Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. [i]Proposed by Evan Chen, Taiwan[/i]

It is given that for some $n \in \mathbb{N}$ there exists a natural number $a$, such that $a^{n-1} \equiv 1 \pmod{n}$ and that for any prime divisor $p$ of $n-1$ we have $a^{\frac{n-1}{p}} \not \equiv 1 \pmod{n}$. Prove that $n$ is a prime.

Let $ABC$ be a scalene acute-angled triangle and $D$ be the point on its circumcircle such that $AD$ is a symmedian of triangle $ABC$. Let $E$ be the reflection of $D$ about $BC$, $C_0$ the reflection of $E$ about $AB$ and $B_0$ the reflection of $E$ about $AC$. Prove that the lines $AD$, $BB_0$ and $CC_0$ are concurrent if and only if $\angle BAC = 60^{\circ}.$

In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of [b]a.)[/b] all vertices $A$ of such triangles; [b]b.)[/b] all vertices $B$ of such triangles; [b]c.)[/b] all vertices $C$ of such triangles.

Through a point in the interior of a sphere we put three pairwise perpendicular planes. Those planes dissect the surface of the sphere in eight curvilinear triangles. Alternately the triangles are coloured black and wide to make the sphere surface look like a checkerboard. Prove that exactly half of the sphere's surface is coloured black.

Suppose one has an unlimited supply of identical tiles in the shape of a right triangle [asy] draw((0,0)--(3,0)--(3,2)--(0,0)); label("$A$",(0,0),SW); label("$B$",(3,0),SE); label("$C$",(3,2),NE); size(100); [/asy] such that, if we measure the sides $AB$ and $AC$ (in inches) their lengths are integers. Prove that one can pave a square completely (without overlaps) with a number of these tiles, exactly when $BC$ has integer length.

In triangle $[ABC]$, the bisector in $C$ and the altitude passing through $B$ intersect at point $D$. Point $E$ is the symmetric of point $D$ wrt $BC$ and lies on the circle circumscribed to the triangle $[ABC]$. Prove that the triangle is $[ABC]$ isosceles.

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

Let $ABCDEFGH$ be a cube of sidelength $a$ and such that $AG$ is one of the space diagonals. Consider paths on the surface of this cube. Then determine the set of points $P$ on the surface for which the shortest path from $P$ to $A$ and from $P$ to $G$ have the same length $l.$ Also determine all possible values of $l$ depending on $a.$

Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles $5$ feet, what is the area in square feet of rectangle $ABCD$? [asy]draw((0,0)--(0,10)--(15,10)--(15,0)--(0,0)); draw((0,5)--(10,5)); draw((10,0)--(10,10)); label("$A$",(0,0),SW); label("$B$",(15,0),SE); label("$C$",(15,10),NE); label("$D$",(0,10),NW); dot((0,10)); dot((15,0)); dot((15,10)); dot((0,0)); [/asy] $\textbf{(A) }45\qquad \textbf{(B) }75\qquad \textbf{(C) }100\qquad \textbf{(D) }125\qquad \textbf{(E) }150\qquad$

Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent. [i]Pavel Kozhevnikov, Russia[/i]

let $ w_1 $ and $ w_2 $ two circles such that $ w_1 \cap w_2 = \{ A , B \} $ let $ X $ a point on $ w_2 $ and $ Y $ on $ w_1 $ such that $ BY \bot BX $ suppose that $ O $ is the center of $ w_1 $ and $ X' = w_2 \cap OX $ now if $ K = w_2 \cap X'Y $ prove $ X $ is the midpoint of arc $ AK $

Let $ABC$ be a triangle with vertices at lattice points. Suppose one of its sides in $\sqrt{n}$, where $n$ is square-free. Prove that $\frac{R}{r}$ is irraational . The symbols have usual meanings.

[b]p1.[/b] A rectangle $ABCD$ is cut from a piece of paper and folded along a straight line so that the diagonally opposite vertices $A$ and $C$ coincide. Find the length of the resulting crease in terms of the length ($\ell$) and width ($w$) of the rectangle. (Justify your answer.) [b]p2.[/b] The residents of Andromeda use only bills of denominations $\$3 $and $\$5$ . All payments are made exactly, with no change given. What whole-dollar payments are not possible? (Justify your answer.) [b]p3.[/b] A set consists of $21$ objects with (positive) weights $w_1, w_2, w_3, ..., w_{21}$ . Whenever any subset of $10$ objects is selected, then there is a subset consisting of either $10$ or $11$ of the remaining objects such that the two subsets have equal fotal weights. Find all possible weights for the objects. (Justify your answer.) [b]p4.[/b] Let $P(x) = x^3 + x^2 - 1$ and $Q(x) = x^3 - x - 1$ . Given that $r$ and $s$ are two distinct solutions of $P(x) = 0$ , prove that $rs$ is a solution of $Q(x) = 0$ [b]p5.[/b] Given: $\vartriangle ABC$ with points $A_1$ and $A_2$ on $BC$ , $B_1$ and $B_2$ on $CA$, and $C_1$ and $C_2$ on $AB$. $A_1 , A_2, B_1 , B_2$ are on a circle, $B_1 , B_2, C_1 , C_2$ are on a circle, and $C_1 , C_2, A_1 , A_2$ are on a circle. The centers of these circles lie in the interior of the triangle. Prove: All six points $A_1$ , $A_2$, $B_1$, $B_2$, $C_1$, $C_2$ are on a circle. [img]https://cdn.artofproblemsolving.com/attachments/7/2/2b99ddf4f258232c910c062e4190d8617af6fa.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

From a point $S$, which lies outside the circle $\Omega$, tangent lines $SA$ and $SB$ to that circle are drawn. On the chord $AB$ an arbitrary point $K$ is chosen. $SK$ intersects $\Omega$ at points $P$ and $Q$, and chords $RT$ and $UW$ pass through $K$ such that $W, Q$ and $T$ lie in the same half-plane with respect to $AB$. Lines $WR$ and $TU$ intersect chord $AB$ at $C$ and $D$, and $M$ is the midpoint of $PQ$. Prove that $\angle AMC = \angle BMD$

A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold: [list=1] [*] each triangle from $T$ is inscribed in $\omega$; [*] no two triangles from $T$ have a common interior point. [/list] Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.

Let $ABC$ be an isosceles triangle with $\angle B = 120^o$ . Points $P$ and $Q$ are chosen on the prolongations of segments $AB$ and $CB$ beyond point $B$ so that the rays $AQ$ and $CP$ intersect and are perpendicular to each other. Prove that $\angle PQB = 2\angle PCQ$. (A.Akopyan, D.Shvetsov)

In a convex quadrilateral $ABCD, AC\perp BD, \angle BCA = 10^o,\angle BDA = 20^o, \angle BAC = 40^o$. Find $\angle BDC$.

Five points are given in the plane. Find the maximum number of similar triangles whose vertices are among those five points.

The point $O$ is the centre of the circumscribed circle of the acute-angled triangle $ABC$. The line $AO$ cuts the side $BC$ in point $N$, and the line $BO$ cuts the side $AC$ at point $M$. Prove that if $CM=CN$, then $AC=BC$.

The pentagon $ABCDE$ is inscirbed in a circle $w$. Suppose that $w_a,w_b,w_c,w_d,w_e$ are reflections of $w$ with respect to sides $AB,BC,CD,DE,EA$ respectively. Let $A'$ be the second intersection point of $w_a,w_e$ and define $B',C',D',E'$ similarly. Prove that \[2\le \frac{S_{A'B'C'D'E'}}{S_{ABCDE}}\le 3,\] where $S_X$ denotes the surface of figure $X$. [i]Proposed by Morteza Saghafian, Ali khezeli[/i]

Show that any positive rational number can be represented as the sum of three positive rational cubes.

Let $\triangle ABC$ be a triangle with sides $a,b,c$ and corresponding angles $A,B,C$ (so $a=BC$ and $A=\angle BAC$ etc.). Suppose that $4A+3C=540^\circ$. Prove that $(a-b)^2(a+b)=bc^2$.

Let $ABC$ be a right triangle with $\angle A=90^o$ and $AB<AC$. Let $AH,AD,AM$ be altitude, angle bisector and median respectively. Prove that $\frac{BD}{CD}<\frac{HD}{MD}.$

A circle inscribed in the square $ABCD$, with side $10$ cm, intersects sides $BC$ and $AD$ at points $M$ and $N$ respectively. The point $I$ is the intersection of $AM$ with the circle different from $M$, and $P$ is the orthogonal projection of $I$ into $MN$. Find the value of segment $PI$.