is $\text{(A)} \ 12 \qquad \text{(B)} \ 4\sqrt{7} \qquad \text{(C)} \ \frac{5\sqrt{65}}{3} \qquad \text{(D)} \ \frac{5\sqrt{22}}{2} \qquad \text{(E)} \ \text{not uniquely determined}$

Through the midpoint $ S $ of the segment $ MN $ with endpoints lying on the legs of an isosceles triangle, a straight line is drawn parallel to the base of the triangle, intersecting its legs at points $ K $ and $ L $. Prove that the orthogonal projection of the segment $ MN $ onto the base of the triangle is equal to the segment $ KL $.

Find the smallest positive $\alpha$ (in degrees) for which all the numbers \[\cos{\alpha},\cos{2\alpha},\ldots,\cos{2^n\alpha},\ldots\] are negative.

$ ABCD$ is a convex quadrilateral for which $ AB$ is the longest side. Points $ M$ and $ N$ are located on sides $ AB$ and $ BC$ respectively, so that each of the segments $ AN$ and $ CM$ divides the quadrilateral into two parts of equal area. Prove that the segment $ MN$ bisects the diagonal $ BD$.

Find all positive integers $n$ such that $n$ has exactly $6$ positive divisors $1<d_{1}<d_{2}<d_{3}<d_{4}<n$ and $1+n=5(d_{1}+d_{2}+d_{3}+d_{4})$.

Let $ ABCD $ be a cyclic quadrilateral and let $ M $ be the set of incenters and excenters of the triangles $ BCD $, $ CDA $, $ DAB $, $ ABC $ (so 16 points in total). Prove that there exist two sets $ \mathcal{K} $ and $ \mathcal{L} $ of four parallel lines each, such that every line in $ \mathcal{K} \cup \mathcal{L} $ contains exactly four points of $ M $.