Let $ABC$ be scalene and acute triangle with $CA>CB$ and let $P$ be an internal point, satisfying $\angle APB=180^{\circ}-\angle ACB$; the lines $AP, BP$ meet $BC, CA$ at $A_1, B_1$. If $M$ is the midpoint of $A_1B_1$ and $(A_1B_1C)$ meets $(ABC)$ at $Q$, show that $\angle PQM=\angle BQA_1$.

[b]p1.[/b] The length of the side $AB$ of the trapezoid with bases $AD$ and $BC$ is equal to the sum of lengths $|AD|+|BC|$. Prove that bisectors of angles $A$ and $B$ do intersect at a point of the side $CD$. [b]p2.[/b] Polynomials $P(x) = x^4 + ax^3 + bx^2 + cx + 1$ and $Q(x) = x^4 + cx^3 + bx^2 + ax + 1$ have two common roots. Find these common roots of both polynomials. [b]p3.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace: $\bullet$ two candies in the box with one chocolate bar, $\bullet$ two muffins in the box with one chocolate bar, $\bullet$ two chocolate bars in the box with one candy and one muffin, $\bullet$ one candy and one chocolate bar in the box with one muffin, $\bullet$ one muffin and one chocolate bar in the box with one candy. Is it possible that after some time it remains only one object in the box? [b]p4.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points? [b]p5.[/b] It is known that $x$ is a real number such that $x+\frac{1}{x}$ is an integer. Prove that $x^n+\frac{1}{x^n}$ is an integer for any positive integer $n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P$ be a point inside a convex quadrilateral $ABCD$. Points $K,L,M,N$ are given on the sides $AB,BC,CD,DA$ respectively such that $PKBL$ and $PMDN$ are parallelograms. Let $S,S_1$, and $S_2$ be the areas of $ABCD, PNAK$, and $PLCM$. Prove that $\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}$.

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let $\ell_a, \ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $\ell_a, \ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$. [i]Proposed by Japan[/i]

In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$, and that $\angle AXO = \angle OXA'$. [i]Proposed by Sammy Luo[/i]

Let the triangle $ABC$ be with $| AB | > | AC |$. Point M is the midpoint of the side $[BC]$, and point $I$ is the center of the circle inscribed in the triangle ABC such that the relation $| AI | = | MI |$. Prove that points $A, B, M, I$ are located on the same circle.

Given a triangle $ABC$. On its sides $BC$ , $CA$ and $AB$ , the points $A_1$ , $B_1$ and $C_1$ are taken, respectively , such that $2 \angle B_1 A_1 C_1 + \angle BAC = 180^o$ , $2 \angle A_1 C_1 B_1 + \angle ACB = 180^o$ , $2 \angle C_1 B_1 A_1 + \angle CBA = 180^o$ . Find the locus of the centers of the circles circumscribed about the triangles $A_1 B_1 C_1$ (all possible such triangles are considered).

A regular tetrahedron of volume $1$ is filled with water of total volume $\frac{7}{16}$. Is it possible that the center of the tetrahedron lies on the surface of the water? How about in a cube of volume $1$?

Show that for an arbitrary tetrahedron there are two planes such that the ratio of the areas of the projections of the tetrahedron onto the two planes is not less than $\sqrt2$.

Phillip is trying to make a two-dimensional donut, but in a fun way: He is trying to make a donut shaped in a way that the inner circle of the donut is inscribed inside a pentagon, and the outer circle of the donut circumscribes the same pentagon. This pentagon has a side length of $6$. The area of Phillip's donut is of the form $a \pi$. Find $a$. (Note that $\sin 54^o= \frac{\sqrt5+1}{4}$ )

Inside of convex quadrilateral $ABCD$ found a point $M$ such that $\angle AMB=\angle ADM+\angle BCM$ and $\angle AMD=\angle ABM+\angle DCM$.Prove that $$AM\cdot CM+BM\cdot DM\ge \sqrt{AB\cdot BC\cdot CD\cdot DA}.$$

A circle and an ellipse with foci $F_{1}$, $F_{2}$ lying inside it are given. Construct a chord $AB$ of the circle touching the ellipse and such that $AF_{1}F_{2}B$ is a cyclic quadrilateral. Proposed by: A.Zaslavsky

The points $D,E$ and $F$ are chosen on the sides $BC,AC$ and $AB$ of triangle $ABC$, respectively. Prove that triangles $ABC$ and $DEF$ have the same centroid if and only if \[\frac{BD}{DC} = \frac{CE}{EA}=\frac{AF}{FB}\]

On the sides of the triangle $ABC$ externally constructed right triangles $ABC_1$, $BCA_1$, $CAB_1$. Prove that the points of intersection of the medians of the triangles $ABC$ and $A_1B_1C_1$ coincide.

Let $BE$ and $CF$ be the medians of an acute triangle $ABC.$ On the line $BC,$ points $K \ne B$ and $L \ne C$ are chosen such that $BE = EK$ and $CF = FL.$ Prove that $AK = AL.$ [i]Proposed by Heorhii Zhilinskyi[/i]

Two of the squares of a $ 7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?

A point $ N $ is marked on the median $ CM $ of the triangle $ ABC $ so that $ MN \cdot MC = AB ^ 2/4 $. Lines $ AN $ and $ BN $ intersect the circumcircle $ \triangle ABC $ for the second time at points $ P $ and $ Q $, respectively. $ R $ is the point of segment $ PQ $, nearest to $ Q $, such that $ \angle NRC = \angle BNC $. $ S $ is the point of the segment $ PQ $ closest to $ P $ such that $ \angle NSC = \angle ANC $. Prove that $ RN = SN $.

Let $n$ be a positive integer. There are $\tfrac{n(n+1)}{2}$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each mark has the black side up. An [i]operation[/i] is to choose a line parallel to the sides of the triangle, and flipping all the marks on that line. A configuration is called [i]admissible [/i] if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration $C$, let $f(C)$ denote the smallest number of operations required to obtain $C$ from the initial configuration. Find the maximum value of $f(C)$, where $C$ varies over all admissible configurations.

$ABCDA'B'C'D'$ is a cube (with $ABCD$ and $A'B'C'D'$ faces, and $AA', BB', CC', DD'$ edges). $L$ is a line which intersects or is parallel to the lines $AA', BC$ and $DB'$. $L$ meets the line $BC$ at $M$ (which may be the point at infinity). Let $m = |BM|$. The plane $MAA'$ meets the line $B'C'$ at $E$. Show that $|B'E| = m$. The plane $MDB'$ meets the line $A'D'$ at $F$. Show that $|D'F| = m$. Hence or otherwise show how to construct the point $P$ at the intersection of $L$ and the plane $A'B'C'D'$. Find the distance between $P$ and the line $A'B'$ and the distance between $P$ and the line $A'D'$ in terms of $m$. Find a relation between these two distances that does not depend on $m$. Find the locus of $M$. Let $S$ be the envelope of the line $L$ as $M$ varies. Find the intersection of $S$ with the faces of the cube.

Let $ABCD$ be a Rhombus and let $w$ be it's incircle. Let $M$ be the midpoint of $AB$ the point $K$ is on $w$ and inside $ABCD$ such that $MK$ is tangent to $w$. Prove that $CDKM$ is cyclic.

Let $E$ be the ellipse lying in the $x, y$ plane centered at $(0, 0)$ with semi-major axis of length $2$ along the $x$-axis and semi-minor axis of length $1$ along the $y$-axis. Let $C$ be a cone created by revolving two perpendicular lines about an angle bisector of the perpendicular angle. There are some points $(x, y, z)$ where the vertex of $C$ could be so that $E$ is the intersection of $C$ with the $x, y$ plane. These points define a convex polygon in the $x, z$ plane. The area of this polygon can be expressed as $\sqrt{n}$ for some positive integer $n.$ Find $n.$ (Some definitions: the semi-major axis is the longest distance from the center of the ellipse to the boundary, and the semi-minor axis is the shortest distance from the center of the ellipse to the boundary.)

A convex hexagon $ABCDEF$ is given, with each two opposite sides of different lengths and parallel ($AB \parallel DE$, $BC \parallel EF$ and $CD \parallel FA$). If $|AE| = |BD|$ and $|BF| = |CE|$, prove that the hexagon $ABCDEF$ is cyclic.

Let $ \omega$ be a circle with center O and radius 10, and let H be a point such that $OH = 6$. A point P is called snug if, for all triangles ABC with circumcircle ω and orthocenter $H$, we have that P lies on $\triangle$ABC or in the interior of $\triangle$ABC. Find the area of the region consisting of all snug points.

A point $P$ lies inside $\usepackage{gensymb} \angle ABC(<90 \degree)$. Show that there exists a point $Q$ inside $\angle ABC$ satisfying the following condition: [center]For any two points $X$ and $Y$ on the rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$ respectively satisfying $\angle XPY = \angle ABC$, it holds that $\usepackage{gensymb} \angle XQY = 180 \degree - 2 \angle ABC.$[/center]

Triangle $ABC$ has an incenter $I$ l its incircle touches the side $BC$ at $T$. The line through $T$ parallel to $IA$ meets the incircle again at $S$ and the tangent to the incircle at $S$ meets $AB , AC$ at points $C' , B'$ respectively. Prove that triangle $AB'C'$ is similar to triangle $ABC$.