Let $ABC$ be an acute triangle. Squares $AA_1A_2A_3$, $BB_1B_2B_3$ and $CC_1C_2C_3$ are located such that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ pass through the points $B$, $C$ and $A$ respectively and the lines $A_2A_3$, $B_2B_3$, $C_2C_3$ pass through the points $C$, $A$ and $B$ respectively. Prove that (a) the lines $AA_2$, $B_1B_2$ and $C_1C_3$ intersect at one point. (b) the lines $AA_2$, $BB_2$ and $CC_2$ intersect at one point. (Mykhailo Plotnikov) [img]https://cdn.artofproblemsolving.com/attachments/3/d/ad2fe12ae2c82d04b48f5e683b7d54e0764baf.png[/img]

Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that \[ [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}].\] Prove that there exists a point $ M$ in the plane of the pentagon such that \[ [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}].\] Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.

Two circles $(O)$ and $(O')$ intersect at $A$ and $B$. Take two points $P,Q$ on $(O)$ and $(O')$, respectively, such that $AP=AQ$. The line $PQ$ intersects $(O)$ and $(O')$ respectively at $M,N$. Let $E,F$ respectively be the centers of the two arcs $BP$ and $BQ$ (which don't contains $A$). Prove that $MNEF$ is a cyclic quadrilateral.

Let $ABCDEF$ be a regular hexagon and $P$ be a point on the shorter arc $EF$ of its circumcircle. Prove that the value of $$\frac{AP+BP+CP+DP}{EP+FP}$$is constant and find its value.

Let $a, b, c$ be the side lengths and $P$ be area of a triangle, respectively. Prove that \[(a^2+b^2+c^2-4\sqrt 3 P) (a^2+b^2+c^2) \geq 2 \left(a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2\right).\]

In a quadrilateral $ABCD$ we have $\angle A = \angle C = 90^o$. Let $E$ be a point in the interior of $ABCD$. Let $M$ be the midpoint of $BE$. Prove that $\angle ADB = \angle EDC$ if and only if $|MA| = |MC|$.

Two unit circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. $M$ is an arbitrary point of $\omega_1$, $N$ is an arbitrary point of $\omega_2$. Two unit circles $\omega_3$ and $\omega_4$ pass through both points $M$ and $N$. Let $C$ be the second common point of $\omega_1$ and $\omega_3$, and $D$ be the second common point of $\omega_2$ and $\omega_4$. Prove that $ACBD$ is a parallelogram.

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Bisectors of $AD$ and $BC$ intersect line segments $BC$ and $AD$ respectively in points $P$ and $Q$. Show that $\angle APD = \angle BQC$.

Points $B,C$ vary on two fixed rays emanating from point $A$ such that $AB+AC$ is constant. Show that there is a point $D$, other than $A$, such that the circumcircle of triangle $ABC$ passes through $D$ for all possible choices of $B, C$.

Prove that the product of the sines of two opposite dihedrals in a tetrahedron is proportional to the product of the lengths of the edges of these dihedrals.

Let $BB_1$ and $CC_1$ be altitudes of triangle $ABC$. The tangents to the circumcircle of $AB_1C_1$ at $B_1$ and $C_1$ meet AB and $AC$ at points $M$ and $N$ respectively. Prove that the common point of circles $AMN$ and $AB_1C_1$ distinct from $A$ lies on the Euler line of $ABC$.

Determine for what values of $x$ the expressions $2x + 2$,$x + 4$, $x + 2$ can represent the sidelengths of a right triangle.

In the triangle $\vartriangle ABC$ we have $| AB |^3 = | AC |^3 + | BC |^3$. Prove that $\angle C> 60^o$ .

Let $ I $ be the incenter of triangle $ ABC $. The incircle touches $ BC, CA, AB$ at points $ P, Q, R $. A circle passing through $ B , C $ is tangent to the circle $I$ at point $ X $, a circle passing through $ C , A $ is tangent to the circle $I$ at point $ Y $, and a circle passing through $ A , B $ is tangent to the circle $I$ at point $ Z $, respectively. Prove that three lines $ PX, QY, RZ $ are concurrent.

Let $ ABCD$ be a convex quadrilateral. Let $ M,N$ be the midpoints of $ AB,AD$ respectively. The foot of perpendicular from $ M$ to $ CD$ is $ K$, the foot of perpendicular from $ N$ to $ BC$ is $ L$. Show that if $ AC,BD,MK,NL$ are concurrent, then $ KLMN$ is a cyclic quadrilateral.

$ABC$ is a triangle with $\angle A = 90^\circ$, $\angle B = 60^\circ$. The points $A_1$, $B_1$, $C_1$ on $BC$, $CA$, $AB$ respectively are such that $A_1B_1C_1$ is equilateral and the perpendiculars (to $BC$ at $A_1$, to $CA$ at $B_1$ and to $AB$ at $C_1$) meet at a point $P$ inside the triangle. Find the ratios $PA_1:PB_1:PC_1$.

Three angle bisectors were drawn in a triangle, and it turned out that the angles between them are $50^o$, $60^o$ and $70^o$. Find the angles of the original triangle.

Let $M$be an interior point of triangle $ABC$. Let the line $AM$ intersect the circumcircle of the triangle $MBC$ for the second time at point $D$, the line $BM$ intersect the circumcircle of the triangle $MCA$ for the second time at point $E$, and the line $CM$ intersect the circumcircle of the triangle $MAB$ for the second time at point $F$. Prove that $\frac{AD}{MD} + \frac{BE}{ME} + \frac{CF}{MF} \geq \frac{9}{2}$. [i]Proposed by Nairy Sedrakyan[/i]

[b]p1.[/b] How many perfect squares are in the set: $\{1, 2, 4, 9, 10, 16, 17, 25, 36, 49\}$? [b]p2.[/b] If $a \spadesuit b = a^b - ab - 5$, what is the value of $2 \spadesuit 11$? [b]p3.[/b] Joe can catch $20$ fish in $5$ hours. Jill can catch $35$ fish in $7$ hours. If they work together, and the number of days it takes them to catch $900$ fish is represented by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, what is $m + n$? Assume that they work at a constant rate without taking breaks and that there are an infinite number of fish to catch. [b]p4.[/b] What is the units digit of $187^{10}$? [b]p5.[/b] What is the largest number of regions we can create by drawing $4$ lines in a plane? [b]p6.[/b] A regular hexagon is inscribed in a circle. If the area of the circle is $2025\pi$, given that the area of the hexagon can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, $c$ where $gcd(a, c) = 1$ and $b$ is not divisible by the square of any number other than $1$, find $a + b + c$. [b]p7.[/b] Find the number of trailing zeroes in the product $3! \cdot 5! \cdot 719!$. [b]p8.[/b] How many ordered triples $(x, y, z)$ of odd positive integers satisfy $x + y + z = 37$? [b]p9.[/b] Let $N$ be a number with $2021$ digits that has a remainder of $1$ when divided by $9$. $S(N)$ is the sum of the digits of $N$. What is the value of $S(S(S(S(N))))$? [b]p10.[/b] Ayana rolls a standard die $10$ times. If the probability that the sum of the $10$ die is divisible by $6$ is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$, what is $m + n$? [b]p11.[/b] In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. The inscribed circle touches the side $BC$ at point $D$. The line $AI$ intersects side $BC$ at point $K$ given that $I$ is the incenter of triangle $ABC$. What is the area of the triangle $KID$? [b]p12.[/b] Given the cubic equation $2x^3+8x^2-42x-188$, with roots $a, b, c$, evaluate $|a^2b+a^2c+ab^2+b^2c+c^2a+bc^2|$. [b]p13.[/b] In tetrahedron $ABCD$, $AB=6$, $BC=8$, $CA=10$, and $DA$, $DB$, $DC=20$. If the volume of $ABCD$ is $a\sqrt{b}$ where $a$, $b$ are positive integers and in simplified radical form, what is $a + b$? [b]p14.[/b] A $2021$-digit number starts with the four digits $2021$ and the rest of the digits are randomly chosen from the set $0$,$1$,$2$,$3$,$4$,$5$,$6$. If the probability that the number is divisible by $14$ is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. what is $m + n$? [b]p15.[/b] Let $ABCD$ be a cyclic quadrilateral with circumcenter $O_1$ and circumradius $20$, Let the intersection of $AC$ and $BD$ be $E$. Let the circumcenter of $\vartriangle EDC$ be $O_2$. Given that the circumradius of 4EDC is $13$; $O_1O_2 = 11$, $BE = 11 \sqrt2$, find $O_1E^2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a positive real number $a$, let $C$ be the cube with vertices at $(\pm a, \pm a, \pm a)$ and let $T$ be the tetrahedron with vertices at $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$. If the intersection of $T$ and $C$ has volume $ka^3$ for some $k$, find $k$.

The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.

Let be three complex numbers $ z,t,u, $ whose affixes in the complex plane form a triangle $ \triangle . $ [b]a)[/b] Let be three non-complex numbers $ a,b,c $ that sum up to $ 0. $ Prove that $$ |az+bt+cu|=|at+bu+cz|=|au+bz+ct| $$ if $ \triangle $ is equilateral. [b]b)[/b] Show that $ \triangle $ is equilateral if $$ |z+2t-3u|=|t+2u-3z|=|u+2z-3t| . $$

On a $55\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out. [asy]size(2.013cm); draw ((0,0)--(0,1)); draw ((0,0)--(1,0)); draw ((0,1)--(.5,1)); draw ((.5,1)--(.5,0)); draw ((0,.5)--(1,.5)); draw ((1,.5)--(1,0)); draw ((1,.5)--(1,0)); [/asy]

Given a triangle $ABC$ with the are $1$. Let $A',B'$ and $C' $ are the midpoints of the sides $[BC], [CA]$ and $[AB]$ respectively. What is the minimal possible area of the common part of two triangles $A'B'C'$ and $KLM$, if the points $K,L$ and $M$ are lying on the segments $[AB'], [CA']$ and $[BC']$ respectively?

$(BEL 2) (a)$ Find the equations of regular hyperbolas passing through the points $A(\alpha, 0), B(\beta, 0),$ and $C(0, \gamma).$ $(b)$ Prove that all such hyperbolas pass through the orthocenter $H$ of the triangle $ABC.$ $(c)$ Find the locus of the centers of these hyperbolas. $(d)$ Check whether this locus coincides with the nine-point circle of the triangle $ABC.$