In triangle $ABC$, $\angle B=90^\circ$, $AB>BC$, and $P$ is the point such that $BP=BC$ and $\angle APB=90^\circ$, where $P$ and $C$ lie on the same side of $AB$. Let $Q$ be the point on $AB$ such that $AP=AQ$, and let $M$ be the midpoint of $QC$. Prove that the line through $M$ parallel to $AP$ passes through the midpoint of $AB$.

Triangle $ABC$ has side lengths $13$, $14$, and $15$. Let $E$ be the ellipse that encloses the smallest area which passes through $A, B$, and $C$. The area of $E$ is of the form $\frac{a \sqrt{b}\pi}{c}$ , where $a$ and $c$ are coprime and $b$ has no square factors. Find $a + b + c$.

We call a permutation $ \left(a_1, a_2, ..., a_n\right)$ of $ \left(1, 2, ..., n\right)$ [i]quadratic[/i] if there exists at least a perfect square among the numbers $ a_1$, $ a_1 \plus{} a_2$, $ ...$, $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n$. Find all natural numbers $ n$ such that all permutations in $ S_n$ are quadratic. [i]Remark.[/i] $ S_{n}$ denotes the $ n$-th symmetric group, the group of permutations on $ n$ elements.

Let $a_1, a_2, \dots, a_n$ be real numbers, and let $b_1, b_2, \dots, b_n$ be distinct positive integers. Suppose that there is a polynomial $f(x)$ satisfying the identity \[ (1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}. \] Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \dots, b_n$ and $n$ (but independent of $a_1, a_2, \dots, a_n$).

Let $ABC$ be an isosceles triangle with $\angle B = 120^o$ . Points $P$ and $Q$ are chosen on the prolongations of segments $AB$ and $CB$ beyond point $B$ so that the rays $AQ$ and $CP$ intersect and are perpendicular to each other. Prove that $\angle PQB = 2\angle PCQ$. (A.Akopyan, D.Shvetsov)

In a convex quadrilateral $ABCD, AC\perp BD, \angle BCA = 10^o,\angle BDA = 20^o, \angle BAC = 40^o$. Find $\angle BDC$.

Given is a triangle $ABC$. The points $P, Q$ lie on the extensions of $BC$ beyond $B, C$, respectively, such that $BP=BA$ and $CQ=CA$. Prove that the circumcenter of triangle $APQ$ lies on the angle bisector of $\angle BAC$.

A convex hexagon $ ABCDEF$ satisfies $ AB\equal{}BC, CD\equal{}DE, EF\equal{}FA$ and: $ \angle ABC\plus{}\angle CDE\plus{}\angle EFA \equal{} 360^{\circ}$. Prove that the perpendiculars from $ A,C$ and $ E$ to $ FB,BD$ and $ DF$ respectively are concurrent.

Consider the square $ABCD$ in which a segment is drawn between each vertex and the midpoints of both opposite sides. Find the ratio of the area of the octagon determined by these segments and the area of the square $ABCD.$

Define $\eta(f)$ to be the number of roots that are repeated of the complex-valued polynomial $f$ (e.g. $\eta((x-1)^3\cdot(x+1)^4)=5$). Prove that for nonconstant, relatively prime $f,g\in\mathbb C[x]$, $$\eta(f)+\eta(g)+\eta(f+g)<\deg f+\deg g$$

Find all $4$ digit number $\overline{abcd}$ such that $4\cdot \overline{abcd}+30=\overline{dcba}$

Construct a point $N$ inside a given right triangle $ABC$ such that the angles $\angle NBC$, $\angle NCA$ and $\angle NAB$ are equal.

Define a function $g: \mathbb{N} \mapsto \mathbb{N}$ by the following rule: (a) $g$ is nondecrasing (b) for each $n$, $g(n)$ i sthe number of times $n$ appears in the range of $g$, Prove that $g(1) = 1$ and $g(n+1) = 1 + g( n +1 - g(g(n)))$ for all $n \in \mathbb{N}$

A positive integer is called [i]sabroso [/i]if when it is added to the number obtained when its digits are interchanged from one side of its written form to the other, the result is a perfect square. For example, $143$ is sabroso, since $143 + 341 =484 = 22^2$. Find all two-digit sabroso numbers.

Let $I$ be the incenter of triangle $ABC$, $O_1$ a circle through $B$ tangent to $CI$, and $O_2$ a circle through $C$ tangent to $BI$. Prove that $O_1$,$O_2$ and the circumcircle of $ABC$ have a common point.

The rails on a railroad are $ 30$ feet long. As the train passes over the point where the rails are joined, there is an audible click. The speed of the train in miles per hour is approximately the number of clicks heard in: $ \textbf{(A)}\ 20\text{ seconds} \qquad\textbf{(B)}\ 2\text{ minutes} \qquad\textbf{(C)}\ 1\frac{1}{2}\text{ minutes} \qquad\textbf{(D)}\ 5\text{ minutes}\\ \textbf{(E)}\ \text{none of these}$

Petya and Vasya are playing a game on an initially empty \(100 \times 100\) grid, taking turns. Petya goes first. On his turn, a player writes an uppercase Russian letter in an empty cell (each cell can contain only one letter). When all cells are filled, Petya is declared the winner if there are four consecutive cells horizontally spelling the word ``ПЕТЯ'' (PETYA) from left to right, or four consecutive cells vertically spelling ``ПЕТЯ'' from top to bottom. Can Petya guarantee a win regardless of Vasya's moves?

It is given that the roots of the polynomial $P(z) = z^{2019} - 1$ can be written in the form $z_k = x_k + iy_k$ for $1\leq k\leq 2019$. Let $Q$ denote the monic polynomial with roots equal to $2x_k + iy_k$ for $1\leq k\leq 2019$. Compute $Q(-2)$.

$ABCD$ is a quadrilateral with $\angle A = 90o$, $AD = a$, $BC = b$, $AB = h$, and area $\frac{(a+b)h}{2}$. What can we say about $\angle B$?

It is known that $ \angle BAC$ is the smallest angle in the triangle $ ABC$. The points $ B$ and $ C$ divide the circumcircle of the triangle into two arcs. Let $ U$ be an interior point of the arc between $ B$ and $ C$ which does not contain $ A$. The perpendicular bisectors of $ AB$ and $ AC$ meet the line $ AU$ at $ V$ and $ W$, respectively. The lines $ BV$ and $ CW$ meet at $ T$. Show that $ AU \equal{} TB \plus{} TC$. [i]Alternative formulation:[/i] Four different points $ A,B,C,D$ are chosen on a circle $ \Gamma$ such that the triangle $ BCD$ is not right-angled. Prove that: (a) The perpendicular bisectors of $ AB$ and $ AC$ meet the line $ AD$ at certain points $ W$ and $ V,$ respectively, and that the lines $ CV$ and $ BW$ meet at a certain point $ T.$ (b) The length of one of the line segments $ AD, BT,$ and $ CT$ is the sum of the lengths of the other two.

[b]p1.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$). [b]p2.[/b] A $5\times 6$ rectangle is drawn on the piece of graph paper (see the figure below). The side of each square on the graph paper is $1$ cm long. Cut the rectangle along the sides of the graph squares in two parts whose areas are equal but perimeters are different by $2$ cm. $\begin{tabular}{|l|l|l|l|l|l|} \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline \end{tabular}$ [b]p3.[/b] Three runners started simultaneously on a $1$ km long track. Each of them runs the whole distance at a constant speed. Runner $A$ is the fastest. When he runs $400$ meters then the total distance run by runners $B$ and $C$ together is $680$ meters. What is the total combined distance remaining for runners $B$ and $C$ when runner $A$ has $100$ meters left? [b]p4.[/b] There are three people in a room. Each person is either a knight who always tells the truth or a liar who always tells lies. The first person said «We are all liars». The second replied «Only you are a liar». Is the third person a liar or a knight? [b]p5.[/b] A $5\times 8$ rectangle is divided into forty $1\times 1$ square boxes (see the figure below). Choose 24 such boxes and one diagonal in each chosen box so that these diagonals don't have common points. $\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A four-digit number $YEAR$ is called [i]very good[/i] if the system \begin{align*} Yx+Ey+Az+Rw& =Y\\ Rx+Yy+Ez+Aw & = E\\\ Ax+Ry+Yz+Ew & = A\\ Ex+Ay+Rz+Yw &= R \end{align*} of linear equations in the variables $x,y,z$ and $w$ has at least two solutions. Find all very good $YEAR$s in the 21st century. (The $21$st century starts in $2001$ and ends in $2100$.) [i]Proposed by Tomáš Bárta, Charles University, Prague[/i]

Let $ ABC $ be a triangle, let $ O $ and $ \gamma $ be its circumcentre and circumcircle, respectively, and let $ P $ and $ Q $ be distinct points in the interior of $ \gamma $ such that $ O, P $ and $ Q $ are not collinear. Reflect $ O $ in the midpoint of the segment $ PQ $ to obtain $R,$ then reflect $R$ in the centre of the nine-point circle of the triangle $ABC$ to obtain $S.$ The circle through $P$ and $Q$ and orthogonal to $ \gamma , $ crosses the rays $OP$ and $OQ,$ emanating from $O,$ again at $P'$ and $Q'$ respectively. Let the lines $PQ'$ and $QP'$ cross at $T.$ Prove that, if $P$ and $Q$ are isogonally conjugate with respect to the triangle $ABC,$ then so are $S$ and $T.$ [i]E.D. Camier[/i]

Evaluate $\lim_{n\to\infty}\left[\left(\prod_{k=1}^{n}\frac{2k}{2k-1}\right)\int_{-1}^{\infty}\frac{(\cos x)^{2n}}{2^x} \, dx\right]$.

For $a_1 = 3$, define the sequence $a_1, a_2, a_3, \ldots$ for $n \geq 1$ as $$na_{n+1}=2(n+1)a_n-n-2.$$ Prove that for any odd prime $p$, there exist positive integer $m,$ such that $p|a_m$ and $p|a_{m+1}.$