Let $\vartriangle ABC$ be an equilateral triangle, and let $M$ and $N$ be points on $AB$ and $AC$, respectively, so that $AN = BM$ and $3MB = AB$. Lines $CM$ and $BN$ intersect at $O$. Find $\angle AOB$.

The diagram shows a semicircle with diameter $20$ and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is $N\pi$, where $N$ is a positive integer. Find $N$.

Find all positive integers $x$ for which there exists a positive integer $y$ such that $\dbinom{x}{y}=1999000$

Given two positives integers $m$ and $n$, prove that there exists a positive integer $k$ and a set $S$ of at least $m$ multiples of $n$ such that the numbers $\frac {2^k{\sigma({s})}} {s}$ are odd for every $s \in S$. $\sigma({s})$ is the sum of all positive integers of $s$ (1 and $s$ included).

Prove that if the natural numbers $ a $, $ b $, $ c $ satisfy the equation $$ a^2 + b^2 = c^2,$$ then: 1) at least one of the numbers $ a $ and $ b $ is divisible by $ 3 $, 2) at least one of the numbers $ a $ and $ b $ is divisible by $ 4 $, 3) at least one of the numbers $ a $, $ b $, $ c $ is divisible by $ 5 $.

In a square $ABCD$ of diagonals $AC$ and $BD$, we call $O$ at the center of the square. A square $PQRS$ is constructed with sides parallel to those of $ABCD$ with $P$ in segment $AO, Q$ in segment $BO, R$ in segment $CO, S$ in segment $DO$. If area of $ABCD$ equals two times the area of $PQRS$, and $M$ is the midpoint of the $AB$ side, calculate the measure of the angle $\angle AMP$.

Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and \[a+b+c-3\sqrt[3]{abc}=1.\] [i]Proposed by usjl[/i]

What is the value of $$(\sqrt{2}-1)^4+\frac{1}{(\sqrt{2}-1)^4}?$$ $\textbf{(A) }32-16\sqrt{2}\qquad\textbf{(B) }30\qquad\textbf{(C) }34\qquad\textbf{(D) }15+15\sqrt{2}\qquad\textbf{(E) }16+16\sqrt{2}$

$a_{0}=2,a_{1}=1$ and for $n\geq 1$ we know that : $a_{n+1}=a_{n}+a_{n-1}$ $m$ is an even number and $p$ is prime number such that $p$ divides $a_{m}-2$. Prove that $p$ divides $a_{m+1}-1$.

Let $f(x)$ be a real-valued function defined on the positive reals such that (1) if $x < y$, then $f(x) < f(y)$, (2) $f\left(2xy\over x+y\right) \geq {f(x) + f(y)\over2}$ for all $x$. Show that $f(x) < 0$ for some value of $x$.

Prove that there exist $78$ lines in the plane such that they have exactly $1992$ points of intersection.

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

Let $ABC$ be a triangle in which $CA>BC>AB$. Let $H$ be its orthocentre and $O$ its circumcentre. Let $D$ and $E$ be respectively the midpoints of the arc $AB$ not containing $C$ and arc $AC$ not containing $B$. Let $D'$ and $E'$ be respectively the reflections of $D$ in $AB$ and $E$ in $AC$. Prove that $O, H, D', E'$ lie on a circle if and only if $A, D', E'$ are collinear.

Prove that the integral part of the decimal representation of the number $(3+\sqrt{5})^n$ is odd, for every positive integer $n.$

Assume that the set of all positive integers is decomposed into $ r$ (disjoint) subsets $ A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \mathbb{N}.$ Prove that one of them, say $ A_i,$ has the following property: There exists a positive $ m$ such that for any $ k$ one can find numbers $ a_1, a_2, \ldots, a_k$ in $ A_i$ with $ 0 < a_{j \plus{} 1} \minus{} a_j \leq m,$ $ (1 \leq j \leq k \minus{} 1)$.

For a positive integer $n$, let $p(n)$ denote the number of prime divisors of $n$, counting multiplicity (i.e. $p(12)=3$). A sequence $a_n$ is defined such that $a_0 = 2$ and for $n > 0$, $a_n = 8^{p(a_{n-1})} + 2$. Compute $$\sum_{n=0}^{\infty} \frac{a_n}{2^n}$$

Let $(a_n)_{n\ge 1}$ be a sequence of real numbers such that \[a_1\binom{n}{1}+a_2\binom{n}{2}+\ldots+a_n\binom{n}{n}=2^{n-1}a_n,\ (\forall)n\in \mathbb{N}^*\] Prove that $(a_n)_{n\ge 1}$ is an arithmetical progression. [i]Lucian Dragomir[/i]

A $2\times 2$ square was cut from a squared sheet of paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.

Show that the sum of the (local) maximum and minimum values of the function $\frac{tan(3x)}{tan^3x}$ on the interval $\big(0, \frac{\pi }{2}\big)$ is rational.

A convex quadrilateral $ABCD$ is given in which $\angle DAB = \angle ABC = 45^o$ and $DA = 3$, $AB = 7\sqrt2$, $BC = 4$. Calculate the length of side $CD$. [img]https://cdn.artofproblemsolving.com/attachments/1/2/046e31a628b3df4d23d3162cb570e1b9cb71e2.png[/img]

Let $f:[0, \infty) \to (0, \infty)$ be an increasing function and $g:[0, \infty) \to \mathbb{R}$ be a two times differentiable function such that $g''$ is continuous and $g''(x)+f(x)g(x) = 0, \: \forall x \geq 0.$ $\textbf{a)}$ Provide an example of such functions, with $g \neq 0.$ $\textbf{b)}$ Prove that $g$ is bounded.

Let $T=\text{TNFTPP}$. For natural numbers $k,n\geq 2$, we define $S(k,n)$ such that \[S(k,n)=\left\lfloor\dfrac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\dfrac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\dfrac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor.\] Compute the value of $S(10,T+55)-S(10,55)+S(10,T-55)$.

For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.

[u]Round 9[/u] [b]p25.[/b] Find the largest prime factor of $1031301$. [b]p26.[/b] Let $ABCD$ be a trapezoid such that $AB \parallel CD$, $\angle ABC = 90^o$ , $AB = 5$, $BC = 20$, $CD = 15$. Let $X$, $Y$ be the intersection of the circle with diameter $BC$ and segment $AD$. Find the length of $XY$. [b]p27.[/b] A string consisting of $1$’s, $2$’s, and $3$’s is said to be a superpermutation of the string $123$ if it contains every permutation of $123$ as a contiguous substring. Find the smallest possible length of such a superpermutation. [u]Round 10[/u] [b]p28.[/b] Suppose that we have a function $f (x) = x^3 -3x^2 +3x$, and for all $n \ge 1$, $f^n(x)$ is defined by the function $f$ applied $n$ times to $x$. Find the remainder when $f^5(2019)$ is divided by $100$. [b]p29.[/b] A function $f : {1,2, . . . ,10} \to {1,2, . . . ,10}$ is said to be happy if it is a bijection and for all $n \in {1,2, . . . ,10}$, $|n - f (n)| \le 1$. Compute the number of happy functions. [b]p30.[/b] Let $\vartriangle LMN$ have side lengths $LM = 15$, $MN = 14$, and $NL = 13$. Let the angle bisector of $\angle MLN$ meet the circumcircle of $\vartriangle LMN$ at a point $T \ne L$. Determine the area of $\vartriangle LMT$ . [u]Round 11[/u] [b]p31.[/b] Find the value of $$\sum_{d|2200} \tau (d),$$ where $\tau (n)$ denotes the number of divisors of $n$, and where $a|b$ means that $\frac{b}{a}$ is a positive integer. [b]p32.[/b] Let complex numbers $\omega_1,\omega_2, ...,\omega_{2019}$ be the solutions to the equation $x^{2019}-1 = 0$. Evaluate $$\sum^{2019}_{i=1} \frac{1}{1+ \omega_i}.$$ [b]p33.[/b] Let $M$ be a nonnegative real number such that $x^{x^{x^{...}}}$ diverges for all $x >M$, and $x^{x^{x^{...}}}$ converges for all $0 < x \le M$. Find $M$. [u]Round 12[/u] [b]p34.[/b] Estimate the number of digits in ${2019 \choose 1009}$. If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$ [b]p35.[/b] You may submit any integer $E$ from $1$ to $30$. Out of the teams that submit this problem, your score will be $$\frac{E}{2 \, (the\,\, number\,\, of\,\, teams\,\, who\,\, chose\,\, E)}$$ [b]p36.[/b] We call a $m \times n$ domino-tiling a configuration of $2\times 1$ dominoes on an $m\times n$ cell grid such that each domino occupies exactly $2$ cells of the grid and all cells of the grid are covered. How many $8 \times 8$ domino-tilings are there? If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166016p28809598]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166019p28809679]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

All angles of $ABC$ are in $(30,90)$. Circumcenter of $ABC$ is $O$ and circumradius is $R$. Point $K$ is projection of $O$ to angle bisector of $\angle B$, point $M$ is midpoint $AC$. It is known, that $2KM=R$. Find $\angle B$