In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin \left( \angle DAE \right)$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$.

Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

[b]p1.[/b] Find the value of the parameter $a$ for which the following system of equations does not have solutions: $$ax + 2y = 1$$ $$2x + ay = 1$$ [b]p2.[/b] Find all solutions of the equation $\cos(2x) - 3 \sin(x) + 1 = 0$. [b]p3.[/b] A circle of a radius $r$ is inscribed into a triangle. Tangent lines to this circle parallel to the sides of the triangle cut out three smaller triangles. The radiuses of the circles inscribed in these smaller triangles are equal to $1,2$ and $3$. Find $r$. [b]p4.[/b] Does there exist an integer $k$ such that $\log_{10}(1 + 49367 \cdot k)$ is also an integer? [b]p5.[/b] A plane is divided by $3015$ straight lines such that neither two of them are parallel and neither three of them intersect at one point. Prove that among the pieces of the plane obtained as a result of such division there are at least $2010$ triangular pieces. PS. You should use hide for answers.

The sequence $\left(a_{n}\right)_{n\in\mathbb{N}}$ is defined recursively as $a_{0}=a_{1}=1$, $a_{n+2}=5a_{n+1}-a_{n}-1$, $\forall n\in\mathbb{N}$ Prove that $$a_{n}\mid a_{n+1}^{2}+a_{n+1}+1$$ for any $n\in\mathbb{N}$

Prove that for every given positive integer $k$, there exist infinitely many $n$, such that $2^{n}+3^{n}-1, 2^{n}+3^{n}-2,\ldots, 2^{n}+3^{n}-k$ are all composite numbers.

Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions: 1) $f$ is surjective 2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$) 3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$). Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.

Show that there exists $n>2$ such that $1991 | 1999 \ldots 91$ (with $n$ 9's).

On a long metal stick, $1000$ red marbles are embedded in the stick so the stick is equally partitioned into $1001$ parts by them. $1001$ blue marbles are embedded in the stick too, so the stick is equally partitioned into $1002$ parts by them. If you cut the metal stick equally into $2003$ smaller parts, how many of the smaller parts would contain at least one embedded marble?

Find all pairs of integers $(x,y)$ satisfying the following condition: [i]each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ [/i] Tournament of Towns

a) Prove that of $5$ consecutive positive integers one that is relatively prime with the other $4$ can always be selected. b) Prove that of $10$ consecutive positive integers one that is relatively prime with the other $9$ can always be selected.

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

Find all prime numbers $p,q,r,k$ such that $pq+qr+rp = 12k+1$

Starting from the number $ 1$ we write down a sequence of numbers where the next number in the sequence is obtained from the previous one either by doubling it, or by rearranging its digits (not allowing the first digit of the rearranged number to be $0$). For instance we might begin: $$1, 2, 4, 8, 16, 61, 122, 212, 424,...$$ Is it possible to construct such a sequence that ends with the number $1,000,000,000$? Is it possible to construct one that ends with the number $9,876,543,210$?

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

Compute gcd $\left( \frac{135^{90}-45^{90}}{90^2} , 90^2 \right)$

The sequence $\{a_n\}_{n\geq 1}$ is defined by $a_{n+2}=7a_{n+1}-a_n$ for positive integers $n$ with initial values $a_1=1$ and $a_2=8$. Another sequence, $\{b_n\}$, is defined by the rule $b_{n+2}=3b_{n+1}-b_n$ for positive integers $n$ together with the values $b_1=1$ and $b_2=2$. Find $\gcd(a_{5000},b_{501})$.

For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.

Given a positive integrer number $n$ ($n\ne 0$), let $f(n)$ be the average of all the positive divisors of $n$. For example, $f(3)=\frac{1+3}{2}=2$, and $f(12)=\frac{1+2+3+4+6+12}{6}=\frac{14}{3}$. [b]a[/b] Prove that $\frac{n+1}{2} \ge f(n)\ge \sqrt{n}$. [b]b[/b] Find all $n$ such that $f(n)=\frac{91}{9}$.

Prove that for each natural number $m$, there is a natural number $N$ such that for each $b$ that $2\leq b\leq1389$ sum of digits of $N$ in base $b$ is larger than $m$.

[u]Round 5[/u] [b]p13.[/b] Vincent the Bug is at the vertex $A$ of square $ABCD$. Each second, he moves to an adjacent vertex with equal probability. The probability that Vincent is again on vertex $A$ after $4$ seconds is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [b]p14.[/b] Let $ABC$ be a triangle with $AB = 2$, $AC = 3$, and $\angle BAC = 60^o$. Let $P$ be a point inside the triangle such that $BP = 1$ and $CP =\sqrt3$, let $x$ equal the area of $APC$. Compute $16x^2$. [b]p15.[/b] Let $n$ be the number of multiples of$ 3$ between $2^{2020}$ and $2^{2021}$. When $n$ is written in base two, how many digits in this representation are $1$? [u]Round 6[/u] [b]p16.[/b] Let $f(n)$ be the least positive integer with exactly n positive integer divisors. Find $\frac{f(200)}{f(50)}$ . [b]p17.[/b] The five points $A, B, C, D$, and $E$ lie in a plane. Vincent the Bug starts at point $A$ and, each minute, chooses a different point uniformly at random and crawls to it. Then the probability that Vincent is back at $A$ after $5$ minutes can be expressed as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [b]p18.[/b] A circle is divided in the following way. First, four evenly spaced points $A, B, C, D$ are marked on its perimeter. Point $P$ is chosen inside the circle and the circle is cut along the rays $PA$, $PB$, $PC$, $PD$ into four pieces. The piece bounded by $PA$, $PB$, and minor arc $AB$ of the circle has area equal to one fifth of the area of the circle, and the piece bounded by $PB$, $PC$, and minor arc $BC$ has area equal to one third of the area of the circle. Suppose that the ratio between the area of the second largest piece and the area of the circle is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [u]Round 7 [/u] [b]p19.[/b] There exists an integer $n$ such that $|2^n - 5^{50}|$ is minimized. Compute $n$. [b]p20.[/b] For nonnegative integers $a = \overline{a_na_{n-1} ... a_2a_1}$, $b = \overline{b_mb_{m-1} ... b_2b_1}$, define their distance to be $$d(a, b) = \overline{|a_{\max\,\,(m,n)} - b_{\max\,\,(m,n)}||a_{\max\,\,(m,n)-1} - b_{\max\,\,(m,n)-1}|...|a_1 - b_1|}$$ where $a_k = 0$ if $k > n$, $b_k = 0$ if $k > m$. For example, $d(12321, 5067) = 13346$. For how many nonnegative integers $n$ is $d(2021, n) + d(12345, n)$ minimized? [b]p21.[/b] Let $ABCDE$ be a regular pentagon and let $P$ be a point outside the pentagon such that $\angle PEA = 6^o$ and $\angle PDC = 78^o$. Find the degree-measure of $\angle PBD$. [u]Round 8[/u] [b]p22.[/b] What is the least positive integer $n$ such that $\sqrt{n + 3} -\sqrt{n} < 0.02$ ? [b]p23.[/b] What is the greatest prime divisor of $20^4 + 21 \cdot 23 - 6$? [b]p24.[/b] Let $ABCD$ be a parallelogram and let $M$ be the midpoint of $AC$. Suppose the circumcircle of triangle $ABM$ intersects $BC$ again at $E$. Given that $AB = 5\sqrt2$, $AM = 5$, $\angle BAC$ is acute, and the area of $ABCD$ is $70$, what is the length of $DE$? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949414p26408213]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.

Let $ m,n > 1$ are integers which satisfy $ n|4^m \minus{} 1$ and $ 2^m|n \minus{} 1$. Is it a must that $ n \equal{} 2^{m} \plus{} 1$?

Determine all positive real numbers $x$ and $y$ satisfying the equation \[x+y+\frac{1}{x}+\frac{1}{y}+4=2\cdot (\sqrt{2x+1}+\sqrt{2y+1})\]