Construct triangle $ABC$, given $h_a$, $h_b$ (the altitudes from $A$ and $B$), and $m_a$, the median from vertex $A$.

Let $ABCD$ be a cyclic quadrilateral (In the same order) inscribed into the circle $\odot (O)$. Let $\overline{AC}$ $\cap$ $\overline{BD}$ $=$ $E$. A randome line $\ell$ through $E$ intersects $\overline{AB}$ at $P$ and $BC$ at $Q$. A circle $\omega$ touches $\ell$ at $E$ and passes through $D$. Given, $\omega$ $\cap$ $\odot (O)$ $=$ $R$. Prove, Points $B,Q,R,P$ are concyclic.

Two circles $K_{1}$ and $K_{2}$, centered at $O_{1}$ and $O_{2}$ with radii $1$ and $\sqrt{2}$ respectively, intersect at $A$ and $B$. Let $C$ be a point on $K_{2}$ such that the midpoint of $AC$ lies on $K_{1}$. Find the length of the segment $AC$ if $O_{1}O_{2}=2$

[b]p33.[/b] Lines $\ell_1$ and $\ell_2$ have slopes $m_1$ and $m_2$ such that $0 < m_2 < m_1$. $\ell'_1$ and $\ell'_2$ are the reflections of $\ell_1$ and $\ell_2$ about the line $\ell_3$ defined by $y = x$. Let $A = \ell_1 \cap \ell_2 = (5, 4)$, $B = \ell_1 \cap \ell_3$, $C = \ell'_1 \cap \ell'_2$ and $D = \ell_2 \cap \ell_3$. If $\frac{4-5m_1}{-5-4m_1} = m_2$ and $\frac{(1+m^2_1)(1+m^2_2)}{(1-m_1)^2(1-m_2)^2} = 41$, compute the area of quadrilateral $ABCD$. [b]p34.[/b] Suppose $S(m, n) = \sum^m_{i=1}(-1)^ii^n$. Compute the remainder when $S(2020, 4)$ is divided by $S(1010, 2)$. [b]p35.[/b] Let $N$ be the number of ways to place the numbers $1, 2, ..., 12$ on a circle such that every pair of adjacent numbers has greatest common divisor $1$. What is $N/144$? (Arrangements that can be rotated to yield each other are the same). [b]p36.[/b] Compute the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{{2n \choose 2}} =\frac{1}{{2 \choose 2}} - \frac{1}{{4 \choose 2}} +\frac{1}{{6 \choose 2}} -\frac{1}{{8 \choose 2}} -\frac{1}{{10 \choose 2}}+\frac{1}{{12 \choose 2}} +...$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$ABC$ is a triangle: $A=(0,0)$, $B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\triangle ABC$ can have? $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \frac{3}{2} \qquad\textbf{(D)}\ \frac{13}{2} \qquad\textbf{(E)}\ \text{there is no minimum} $

It is given a regular hexagon in the plane. Let P be a point of the plane. Define s(P) as the sum of the distances from P to each side of the hexagon, and v(P) as the sum of the distances from P to each vertex. a) Find the locus of points P that minimize s(P) b) Find the locus of points P that minimize v(P)

Using only angle with angle $\frac{\pi}{7}$ and a ruler, constuct angle $\frac{\pi}{14}$

Let $ABC$ be an acute triangle with $| AB | > | AC |$. Let $D$ be the foot of the altitude from $A$ to $BC$, let $K$ be the intersection of $AD$ with the internal bisector of angle $B$, Let $M$ be the foot of the perpendicular from $B$ to $CK$ (it could be in the extension of segment $CK$) and$ N$ the intersection of $BM$ and $AK$ (it could be in the extension of the segments). Let $T$ be the intersection of$ AC$ with the line that passes through $N$ and parallel to $DM$. Prove that $BM$ is the internal bisector of the angle $\angle TBC$

Let $O$ and $H$ be the circumcenter and orthocenter, respectively, of an acute scalene triangle $ABC$. The perpendicular bisector of $\overline{AH}$ intersects $\overline{AB}$ and $\overline{AC}$ at $X_A$ and $Y_A$ respectively. Let $K_A$ denote the intersection of the circumcircles of triangles $OX_AY_A$ and $BOC$ other than $O$. Define $K_B$ and $K_C$ analogously by repeating this construction two more times. Prove that $K_A$, $K_B$, $K_C$, and $O$ are concyclic. [i]Hongzhou Lin[/i]

[b]p1.[/b] Compute the sum of sharp angles at all five nodes of the star below. ( [url=http://www.math.msu.edu/~mshapiro/NewOlympiad/Olymp2009/10_12_2009.pdf]figure missing[/url] ) [b]p2.[/b] Arrange the integers from $1$ to $15$ in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways this can be done? [b]p3.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 -q^2$ is divisible by $ 24$. [b]p4.[/b] A city in a country is called Large Northern if comparing to any other city of the country it is either larger or farther to the North (or both). Similarly, a city is called Small Southern. We know that in the country all cities are Large Northern city. Show that all the cities in this country are simultaneously Small Southern. [b]p5.[/b] You have four tall and thin glasses of cylindrical form. Place on the flat table these four glasses in such a way that all distances between any pair of centers of the glasses' bottoms are equal. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Triangle $ABC$ has $BC=20$. The incircle of the triangle evenly trisects the median $AD$. If the area of the triangle is $m \sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n$.

Let $ABC$ be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid $G$ and the circumcentre $O$ of $ABC$ in its sides $BC,CA,AB$ are denoted by $G_1,G_2,G_3$ and $O_1,O_2,O_3$, respectively. Show that the circumcircles of triangles $G_1G_2C$, $G_1G_3B$, $G_2G_3A$, $O_1O_2C$, $O_1O_3B$, $O_2O_3A$ and $ABC$ have a common point. [i]The centroid of a triangle is the intersection point of the three medians. A median is a line connecting a vertex of the triangle to the midpoint of the opposite side.[/i]

For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$. (We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.) [i](United Kingdom) Luke Betts[/i]

Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given. a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$; b) Calculate the distance of $p$ from either base; c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.

In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$. [b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$. [b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.

Squares of side lengths $1,$ $2,$ $3,$ and $4,$ are placed on a line segment $\ell$ from left to right, respectively, and these squares lie on the same side of $\ell,$ forming a polygon $P.$ An equilateral triangle whose base is $\ell$ is drawn around the squares such that its other two sides intersect $P$ at its leftmost and rightmost vertices (that are not on $\ell$). The area of the triangle can be written in the form $\tfrac{a + b\sqrt{3}}{c},$ where $a,$ $b,$ and $c$ are positive integers, and $b$ and $c$ are relatively prime. Find $a + b + c.$

There are three squares in the picture. Find the sum of angles $ADC$ and $BDC$. [img]https://cdn.artofproblemsolving.com/attachments/c/9/885a6c6253fca17e24528f8ba8a5d31a18c845.png[/img]

Let $C$ be the circle of radius $12$ centered at $(0, 0)$. What is the length of the shortest path in the plane between $(8\sqrt3, 0)$ and $(0, 12 \sqrt2)$ that does not pass through the interior of $C$?

Andy has 2010 square tiles, each of which has a side length of one unit. He plans to arrange the tiles in an m x n rectangle, where mn = 2010. Compute the sum of the perimeters of all of the different possible rectangles he can make. Two rectangles are considered to be the same if one can be rotated to become the other, so, for instance, a 1 x 2010 rectangle is considered to be the same as a 2010 x 1 rectangle.

Consider triangle $\vartriangle ABC$ with circumradius $R = 10$, inradius $r = 2$ and semi-perimeter $S = 18$. Let $I$ be the incenter, and we extend $AI$, $BI$ and $CI$ to intersect the circumcircle at $D, E$ and $F$ respectively. Compute the area of $\vartriangle DEF$.

Triangle $ ABC$ has $ \angle C \equal{} 60^{\circ}$ and $ BC \equal{} 4$. Point $ D$ is the midpoint of $ BC$. What is the largest possible value of $ \tan{\angle BAD}$? $ \textbf{(A)} \ \frac {\sqrt {3}}{6} \qquad \textbf{(B)} \ \frac {\sqrt {3}}{3} \qquad \textbf{(C)} \ \frac {\sqrt {3}}{2\sqrt {2}} \qquad \textbf{(D)} \ \frac {\sqrt {3}}{4\sqrt {2} \minus{} 3} \qquad \textbf{(E)}\ 1$

Symedians of an acute-angled non-isosceles triangle $ABC$ intersect at a point at point $L$, and $AA_1$, $BB_1$ and $CC_1$ are its altitudes. Prove that you can construct equilateral triangles $A_1B_1C'$, $B_1C_1A'$ and $C_1A_1B'$ not lying in the plane $ABC$, so that lines $AA' , BB'$ and $CC'$ and also perpendicular to the plane $ABC$ at point $L$ intersected at one point.

[b]p16.[/b] Let $P(x)$ be a quadratic such that $P(-2) = 10$, $P(0) = 5$, $P(3) = 0$. Then, find the sum of the coefficients of the polynomial equal to $P(x)P(-x)$. [b]p17.[/b] Suppose that $a < b < c < d$ are positive integers such that the pairwise differences of $a, b, c, d$ are all distinct, and $a + b + c + d$ is divisible by $2023$. Find the least possible value of $d$. [b]p18.[/b] Consider a right rectangular prism with bases $ABCD$ and $A'B'C'D'$ and other edges $AA'$, $BB'$, $CC'$ and $DD'$. Suppose $AB = 1$, $AD = 2$, and $AA' = 1$. $\bullet$ Let $X$ be the plane passing through $A$, $C'$, and the midpoint of $BB'$. $\bullet$ Let $Y$ be the plane passing through $D$, $B'$, and the midpoint of $CC'$. Then the intersection of $X$, $Y$ , and the prism is a line segment of length $\ell$. Find $\ell$. PS. You should use hide for answers.

Given is a pyramid $SA_1A_2A_3\ldots A_n$ whose base is convex polygon $A_1A_2A_3\ldots A_n$. For every $i=1,2,3,\ldots ,n$ there is a triangle $X_iA_iA_{i+1} $ congruent to triangle $SA_iA_{i+1}$ that lies on the same side from $A_iA_{i+1}$ as the base of that pyramid. (You can assume $a_1$ is the same as $a_{n+1}$.) Prove that these triangles together cover the entire base.