Let $n{}$ be a non-negative integer and consider the standard power expansion of the following polynomial \[\sum_{k=0}^n\binom{n}{k}^2(X+1)^{2k}(X-1)^{2(n-k)}=\sum_{k=0}^{2n}a_kX^k.\]The coefficients $a_{2k+1}$ all vanish since the polynomial is invariant under the change $X\mapsto -X.$ Prove that the coefficients $a_{2k}$ are all positive.

Let $\omega$ be a semicircle and $AB$ its diameter. $\omega_1$ and $\omega_2$ are two different circles, both tangent to $\omega$ and to $AB$, and $\omega_1$ is also tangent to $\omega_2$. Let $P,Q$ be the tangent points of $\omega_1$ and $\omega_2$ to $AB$ respectively, and $P$ is between $A$ and $Q$. Let $C$ be the tangent point of $\omega_1$ and $\omega$. Find $\tan\angle ACQ$.

The matrix \[A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}\] satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that \[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]

Given three cocentric circles $\omega_1$,$\omega_2$,$\omega_3$ with radius $r_1,r_2,r_3$ such that $r_1+r_3\geq {2r_2}$.Constrat a line that intersects $\omega_1$,$\omega_2$,$\omega_3$ at $A,B,C$ respectively such that $AB=BC$.

$\boxed{\text{A7}}$ Let $a,b,c$ be positive reals such that $abc=1$.Prove the inequality $\sum\frac{2a^2+\frac{1}{a}}{b+\frac{1}{a}+1}\geq 3$

Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is: \[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\] Find the least possible value of $n$.

$n\geq2$ and $E=\left \{ 1,2,...,n \right \}. A_1,A_2,...,A_k$ are subsets of $E$, such that for all $1\leq{i}<{j}\leq{k}$ Exactly one of $A_i\cap{A_j},A_i'\cap{A_j},A_i\cap{A_j'},A_i'\cap{A_j'}$ is empty set. What is the maximum possible $k$?

Given positive real numbers $a, b, c, d$ that satisfy equalities \[a^2 + d^2 - ad = b^2 + c^2 + bc \ \ \text{and} \ \ a^2 + b^2 = c^2 + d^2\] find all possible values of the expression $\frac{ab+cd}{ad+bc}.$

Let $n$ and $p$ be two integers with $p$ positive prime, such that $pn + 1$ is a perfect square. Show that $n + 1$ is the sum of $p$ perfect squares, not necessarily distinct.

Let $A_1,A_2,\ldots , A_n$ $(n\geq 3)$ be finite sets of positive integers. Prove, that \[ \displaystyle \frac{1}{n} \left( \sum_{i=1}^n |A_i|\right) + \frac{1}{{{n}\choose{3}}}\sum_{1\leq i < j < k \leq n} |A_i \cap A_j \cap A_k| \geq \frac{2}{{{n}\choose{2}}}\sum_{1\leq i < j \leq n}|A_i \cap A_j| \] holds, where $|E|$ is the cardinality of the set $E$

Prove that a binomial coefficient $\binom nk$ is odd if and only if all digits $1$ of $k$, when $k$ is written in binary, are on the same positions when $n$ is written in binary. [i]I. Dimovski[/i]

Is there a six-link broken line in space that passes through all the vertices of a given cube?

Let $n \ge 3$ be a positive integer. Suppose that $\Gamma$ is a unit circle passing through a point $A$. A regular $3$-gon, regular $4$-gon, \dots, regular $n$-gon are all inscribed inside $\Gamma$ such that $A$ is a common vertex of all these regular polygons. Let $Q$ be a point on $\Gamma$ such that $Q$ is a vertex of the regular $n$-gon, but $Q$ is not a vertex of any of the other regular polygons. Let $\mathcal{S}_n$ be the set of all such points $Q$. Find the number of integers $3 \le n \le 100$ such that \[ \prod_{Q \in \mathcal{S}_n} |AQ| \le 2. \]

Let $a_1,\ldots, a_n$ be real numbers. Define polynomials $f,g$ by $$f(x)=\sum_{k=1}^n a_kx^k,\ g(x)=\sum_{k=1}^n \frac{a_k}{2^k-1}x^k.$$ Assume that $g(2016)=0$. Prove that $f(x)$ has a root in $(0;2016)$.

Two infinite arithmetic sequences with positive integers are given:$$a_1<a_2<a_3<\cdots ; b_1<b_2<b_3<\cdots$$ It is known that there are infinitely many pairs of positive integers $(i,j)$ for which $i\leq j\leq i+2021$ and $a_i$ divides $b_j$. Prove that for every positive integer $i$ there exists a positive integer $j$ such that $a_i$ divides $b_j$.

Problem Shortlist BMO 2017 Let $ a $,$ b$,$ c$, be positive real numbers such that $abc= 1 $. Prove that $$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+b^{5}+b^{2}}\leq 1 . $$

Let $H$ be a regular hexagon with side length $1$. The sum of the areas of all triangles whose vertices are all vertices of $H$ can be expressed as $A\sqrt{B}$ for positive integers $A$ and $B$ such that $B$ is square-free. What is $1000A +B$?

For how many positive integers $n \le 500$ is $n!$ divisible by $2^{n-2}$? [i]Proposed by Eugene Chen[/i]

We are given $n (n \ge 5)$ circles in a plane. Suppose that every three of them have a common point. Prove that all $n$ circles have a common point.

[b]p1.[/b] James has $8$ Instagram accounts, $3$ Facebook accounts, $4$ QQ accounts, and $3$ YouTube accounts. If each Instagram account has $19$ pictures, each Facebook account has $5$ pictures and $9$ videos, each QQ account has a total of $17$ pictures, and each YouTube account has $13$ videos and no pictures, how many pictures in total does James have in all these accounts? [b]p2.[/b] If Poonam can trade $7$ shanks for $4$ shinks, and she can trade $10$ shinks for $17$ shenks. How many shenks can Poonam get if she traded all of her $105$ shanks? [b]p3.[/b] Jerry has a bag with $3$ red marbles, $5$ blue marbles and $2$ white marbles. If Jerry randomly picks two marbles from the bag without replacement, the probability that he gets two different colors can be expressed as a fraction $\frac{m}{n}$ in lowest terms. What is $m + n$? [b]p4.[/b] Bob's favorite number is between $1200$ and $4000$, divisible by $5$, has the same units and hundreds digits, and the same tens and thousands digits. If his favorite number is even and not divisible by $3$, what is his favorite number? [b]p5.[/b] Consider a unit cube $ABCDEFGH$. Let $O$ be the center of the face $EFGH$. The length of $BO$ can be expressed in the form $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are simplified to lowest terms. What is $a + b$? [b]p6.[/b] Mr. Eddie Wang is a crazy rich boss who owns a giant company in Singapore. Even though Mr. Wang appears friendly, he finds great joy in firing his employees. His immediately fires them when they say "hello" and/or "goodbye" to him. It is well known that $1/2$ of the total people say "hello" and/or "goodbye" to him everyday. If Mr. Wang had $2050$ employees at the end of yesterday, and he hires $2$ new employees at the beginning of each day, in how many days will Mr. Wang first only have $6$ employees left? [b]p7.[/b] In $\vartriangle ABC$, $AB = 5$, $AC = 6$. Let $D,E,F$ be the midpoints of $\overline{BC}$, $\overline{AC}$, $\overline{AB}$, respectively. Let $X$ be the foot of the altitude from $D$ to $\overline{EF}$. Let $\overline{AX}$ intersect $\overline{BC}$ at $Y$ . Given $DY = 1$, the length of $BC$ is $\frac{p}{q}$ for relatively prime positive integers $p, q$: Find $p + q$. [b]p8.[/b] Given $\frac{1}{2006} = \frac{1}{a} + \frac{1}{b}$ where $a$ is a $4$ digit positive integer and $b$ is a $6$ digit positive integer, find the smallest possible value of $b$. [b]p9.[/b] Pocky the postman has unlimited stamps worth $5$, $6$ and $7$ cents. However, his post office has two very odd requirements: On each envelope, an odd number of $7$ cent stamps must be used, and the total number of stamps used must also be odd. What is the largest amount of postage money Pocky cannot make with his stamps, in cents? [b]p10.[/b] Let $ABCDEF$ be a regular hexagon with side length $2$. Let $G$ be the midpoint of side $DE$. Now let $O$ be the intersection of $BG$ and $CF$. The radius of the circle inscribed in triangle $BOC$ can be expressed in the form $\frac{a\sqrt{b}-\sqrt{c}}{d} $ where $a$, $b$, $c$, $d$ are simplified to lowest terms. What is $a + b + c + d$? [b]p11.[/b] Estimation (Tiebreaker): What is the total number of characters in all of the participants' email addresses in the Accuracy Round? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a convex polygon $P$ some diagonals have been drawn, without intersections inside $P$. Show that there exist at least two vertices of $P$, neither one of them being an endpoint of any one of those diagonals.

If $P(x)=x^n+a_1x^{n-1}+...+a_{n-1}$ be a polynomial with real coefficients and $a_1^2<a_2$ then prove that not all roots of $P(x)$ are real.

Given a triangle $PQX$ in the plane, with $PQ = 3, PX = 2.6$ and $QX = 3.8$. Construct a right-angled triangle $ABC$ such that the incircle of $\vartriangle ABC$ touches $AB$ at $P$ and $BC$ at $Q$, and point $X$ lies on the line $AC$.

Let $n$ be a natural number. Solve in whole numbers the equation \[x^{n}+y^{n}=(x-y)^{n+1}.\]

$\odot A$, centered at point $A$, has radius $14$ and $\odot B$, centered at point $B$, has radius $15$. $AB = 13$. The circles intersect at points $C$ and $D$. Let $E$ be a point on $\odot A$, and $F$ be the point where line $EC$ intersects $\odot B$, again. Let the midpoints of $DE$ and $DF$ be $M$ and $N$, respectively. Lines $AM$ and $BN$ intersect at point $G$. If point $E$ is allowed to move freely on $\odot A$, what is the radius of the locus of $G$?