Let $ABC$ be a triangle. Let $D,E$ be the points outside of the triangle so that $AD=AB,AC=AE$ and $\angle DAB =\angle EAC =90^o$. Let $F$ be at the same side of the line $BC$ as $A$ such that $FB = FC$ and $\angle BFC=90^o$. Prove that the triangle $DEF$ is a right- isosceles triangle.

[b]p10 [/b]Kathy has two positive real numbers, $a$ and $b$. She mistakenly writes $$\log (a + b) = \log (a) + \log( b),$$ but miraculously, she finds that for her combination of $a$ and $b$, the equality holds. If $a = 2022b$, then $b = \frac{p}{q}$ , for positive integers $p, q$ where $gcd(p, q) = 1$. Find $p + q$. [b]p11[/b] Points $X$ and $Y$ lie on sides $AB$ and $BC$ of triangle $ABC$, respectively. Ray $\overrightarrow{XY}$ is extended to point $Z$ such that $A, C$, and $Z$ are collinear, in that order. If triangle$ ABZ$ is isosceles and triangle $CYZ$ is equilateral, then the possible values of $\angle ZXB$ lie in the interval $I = (a^o, b^o)$, such that $0 \le a, b \le 360$ and such that $a$ is as large as possible and $b$ is as small as possible. Find $a + b$. [b]p12[/b] Let $S = \{(a, b) | 0 \le a, b \le 3, a$ and $b$ are integers $\}$. In other words, $S$ is the set of points in the coordinate plane with integer coordinates between $0$ and $3$, inclusive. Prair selects four distinct points in $S$, for each selected point, she draws lines with slope $1$ and slope $-1$ passing through that point. Given that each point in $S$ lies on at least one line Prair drew, how many ways could she have selected those four points?

Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$. If $AI=3$ and the distance from $I$ to $BC$ is $2$, what is the square of length on $BC$?

The line passing through the center of the equilateral triangle $ ABC $ intersects the lines $ AB $, $ BC $ and $ CA $ at the points $ {{C} _ {1}} $, $ {{A} _ {1}} $ and $ {{B} _ {1}} $, respectively. Let $ {{A} _ {2}} $ be a point that is symmetric $ {{A} _ {1}} $ with respect to the midpoint of $ BC $; the points $ {{B} _ {2}} $ and $ {{C} _ {2}} $ are defined similarly. Prove that the points $ {{A} _ {2}} $, $ {{B} _ {2}} $ and $ {{C} _ {2}} $ lie on the same line tangent to the inscribed circle of the triangle $ ABC $. (Serdyuk Nazar)

Consider $ABC$ an acute-angled triangle with $AB \ne AC$. Denote by $M$ the midpoint of $BC$, by $D, E$ the feet of the altitudes from $B, C$ respectively and let $P$ be the intersection point of the lines $DE$ and $BC$. The perpendicular from $M$ to $AC$ meets the perpendicular from $C$ to $BC$ at point $R$. Prove that lines $PR$ and $AM$ are perpendicular.

Let $\Gamma$ be the circumcircle of triangle $ABC$. One circle $\omega$is tangent to $\Gamma$ at $A$ and tangent to $BC$ at $N$. Suppose that the extension of $AN$ crosses $\Gamma$ again at $E$. Let $AD$ and $AF$ be respectively the line of altitude $ABC$ and diameter of $\Gamma$, show that $AN \times AE = AD \times AF = AB \times AC$

Two triangles $ ABC$and $ A'B'C'$ are positioned in the space such that the length of every side of $ \triangle ABC$ is not less than $ a$, and the length of every side of $ \triangle A'B'C'$ is not less than $ a'$. Prove that one can select a vertex of $ \triangle ABC$ and a vertex of $ \triangle A'B'C'$ so that the distance between the two selected vertices is not less than $ \sqrt {\frac {a^2 \plus{} a'^2}{3}}$.

A [i]diagonal line[/i] of a (not necessarily convex) polygon with at least four sides is any line through two non-adjacent vertices of that polygon. Determine all polygons with at least four sides satisfying the following condition: The reflexion of each vertex in each diagonal line lies inside or on the boundary of the polygon. [i]The Problem Selection Committee[/i]

Let $ABCD$ be a convex quadrilateral, where $R$ and $S$ are points in $DC$ and $AB$, respectively, such that $AD=RC$ and $BC=SA$. Let $P$, $Q$ and $M$ be the midpoints of $RD$, $BS$ and $CA$, respectively. If $\angle MPC + \angle MQA = 90$, prove that $ABCD$ is cyclic.

The convex pentagon $ ABCDE $ has the following properties: $ \text{(i)} AB=BC $ $ \text{(ii)} \angle ABE+\angle CBD =\angle DBE $ $ \text{(iii)} \angle AEB +\angle BDC=180^{\circ} $ Prove that the orthocenter of $ BDE $ lies on $ AC. $

Let $C$ be a circle centered at point $O$, and let $P$ be a point in the interior of $C$. Let $Q$ be a point on the circumference of $C$ such that $PQ \perp OP$, and let $D$ be the circle with diameter $PQ$. Consider a circle tangent to $C$ whose circumference passes through point $P$. Let the curve $\Gamma$ be the locus of the centers of all such circles. If the area enclosed by $\Gamma$ is $1/100$ the area of $C$, then what is the ratio of the area of $C$ to the area of $D$?

A side of an arbitrary triangle has a length greater than $1$. Prove that the given triangle it can be cut into at least $2$ triangles, so that each of them has a side of length equal to $1$.

Let $ABCD$ be a trapezium $(AB \parallel CD)$ and $M,N$ be the intersection points of the circles of diameters $AD$ and $BC$. Prove that $O \in MN$, where $O \in AC \cap BD$.

Given four points $A,B,C,D$ in space, find a plane, $S$, equidistant from all four points and having $A$ and $C$ on one side, $B$ and $D$ on the other.

Determine all natural numbers $n$ for which it is possible to construct a rectangle of sides $15$ and $n$, with pieces congruent to: [asy] unitsize(0.6 cm); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(1,2)); draw((2,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); draw((5,-0.5)--(6,-0.5)); draw((4,0.5)--(7,0.5)); draw((4,1.5)--(7,1.5)); draw((5,2.5)--(6,2.5)); draw((4,0.5)--(4,1.5)); draw((5,-0.5)--(5,2.5)); draw((6,-0.5)--(6,2.5)); draw((7,0.5)--(7,1.5)); [/asy] The squares of the pieces have side $1$ and the pieces cannot overlap or leave free spaces

A tetrahedron $ABCD$ satisfies $\angle BAC=\angle CAD=\angle DAB=90^o$. Show that the areas of its faces satisfy the equation $area(BAC)^2 + area(CAD)^2 + area(DAB)^2 = area(BCD)^2$. .

The sides of a triangle are consecutive natural numbers, and the radius of the inscribed circle is $4$. Find the radius of the circumscribed circle.

Let $\Gamma_1$ and $\Gamma_2$ be circles - with respective centres $O_1$ and $O_2$ - that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\Gamma_2$ in $A$ and $C$ and the line $O_2A$ intersects $\Gamma_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\Gamma_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.

Triangle $ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$: $ \textbf{(A)}\ s^2\le8r^2$ $\qquad\textbf{(B)}\ s^2=8r^2$ $\qquad\textbf{(C)}\ s^2 \ge 8r^2$ ${\qquad\textbf{(D)}\ s^2\le4r^2 }$ ${\qquad\textbf{(E)}\ x^2=4r^2 } $

[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$. [b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral. [b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms? [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Amy has divided a square into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates $x$, the sum of these numbers. If the total area of white equals the total area of red, determine the minimum of $x$.

A diagonal is drawn in an isosceles trapezoid. By the contour of each of the resulting two triangles creeps its own beetle. The velocities of the beetles are constant and identical. Beetles don't change directions around their contours, and along the diagonal of the trapezoid they crawl in different directions. Prove that for any starting positions of the beetles they will ever meet.

In a triangle $ABC$ with a right angle at $A$, $H$ is the foot of the altitude from $A$. Prove that the sum of the inradii of the triangles $ABC$, $ABH$, and $AHC$ is equal to $AH$.

Let $I$ be the incenter of the triangle $ABC$. Let $A_1,A_2$ be two distinct points on the line $BC$, let $B_1,B_2$ be two distinct points on the line $CA$, and let $C_1,C_2$ be two distinct points on the line $BA$ such that $AI = A_1I = A_2I$ and $BI = B_1I = B_2I$ and $CI = C_1I = C_2I$. Prove that $A_1A_2+B_1B_2+C_1C_2 = p$ where $p$ denotes the perimeter of $ABC.$ Pierre.

The number of ordered pairs of integers $ (m,n)$ for which $ mn \ge 0$ and \[m^3 \plus{} n^3 \plus{} 99mn \equal{} 33^3\] is equal to $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 33\qquad \textbf{(D)}\ 35\qquad \textbf{(E)}\ 99$