In a perpendicular triangle the perimeter is 60 and the altitude on the hypotenuse is 12. Then, the length of the hypotenuse is $ \text{(A)}\ 24 \qquad \text{(B)}\ 25 \qquad \text{(C)}\ 26 \qquad \text{(D)}\ 27 \qquad \text{(E)}\ 28$

Let $ ABC$ be a triangle with $ \angle A \equal{} 45^\circ$. Let $ P$ be a point on side $ BC$ with $ PB \equal{} 3$ and $ PC \equal{} 5$. Let $ O$ be the circumcenter of $ ABC$. Determine the length $ OP$.

The letter [b]M[/b] in the figure below is first reflected over the line $q$ and then reflected over the line $p$. What is the resulting image? [asy] // pog diagram usepackage("newtxtext"); size(3cm); draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label("$\textbf{\textsf{M}}$",(0.25,0.6)); draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("$p$", (-1,0),NE); label("$q$", (-0.75,-0.75), N*1.5); [/asy] [asy] // pog diagram usepackage("newtxtext"); size(12.5cm); draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(90)*"$\textbf{\textsf{M}}$",(0.6,-0.25)); draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("$\textbf{(A)}$",(-1,1),W); draw((2,0)--(4,0)); draw((3,-1)--(3,1)); label(rotate(270)*"$\textbf{\textsf{M}}$",(2.8,0.7)); draw((2.2,-0.8)--(3.8,0.8),linewidth(1.1)); label("$\textbf{(B)}$",(2,1),W); draw((5,0)--(7,0)); draw((6,-1)--(6,1)); label(rotate(90)*"$\textbf{\textsf{M}}$",(5.4,0.2)); draw((5.2,-0.8)--(6.8,0.8),linewidth(1.1)); label("$\textbf{(C)}$",(5,1),W); draw((-1,-2.5)--(1,-2.5)); draw((0,-3.5)--(0,-1.5)); label(rotate(180)*"$\textbf{\textsf{M}}$",(-0.25,-3.1)); draw((-0.8,-3.3)--(0.8,-1.7),linewidth(1.1)); label("$\textbf{(D)}$",(-1,-1.5),W); draw((2,-2.5)--(4,-2.5)); draw((3,-3.5)--(3,-1.5)); label(rotate(270)*"$\textbf{\textsf{M}}$",(3.6,-2.75)); draw((2.2,-3.3)--(3.8,-1.7),linewidth(1.1)); label("$\textbf{(E)}$",(2,-1.5),W); [/asy]

Denote in the triangle $ABC$ by $h$ the length of the height drawn from vertex $A$, and by $\alpha = \angle BAC$. Prove that the inequality $AB + AC \ge BC \cdot \cos \alpha + 2h \cdot \sin \alpha$ . Are there triangles for which this inequality turns into equality?

In triangle $\triangle ABC$ points $M,N$ lie on $BC$ st : $\angle BAM= \angle MAN= \angle NAC$ . Points $P,Q$ are on the angle bisector of $BAC$, on the same side of $BC$ as A , st : $$\frac{1}{3} \angle BAC = \frac{1}{2} \angle BPC = \angle BQC$$ Let $E = AM \cap CQ$ and $F = AN \cap BQ$ . Prove that the common tangents to $(EPF), (EQF)$ and the circumcircle of $\triangle ABC$ , are concurrent.

[b]p1.[/b] A function $f$ defined on the set of positive numbers satisfies the equality $$f(xy) = f(x) + f(y), x, y > 0.$$ Find $f(2007)$ if $f\left( \frac{1}{2007} \right) = 1$. [b]p2.[/b] The plane is painted in two colors. Show that there is an isosceles right triangle with all vertices of the same color. [b]p3.[/b] Show that the number of ways to cut a $2n \times 2n$ square into $1\times 2$ dominoes is divisible by $2$. [b]p4.[/b] Two mirrors form an angle. A beam of light falls on one mirror. Prove that the beam is reflected only finitely many times (even if the angle between mirrors is very small). [b]p5.[/b] A sequence is given by the recurrence relation $a_{n+1} = (s(a_n))^2 +1$, where $s(x)$ is the sum of the digits of the positive integer $x$. Prove that starting from some moment the sequence is periodic. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[i](9 pts)[/i] Let $ABC$ be a triangle with circumcenter $O$. The interior bisector of $\angle BAC$ intersects $BC$ at $D$. Circle $\omega_A$ is tangent to segments $AB$ and $AC$ and internally tangent to the circumcircle of $ABC$ at the point $P$. Let $E$ and $F$ be the respective points at which the $B$-excircle and $C$-excircle of $ABC$ are tangent to $AC$ and $AB$. Suppose that lines $BE$ and $CF$ pass through a common point $N$ on the circumcircle of $AEF$. [i]Note: for a triangle $ABC$, the $A$-excircle is the circle lying outside triangle $ABC$ that is tangent to side $BC$ and the extensions of sides $AB, AC$. The $B, C$-excircles are defined similarly.[/i] (a) [i](7 pts)[/i] Prove that the circumcircle of $PDO$ passes through $N$. (b) [i](2 pts)[/i] Suppose that $\frac{PD}{BC} = \frac{2}{7}$. Find, with proof, the value of $\cos (\angle BAC)$.

$WHY$ is a triangle with angle $W \ge 90$ degrees. On the side $HY$, two distinct points $M$ and $E$ are chosen such that angle $HWM$ is equivalent to angle $ MWE$ and $HM * YE = HY * ME$. Find the angle $MWY$.

It is given a tetrahedron $ABCD$ for which two points of opposite edges are mutually perpendicular. Prove that: (a) the four altitudes of $ABCD$ intersects at a common point $H$; (b) $AH+BH+CH+DH<p+2R$, where $p$ is the sum of the lengths of all edges of $ABCD$ and $R$ is the radii of the sphere circumscribed around $ABCD$. [i]H. Lesov[/i]

Let $ABCDE$ be a convex pentagon with $\angle AEB=\angle BDC=90^o$ and line $AC$ bisects $\angle BAE$ and $\angle DCB$ internally. The circumcircle of $ABE$ intersects line $AC$ again at $P$. (a) Show that $P$ is the circumcenter of $BDE$. (b) Show that $A, C, D, E$ are concyclic.

Prove that if the sides $ a $, $ b $, $ c $ of a triangle satisfy the inequality $$a < b < c$$then the angle bisectors $ d_a $, $ d_b $, $ d_c $ of opposite angles satisfy the inequality $$ d_a > d_b > d_c.$$

A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of $134$ centimeters, how many centimeters above the base of the pyramid is the mouse?

Let $O$ be the circumcenter and $H$ the orthocenter of an acute-triangle $ABC$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Let $L$ be the midpoint of $OH$. Prove that $\angle OAH = \angle LSA$.

On a line $l$ there are three different points $A$, $B$ and $P$ in that order. Let $a$ be the line through $A$ perpendicular to $l$, and let $b$ be the line through $B$ perpendicular to $l$. A line through $P$, not coinciding with $l$, intersects $a$ in $Q$ and $b$ in $R$. The line through $A$ perpendicular to $BQ$ intersects $BQ$ in $L$ and $BR$ in $T$. The line through $B$ perpendicular to $AR$ intersects $AR$ in $K$ and $AQ$ in $S$. (a) Prove that $P$, $T$, $S$ are collinear. (b) Prove that $P$, $K$, $L$ are collinear. [i](2nd Benelux Mathematical Olympiad 2010, Problem 3)[/i]

Let $ABCD$ be a regular tetrahedron and $\mathbf{Z}$ an isometry mapping $A,B,C,D$ into $B,C,D,A$, respectively. Find the set $M$ of all points $X$ of the face $ABC$ whose distance from $\mathbf{Z}(X)$ is equal to a given number $t$. Find necessary and sufficient conditions for the set $M$ to be nonempty.

Let us call a triangle rational if each of its angles is a rational number when measured in degrees. Let us call a point inside triangle rational if joining it to the three vertices of the triangle we get three rational triangles. Show that any acute rational triangle contains at least three distinct rational points.

In a set of consecutive positive integers, there are exactly $100$ perfect cubes and $10$ perfect fourth powers. Prove that there are at least $2000$ perfect squares in the set.

Let $k$ be a circle and $l$–line that is tangent to $k$ in point $P$. On $l$ from the two sides of $P$ are chosen arbitrary points $A$ and $B$. The tangents through $A$ and $B$ to $k$, different than $l$, intersect in point $C$. Find the geometric place of points $C$, when $A$ and $B$ change in such way so that $AP.BP$ is a constant.

Let $ n\ge 3 $ be a natural number and $ A_0A_1...A_{n-1} $ a regular polygon. Consider $ B_0 $ on the segment $ A_0A_1 $ such that $ A_0B_0<\frac{1}{2}A_0A_1; B_1 $ on $ A_1A_2 $ so that $ A_1B_1<\frac{1}{2} A_1A_2; $ etc.; $ B_{n-2} $ on $ A_{n-2}A_{n-1} $ so that $ A_{n-2}B_{n-2} <\frac{1}{2} A_{n-2}A_{n-1} , $ and $ B_{n-1} $ on $ A_{n-1}A_0 $ with $ A_{n-1}B_{n-1} <\frac{1}{2} A_{n-1}A_{0} . $ Show that the perimeter of any ploygon that has its vertices on the segments $ A_1B_1,A_2B_2,...,A_{n-1}B_{n-1}, $ is equal or greater than the perimeter of $ B_0B_1...B_{n-1} . $

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Let $H$ be the orthocenter of $ABC$. Find the radius of the circle with nonzero radius tangent to the circumcircles of $AHB$, $BHC$, $CHA$.

Let $ABC$ be an acute triangle with $\angle ABC=60^{\circ}.$ Suppose points $D$ and $E$ are on lines $AB$ and $CB,$ respectively, such that $CDB$ and $AEB$ are equilateral triangles. Given that the positive difference between the perimeters of $CDB$ and $AEB$ is $60$ and $DE=45,$ what is the value of $AB \cdot BC?$ [i]Proposed by Kyle Lee[/i]

In a triangle $ABC$, the angle bisector at $A,B,C$ meet the opposite sides at $A_1,B_1,C_1$, respectively. Prove that if the quadrilateral $BA_1B_1C_1$ is cyclic, then $$\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.$$

A rectangle with sides of lengths $a$ and $b$ ($a > b$) is cut into rightangled triangles so that any two of these triangles either have a common side, a common vertex or no common points. Moreover, any common side of two triangles is a leg of one of them and the hypotenuse of the other. Prove that $a > 2b$. (A Shapovalov)

Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.

Let $ABC$ be a triangle, and $M$ the midpoint of its side $BC$. Let $\gamma$ be the incircle of triangle $ABC$. The median $AM$ of triangle $ABC$ intersects the incircle $\gamma$ at two points $K$ and $L$. Let the lines passing through $K$ and $L$, parallel to $BC$, intersect the incircle $\gamma$ again in two points $X$ and $Y$. Let the lines $AX$ and $AY$ intersect $BC$ again at the points $P$ and $Q$. Prove that $BP = CQ$.