How many triangles are there for which the perimeters are equal to $30$ cm and the lengths of sides are integers in centimeters? A. $16$ B. $17$ C. $18$ D. $19$ E. $20$

Let $ m,n\in \mathbb{N}^*$. Find the least $ n$ for which exists $ m$, such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$, $ n \minus{} 1$ of length $ 2$, $ ...$, $ 1$ square of length $ n$. For the found value of $ n$ give the example of covering.

In convex quadrilateral $ABCD$, let diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. Let the circumcircles of $ADE$ and $BCE$ intersect $\overline{AB}$ again at $P \neq A$ and $Q \neq B$, respectively. Let the circumcircle of $ACP$ intersect $\overline{AD}$ again at $R \neq A$, and let the circumcircle of $BDQ$ intersect $\overline{BC}$ again at $S \neq B$. Prove that $A$, $B$, $R$, and $S$ are concyclic. [i]Tiger Zhang[/i]

The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). The points of tangency of the circumcircles of triangles \(ABE\) and \(CDE\) with their common external tangents lie on a circle \(\omega\). The points of tangency of the circumcircles of triangles \(ADE\) and \(BCE\) with their common external tangents lie on a circle \(\gamma\). Prove that the centers of circles \(\omega\) and \(\gamma\) coincide.

Let $\omega$ is tangent to the sides of an acute angle with vertex $A$ at points $B$ and $C$. Let $D$ be an arbitrary point onn the major arc $BC$ of the circle $\omega$. Points $E$ and $F$ are chosen inside the angle $DAC$ so that quadrilaterals $ABDF$ and $ACED$ are inscribed and the points $A,E,F$ lie on the same straight line. Prove that lines $BE$ and $CF$ intersectat $\omega$.

The medians of a right triangle $ABC$ ($\angle C = 90^{\circ}$) intersect at $M$. Point $L$ lies on the $AC$ such that $\angle ABL=\angle CBL$. It turned out that $\angle BML = 90^{\circ}$. Find the ration $AB : BC$.

The quadrangle $ABCD$ is inscribed in a circle whose center $O$ lies inside it. Prove that if $\angle BAO = \angle DAC$, then the diagonals of the quadrilateral are perpendicular.

The unit square $ABCD$ has $E$ as midpoint of $AD$ and a circle of radius $r$ tangent to $AB$, $BC$, and $CE$. Determine $r$.

$ABC$ is an equilateral triangle. $D$ and$ D'$ are points on opposite sides of the plane $ABC$ such that the two tetrahedra $ABCD$ and $ABCD'$ are congruent (but not necessarily with the vertices in that order). If the polyhedron with the five vertices $A, B, C, D, D'$ is such that the angle between any two adjacent faces is the same, find $DD'/AB$ .

Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that \[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]

Let $ABC$ be a fixed triangle. Three similar (by point order) isosceles trapezoids are built on its sides: $ABXY, BCZW, CAUV$, such that the sides of the triangle are bases of the respective trapezoids. The circumcircles of triangles $XZU, YWV$ meet at two points $P, Q$. Prove that the line $PQ$ passes through a fixed point independent of the choice of trapezoids.

Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.

$n$ points are given on a circle $\omega$. There is a circle with radius smaller than $\omega$ such that all these points lie inside or on the boundary of this circle. Prove that we can draw a diameter of $\omega$ with endpoints not belonging to the given points such that all the $n$ given points remain in one side of the diameter.

The perimeter of an isosceles right triangle is $ 2p$. Its area is: $ \textbf{(A)}\ (2\plus{}\sqrt{2})p \qquad\textbf{(B)}\ (2\minus{}\sqrt{2})p \qquad\textbf{(C)}\ (3\minus{}2\sqrt{2})p^2\\ \textbf{(D)}\ (1\minus{}2\sqrt{2})p^2 \qquad\textbf{(E)}\ (3\plus{}2\sqrt{2})p^2$

Let $T$ be the intersection of the common internal tangents of circles $C_1$, $C_2$ with centers $O_1$, $O_2$ respectively. Let $P$ be one of the points of tangency on $C_1$ and let line $\ell$ bisect angle $O_1TP$ . Label the intersection of $\ell$ with $C_1$ that is farthest from $T$, $R$, and label the intersection of $\ell$ with $C_2$ that is closest to $T$, $S$. If $C_1$ has radius $4$, $C_2$ has radius $6$, and $O_1O_2= 20$ , calculate $(TR)(TS) $. [img]https://cdn.artofproblemsolving.com/attachments/3/c/284f17bb0dd73eab93132e41f27ecc121f496d.png[/img]

Is there a set $S\subset\mathbb{R}^3$ of $2006$ points such that not all its points are coplanar, no three of the points are collinear, and for any $A,B\in S$ we can find points $C,D\in S$ for which $AB||CD$?

Find all triangles that can be divided into congruent right-angled isosceles triangles with side lengths $1, 1, \sqrt{2}$.

Let $\triangle ABC$ be a acute triangle. Let $H$ the foot of the C-altitude in $AB$ such that $AH=3BH$, let $M$ and $N$ the midpoints of $AB$ and $AC$ and let $P$ be a point such that $NP=NC$ and $CP=CB$ and $B$, $P$ are located on different sides of the line $AC$. Prove that $\measuredangle APM=\measuredangle PBA$.

One of the external common tangent lines of the two externally tangent circles with center $O_1$ and $O_2$ touches the circles at $B$ and $C$, respectively. Let $A$ be the common point of the circles. The line $BA$ meets the circle with center $O_2$ at $A$ and $D$. If $|BA|=5$ and $|AD|=4$, then what is $|CD|$? $ \textbf{(A)}\ \sqrt{20} \qquad\textbf{(B)}\ \sqrt{27} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ \frac{15}2 \qquad\textbf{(E)}\ 4\sqrt5 $

The number $0$ or $1$ is to be assigned to each of the $n$ vertices of a regular polygon. In how many different ways can this be done (if we consider two assignments that can be obtained one from the other through rotation in the plane of the polygon to be identical)?

In the diagram below, three squares are inscribed in right triangles. Their areas are $A$, $M$, and $N$, as indicated in the diagram. If $M = 5$ and $N = 12$, then $A$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$. [asy] size(250); defaultpen (linewidth (0.7) + fontsize (10)); pair O = origin, A = (1, 1), B = (4/3, 1/3), C = (2/3, 5/3), P = (3/2, 0), Q = (0,3); draw (P--O--Q--cycle^^(0, 5/3)--C--(2/3,1)^^(0,1)--A--(1,0)^^(1,1/3)--B--(4/3,0)); label("$A$", (.5,.5)); label("$M$", (7/6, 1/6)); label("$N$", (1/3, 4/3));[/asy] [i]Proposed by Aaron Lin[/i]

There are $3$ sticks of each color between blue, red and green, such that we can make a triangle $T$ with sides sticks with all different colors. Dana makes $2$ two arrangements, she starts with $T$ and uses the other six sticks to extend the sides of $T$, as shown in the figure. This leads to two hexagons with vertex the ends of these six sticks. Prove that the area of the both hexagons it´s the same. [asy]size(300); pair A, B, C, D, M, N, P, Q, R, S, T, U, V, W, X, Y, Z, K; A = (0, 0); B = (1, 0); C=(-0.5,2); D=(-1.1063,4.4254); M=(-1.7369,3.6492); N=(3.5,0); P=(-2.0616,0); Q=(0.2425,-0.9701); R=(1.6,-0.8); S=(7.5164,0.8552); T=(8.5064,0.8552); U=(7.0214,2.8352); V=(8.1167,-1.546); W=(9.731,-0.7776); X=(10.5474,0.8552); Y=(6.7813,3.7956); Z=(6.4274,3.6272); K=(5.0414,0.8552); draw(A--B, blue); label("$b$", (A + B) / 2, dir(270), fontsize(10)); label("$g$", (B+C) / 2, dir(10), fontsize(10)); label("$r$", (A+C) / 2, dir(230), fontsize(10)); draw(B--C,green); draw(D--C,green); label("$g$", (C + D) / 2, dir(10), fontsize(10)); draw(C--A,red); label("$r$", (C + M) / 2, dir(200), fontsize(10)); draw(B--N,green); label("$g$", (B + N) / 2, dir(70), fontsize(10)); draw(A--P,red); label("$r$", (A+P) / 2, dir(70), fontsize(10)); draw(A--Q,blue); label("$b$", (A+Q) / 2, dir(540), fontsize(10)); draw(B--R,blue); draw(C--M,red); label("$b$", (B+R) / 2, dir(600), fontsize(10)); draw(Q--R--N--D--M--P--Q, dashed); draw(Y--Z--K--V--W--X--Y, dashed); draw(S--T,blue); draw(U--T,green); draw(U--S,red); draw(T--W,red); draw(T--X,red); draw(S--K,green); draw(S--V,green); draw(Y--U,blue); draw(U--Z,blue); label("$b$", (Y+U) / 2, dir(0), fontsize(10)); label("$b$", (U+Z) / 2, dir(200), fontsize(10)); label("$b$", (S+T) / 2, dir(100), fontsize(10)); label("$r$", (S+U) / 2, dir(200), fontsize(10)); label("$r$", (T+W) / 2, dir(70), fontsize(10)); label("$r$", (T+X) / 2, dir(70), fontsize(10)); label("$g$", (U+T) / 2, dir(70), fontsize(10)); label("$g$", (S+K) / 2, dir(70), fontsize(10)); label("$g$", (V+S) / 2, dir(30), fontsize(10)); [/asy]

$ABC$ is a right triangle with $\angle A = 30^o$ and circumcircle $O$. Circles $\omega_1$, $\omega_2$, and $\omega_3$ lie outside $ABC$ and are tangent to $O$ at $T_1$, $T_2$, and $T_3$ respectively and to $AB$, $BC$, and $CA$ at $S_1$, $S_2$, and $S_3$, respectively. Lines $T_1S_1$, $T_2S_2$, and $T_3S_3$ intersect $O$ again at $A'$, $B'$, and $C'$, respectively. What is the ratio of the area of $A'B'C'$ to the area of $ABC$?

Given are triangle $ABC$ and line $\ell$. The reflections of $\ell$ in $AB$ and $AC$ meet at point $A_1$. Points $B_1, C_1$ are defined similarly. Prove that a) lines $AA_1, BB_1, CC_1$ concur, b) their common point lies on the circumcircle of $ABC$ c) two points constructed in this way for two perpendicular lines are opposite.

Let $ABC$ be a triangle, and let $D$ be a point on the internal angle bisector of $BAC$. Let $x$ be the ellipse with foci $B$ and $C$ passing through $D$, $y$ be the ellipse with foci $A$ and $C$ passing through $D$, and $z$ be the ellipse with foci $A$ and $B$ passing through $D$. Ellipses $x$ and $z$ intersect at distinct points $D$ and $E$, and ellipses $x$ and $y$ intersect at distinct points $D$ and $F$. Prove that $AD$ bisects angle $EAF$. [i]Andrew Carratu[/i]