The radii of the circumcircle and the incircle of a right triangle are given. Cconstruct that triangle with compass and ruler, describe the construction and justify why it is correct.

Let $C$ be a right circular cone with apex $A$. Let $P_1, P_2, P_3, P_4$ and $P_5$ be points placed evenly along the circular base in that order, so that $P_1P_2P_3P_4P_5$ is a regular pentagon. Suppose that the shortest path from $P_1$ to $P_3$ along the curved surface of the cone passes through the midpoint of $AP_2$. Let $h$ be the height of $C$, and $r$ be the radius of the circular base of $C$. If $\left(\frac{h}{r}\right)^2$ can be written in simplest form as $\frac{a}{b}$ , find $a + b$.

Find the smallest positive real constant $a$, such that for any three points $A,B,C$ on the unit circle, there exists an equilateral triangle $PQR$ with side length $a$ such that all of $A,B,C$ lie on the interior or boundary of $\triangle PQR$.

In the diagram line segments $AB$ and $CD$ are of length 1 while angles $ABC$ and $CBD$ are $90^\circ$ and $30^\circ$ respectively. Find $AC$. [asy] import geometry; import graph; unitsize(1.5 cm); pair A, B, C, D; B = (0,0); D = (3,0); A = 2*dir(120); C = extension(B,dir(30),A,D); draw(A--B--D--cycle); draw(B--C); draw(arc(B,0.5,0,30)); label("$A$", A, NW); label("$B$", B, SW); label("$C$", C, NE); label("$D$", D, SE); label("$30^\circ$", (0.8,0.2)); label("$90^\circ$", (0.1,0.5)); perpendicular(B,NE,C-B); [/asy]

Circle $\omega$ is inscribed in quadrilateral $ABCD$ and is tangent to segments $BC, AD$ at $E,F$ , respectively.$DE$ intersects $\omega$ for the second time at $X$. if the circumcircle of triangle $DFX$ is tangent to lines $AB$ and $CD$ , prove that quadrilateral $AFXC$ is cyclic.

Points $P$ and $Q$ lie respectively on sides $AB$ and $AC$ of a triangle $ABC$ and $BP=CQ$. Segments $BQ$ and $CP$ cross at $R$. Circumscribed circles of triangles $BPR$ and $CQR$ cross again at point $S$ different from $R$. Prove that point $S$ lies on the bisector of angle $BAC$.

In the triangle $ ABC$, $ \angle B$ is greater than $ \angle C$. $ T$ is the midpoint of the arc $ BAC$ from the circumcircle of $ ABC$ and $ I$ is the incenter of $ ABC$. $ E$ is a point such that $ \angle AEI\equal{}90^\circ$ and $ AE\parallel BC$. $ TE$ intersects the circumcircle of $ ABC$ for the second time in $ P$. If $ \angle B\equal{}\angle IPB$, find the angle $ \angle A$.

Let $ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$ and $\angle B = 30^{\circ}$. The perpendicular at the midpoint $M$ of $BC$ meets the bisector $BK$ of the angle $B$ at the point $E$. The perpendicular bisector of $EK$ meets $AB$ at $D$. Prove that $KD$ is perpendicular to $DE$. [i]Proposed by Greece[/i]

Let $C_1$ and $C_2$ be two concentric circles and $C_3$ an outer circle to $C_1$ inner to $C_2$ and tangent to both. If the radius of $C_2$ is equal to $ 1$, how much must the radius of $C_1$ be worth, so that the area of is twice that of $C_3$?

An acute, non-isosceles triangle $ABC$ is inscribed in a circle with centre $O$. A line go through $O$ and midpoint $I$ of $BC$ intersects $AB, AC$ at $E, F$ respectively. Let $D, G$ be reflections to $A$ over $O$ and circumcentre of $(AEF)$, respectively. Let $K$ be the reflection of $O$ over circumcentre of $(OBC)$. $a)$ Prove that $D, G, K$ are collinear. $b)$ Let $M, N$ are points on $KB, KC$ that $IM\perp AC$, $IN\perp AB$. The midperpendiculars of $IK$ intersects $MN$ at $H$. Assume that $IH$ intersects $AB, AC$ at $P, Q$ respectively. Prove that the circumcircle of $\triangle APQ$ intersects $(O)$ the second time at a point on $AI$.

Anita plays the following single-player game: She is given a circle in the plane. The center of this circle and some point on the circle are designated “known points”. Now she makes a series of moves, each of which takes one of the following forms: (i) She draws a line (infinite in both directions) between two “known points”; or (ii) She draws a circle whose center is a “known point” and which intersects another “known point”. Once she makes a move, all intersections between her new line/circle and existing lines/circles become “known points”, unless the new/line circle is identical to an existing one. In other words, Anita is making a ruler-and-compass construction, starting from a circle. What is the smallest number of moves that Anita can use to construct a drawing containing an equilateral triangle inscribed in the original circle?

Let $ A$ be one of the two points of intersection of two circles with centers $ X, Y$ respectively.The tangents at $ A$ to the two circles meet the circles again at $ B, C$. Let a point $ P$ be located so that $ PXAY$ is a parallelogram. Show that $ P$ is also the circumcenter of triangle $ ABC$.

Let $ABC$ be a right triangle with $AB = BC = 2$. Construct point $D$ such that $\angle DAC = 30^o$ and $\angle DCA = 60^o$, and $\angle BCD > 90^o$. Compute the area of triangle $BCD$.

Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$

Given a triangle with $a,b,c$ sides and with the area $1$ ($a \ge b \ge c$). Prove that $b^2 \ge 2$.

Let $\vartriangle ABC$ be a triangle with $BC = 7$, $CA = 6$, and, $AB = 5$. Let $I$ be the incenter of $\vartriangle ABC$. Let the incircle of $\vartriangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D,E$, and $F$. Let the circumcircle of $\vartriangle AEF$ meet the circumcircle of $\vartriangle ABC$ for a second time at point $X\ne A$. Let $P$ denote the intersection of $XI$ and $EF$. If the product $XP \cdot IP$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.

A convex $2019$-gon $A_1A_2...A_{2019}$ is cut into smaller pieces along its $2019$ diagonals of the form $A_iA_{i+3}$ for $1 \le i \le2019$, where $A_{2020} = A_1$, $A_{2021} = A_2$, and $A_{2022} = A_3$. What is the least possible number of resulting pieces?

In quadrilateral $ABCD$, $S_{\triangle ABD}:S_{\triangle BCD}:S_{\triangle ABC}=3:4:1$. $M\in AC,N\in CD$, satisfying that $\frac{AM}{AC}=\frac{CN}{CD}$. If $B,M,N$ are collinear, prove that $M,N$ are mid points of $AC,CD$.

Let $S$ be a finite subset of a straight line. Say that $S$ has the [i]repeated distance property [/i] if every value of the distance between two points of $S$ (except the longest) occurs at least twice. Show that if $S$ has the [i]repeated distance property [/i] then the ratio of any two distances between two points of $S$ is rational.

Let $ABC$ denote a triangle with $AC\ne BC$. Let $I$ and $U$ denote the incenter and circumcenter of the triangle $ABC$, respectively. The incircle touches $BC$ and $AC$ in the points $D$ and E, respectively. The circumcircles of the triangles $ABC$ and $CDE$ intersect in the two points $C$ and $P$. Prove that the common point $S$ of the lines $CU$ and $P I$ lies on the circumcircle of the triangle $ABC$. [i](Karl Czakler)[/i]

If the length of a rectangle is increased by $20\% $ and its width is increased by $50\% $, then the area is increased by $\text{(A)}\ 10\% \qquad \text{(B)}\ 30\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 80\% \qquad \text{(E)}\ 100\% $

Let $ABC$ be a triangle. Prove that \[\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab} \ge 4\left(\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}\right).\]

The circumference inscribed on the triangle $ABC$ is tangent to the sides $BC$, $CA$ and $AB$ on the points $D$, $E$ and $F$, respectively. $AD$ intersect the circumference on the point $Q$. Show that the line $EQ$ meet the segment $AF$ at its midpoint if and only if $AC=BC$.

Cut the $9 \times 10$ grid rectangle along the grid lines into several squares so that there are exactly two of them with odd sidelengths.

assume that we have a n*n table we fill it with 1,...,n such that each number exists exactly n times prove that there exist a row or column such that at least $\sqrt{n}$ diffrent number are contained.