Let $S=\{1,2,\ldots,10\}$. Three numbers are chosen with replacement from $S$. If the chosen numbers denote the lengths of sides of a triangle, then the probability that they will form a triangle is: $\textbf{(A)}~\frac{101}{200}$ $\textbf{(B)}~\frac{99}{200}$ $\textbf{(C)}~\frac12$ $\textbf{(D)}~\frac{110}{200}$

The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of: [b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes; [b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes; [b](c)[/b] squares in total, with vertices on the lattice.

Let $ABCD$ be a rectangle. The point $E$ lies on side $ \overline{AB}$ and the point $F$ is lies side $ \overline{AD}$, such that $\angle FEC=\angle CEB$ and $\angle DFC=\angle CFE$. Determine the measure of the angle $\angle FCE$ and the ratio $AD/AB$.

[b]p1.[/b] The edges of a tetrahedron are all tangent to a sphere. Prove that the sum of the lengths of any pair of opposite edges equals the sum of the lengths of any other pair of opposite edges. (Two edges of a tetrahedron are said to be opposite if they do not have a vertex in common.) [b]p2.[/b] Find the simplest formula possible for the product of the following $2n - 2$ factors: $$\left(1+\frac12 \right),\left(1-\frac12 \right), \left(1+\frac13 \right) , \left(1-\frac13 \right),...,\left(1+\frac{1}{n} \right), \left(1-\frac{1}{n} \right)$$. Prove that your formula is correct. [b]p3.[/b] Solve $$\frac{(x + 1)^2+1}{x + 1} + \frac{(x + 4)^2+4}{x + 4}=\frac{(x + 2)^2+2}{x + 2}+\frac{(x + 3)^2+3}{x + 3}$$ [b]p4.[/b] Triangle $ABC$ is inscribed in a circle, $BD$ is tangent to this circle and $CD$ is perpendicular to $BD$. $BH$ is the altitude from $B$ to $AC$. Prove that the line $DH$ is parallel to $AB$. [img]https://cdn.artofproblemsolving.com/attachments/e/9/4d0b136dca4a9b68104f00300951837adef84c.png[/img] [b]p5.[/b] Consider the picture below as a section of a city street map. There are several paths from $A$ to $B$, and if one always walks along the street, the shortest paths are $15$ blocks in length. Find the number of paths of this length between $A$ and $B$. [img]https://cdn.artofproblemsolving.com/attachments/8/d/60c426ea71db98775399cfa5ea80e94d2ea9d2.png[/img] [b]p6.[/b] A [u]finite [/u] [u]graph [/u] is a set of points, called [u]vertices[/u], together with a set of arcs, called [u]edges[/u]. Each edge connects two of the vertices (it is not necessary that every pair of vertices be connected by an edge). The [u]order [/u] of a vertex in a finite graph is the number of edges attached to that vertex. [u]Example[/u] The figure at the right is a finite graph with $4$ vertices and $7$ edges. [img]https://cdn.artofproblemsolving.com/attachments/5/9/84d479c5dbd0a6f61a66970e46ab15830d8fba.png[/img] One vertex has order $5$ and the other vertices order $3$. Define a finite graph to be [u]heterogeneous [/u] if no two vertices have the same order. Prove that no graph with two or more vertices is heterogeneous. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let the circles $\omega$ and $\omega'$ intersect in points $A$ and $B$. The tangent to circle $\omega$ at $A$ intersects $\omega'$ at $C$ and the tangent to circle $\omega'$ at $A$ intersects $\omega$ at $D$. Suppose that the internal bisector of $\angle CAD$ intersects $\omega$ and $\omega'$ at $E$ and $F$, respectively, and the external bisector of $\angle CAD$ intersects $\omega$ and $\omega'$ at $X$ and $Y$, respectively. Prove that the perpendicular bisector of $XY$ is tangent to the circumcircle of triangle $BEF$. [i]Proposed by Mahdi Etesami Fard[/i]

Let $ABC$ be an acute triangle with $\angle BAC=30^{\circ}$. The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$, respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$, respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$. Prove that $\angle BPC=90^{\circ}$ .

Let $\triangle ABC$ be an acute triangle, $O$ be its circumcenter, and $D$ be a point different that $A$ and $C$ on the smaller $AC$ arc of its circumcircle. Let $P$ be a point on $[AB]$ satisfying $\widehat{ADP} = \widehat {OBC}$ and $Q$ be a point on $[BC]$ satisfying $\widehat{CDQ}=\widehat {OBA}$. Show that $\widehat {DPQ} = \widehat {DOC}$.

Let $ABCD$ be a tetrahedron such that edges $AB$, $AC$, and $AD$ are mutually perpendicular. Let the areas of triangles $ABC$, $ACD$, and $ADB$ be denoted by $x$, $y$, and $z$, respectively. In terms of $x$, $y$, and $z$, find the area of triangle $BCD$.

Inside the triangle $ABC$, the point $P$ is chosen. Let $ a, b, c $ be the lengths of the sides $ BC $, $ CA $, $ AB $, respectively, and $ x, y, z $ the distances of the point $ P $ from the vertices $ B, C, A $. Prove that if $$ x^2 + xy + y^2 = a^2 $$ $$y^2 + yz + z^2 = b^2 $$ $$z^2 + zx + x^2 = c^2$$ this $$ a^2 + ab + b^2 > c^2.$$

Consider the cyclic quadrilateral with side lengths $1$, $4$, $8$, $7$ in that order. What is its circumdiameter? Let the answer be of the form $a\sqrt b+c$, for $b$ squarefree. Find $a+b+c$.

Given triangle $ABC$. Point $O_1$ is the center of the $BCDE$ rectangle, constructed so that the side $DE$ of the rectangle contains the vertex $A$ of the triangle. Points $O_2$ and $O_3$ are the centers of rectangles constructed in the same way on the sides $AC$ and $AB$, respectively. Prove that lines $AO_1, BO_2$ and $CO_3$ meet at one point.

Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$. The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$, and argue the existence of such a triangle.

Given a line segment $AB=7$, $C$ is constructed on $AB$ so that $AC=5$. Two equilateral triangles are constructed on the same side of $AB$ with $AC$ and $BC$ as a side. Find the length of the segment connecting their two circumcenters.

What is the smallest $r$ such that three disks of radius $r$ can completely cover up a unit disk?

Let $K, T$ be the points of tangency of inscribed and exscribed circles to the side $BC$ triangle $ABC$, $M$ is the midpoint of the side $BC$. Using a compass and a ruler, construct triangle ABC given rays $AK$ and $AT$ (points $K, T$ are not marked on them) and point $M$.

$ AL$, $ BM$, and $ CN$ are the medians of $ \triangle ABC$. $ K$ is the intersection of medians. If $ C,K,L,M$ are concyclic and $ AB \equal{} \sqrt 3$, then the median $ CN$ = ? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ \sqrt 3 \qquad\textbf{(C)}\ \frac {3\sqrt3}{2} \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

Let $ABC$ be a triangle with an interior point $P$ such that $\angle APB = \angle APC = \alpha$ and $\alpha > 180^o-\angle BAC$. The circumcircle of triangle $APB$ cuts $AC$ at $E$, the circumcircle of triangle $APC$ cuts $AB$ at $F$. Let $Q$ be the point in the triangle $AEF$ such that $\angle AQE = \angle AQF =\alpha$. Let $D$ be the symmetric point of $Q$ wrt $EF$. Angle bisector of $\angle EDF$ cuts $AP$ at $T$. a) Prove that $\angle DET = \angle ABC, \angle DFT = \angle ACB$. b) Straight line $PA$ cuts straight lines $DE, DF$ at $M, N$ respectively. Denote $I, J$ the incenters of the triangles $PEM, PFN$, and $K$ the circumcenter of the triangle $DIJ$. Straight line $DT$ cut $(K)$ at $H$. Prove that $HK$ passes through the incenter of the triangle $DMN$.

Suppose that $ ABC$ is an isosceles triangle with $ AB \equal{} AC$. Let $ P$ be the point on side $ AC$ so that $ AP \equal{} 2CP$. Given that $ BP \equal{} 1$, determine the maximum possible area of $ ABC$.

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]

Prove that a triangle with angles $\alpha, \beta, \gamma$, circumradius $R$, and area $A$ satisfies \[\tan \frac{ \alpha}{2}+\tan \frac{ \beta}{2}+\tan \frac{ \gamma}{2} \leq \frac{9R^2}{4A}.\] [hide="Remark."]Remark. Can we determine [i]all[/i] of equality cases ?[/hide]

The circles $\omega_1$ and $\omega_2$ intersect at two points $A$ and $B$. On the circle $\omega_2$, point $C$ is taken in such a way that $CA$ is tangent to the circle $\omega_1$. Through point $A$, a straight line is drawn that intersects the circles $\omega_1$, and $\omega_2$ at points $M$ and $N$, respectively , different from point $A$. Point $P$ is the midpoint of the segment $AC$, $Q$ is the midpoint of $MN$, and $S$ is the intersection point of the line $BQ$ with the circle $\omega_1$, different from point $B$. Prove that the lines $AS$ and $PQ$ are parallel.

Let $ABC$ be a triangle. The ex-circles touch sides $BC, CA$ and $AB$ at points $U, V$ and $W$, respectively. Be $r_u$ a straight line that passes through $U$ and is perpendicular to $BC$, $r_v$ the straight line that passes through $V$ and is perpendicular to $AC$ and $r_w$ the straight line that passes through W and is perpendicular to $AB$. Prove that the lines $r_u$, $r_v$ and $r_w$ pass through the same point.

Let $ABC$ be a triangle such that $\vert AB\vert \ne \vert AC\vert$. Prove that there exists a point $D \ne A$ on its circumcircle satisfying the following property: For any points $M, N$ outside the circumcircle on the rays $AB$ and $AC$, respectively, satisfying $\vert BM\vert=\vert CN\vert$, the circumcircle of $AMN$ passes through $D$.

In triangle $ABC$ the angle bisector $AK$ is perpendicular on the median is $CL$. Prove that in the triangle $BKL$ also one of angle bisectors are perpendicular to one of the medians.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.