Let $I$ be the incenter and $AB$ the shortest side of the triangle $ABC$. The circle centered at $I$ passing through $C$ intersects the ray $AB$ in $P$ and the ray $BA$ in $Q$. Let $D$ be the point of tangency of the $A$-excircle of the triangle $ABC$ with the side $BC$. Let $E$ be the reflection of $C$ with respect to the point $D$. Prove that $PE\perp CQ$.

Given a triangle$ABC$ with $AB<BC<AC$ inscribed in circle $(c)$. The circle $c(A,AB)$ (with center $A$ and radius $AB$) interects the line $BC$ at point $D$ and the circle $(c)$ at point $H$. The circle $c(A,AC)$ (with center $A$ and radius $AC$) interects the line $BC$ at point $Z$ and the circle $(c)$ at point $E$. Lines $ZH$ and $ED$ intersect at point $T$. Prove that the circumscribed circles of triangles $TDZ$ and $TEH$ are equal.

Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$ [i]Proposed by Christopher Bradley, United Kingdom[/i]

Let $T$ be a point on a line segment $AB$ such that $T$ is closer to $B$ than to $A$. Show that for each point $C \ne T$ on the line through $T$ perpendicular to $AB$ there is exactly one point $D$ on the line segment $AC$ with $\angle CBD=\angle BAC$. Moreover, show that the line through $D$ perpendicular to $AC$ intersects the line $AB$ in a point $E$ which is independent of the position of $C$.

Problems 15, 16, and 17 all refer to the following: In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles. 17. In how many years, approximately, from 1998 will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support? $ \text{(A)}\ 50\text{ yrs.}\qquad\text{(B)}\ 75\text{ yrs.}\qquad\text{(C)}\ 100\text{ yrs.}\qquad\text{(D)}\ 125\text{ yrs.}\qquad\text{(E)}\ 150\text{ yrs.} $

In the figure below points $A,B$ are the centers of the circles $\omega_1, \omega_2$. Starting from the line $BC$ points $E,F,G,H,I$ are obtained respectively. Find the angle $\angle IBE$.

Given a cyclic quadrilateral $ABCD$ with the circumcenter $O$, with $BC$ and $AD$ not parallel. Let $P$ be the intersection of $AC$ and $BD$. Let $E$ be the intersection of the rays $AB$ and $DC$. Let $I$ be the incenter of $EBC$ and the incircle of $EBC$ touches $BC$ at $T_1$. Let $J$ be the excenter of $EAD$ that touches $AD$ and the excircle of $EAD$ that touches $AD$ touches $AD$ at $T_2$. Let $Q$ be the intersection between $IT_1$ and $JT_2$. Prove that $O,P,Q$ are collinear.

Consider the two mutually tangent parabolas $y=x^2$ and $y=-x^2$. The upper parabola rolls without slipping around the fixed lower parabola. Find the locus of the focus of the moving parabola.

Let $O$ be the point of intersection of the diagonals of the trapezoid $ABCD$ with the bases $AB$ and $CD$. It is known that $\angle AOB = \angle DAB = 90^o$. On the sides $AD$ and $BC$ take the points $E$ and $F$ so that $EF\parallel AB$ and $EF = AD$. Find the angle $\angle AOE$.

In $\triangle ABC$ with $AB\neq{AC}$ let $M$ be the midpoint of $AB$, let $K$ be the midpoint of the arc $BAC$ in the circumcircle of $\triangle ABC$, and let the perpendicular bisector of $AC$ meet the bisector of $\angle BAC$ at $P$ . Prove that $A, M, K, P$ are concyclic.

In an acute triangle $ABC$, $\angle C = 60^o$. If $AA'$ and $BB'$ are two of the altitudes and $C_1$ is the midpoint of $AB$, prove that triangle $C_1A'B'$ is equilateral.

Prove the inequality for all positive integer $n$ : \[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]

A square's area is equal to the perimeter of a $15$ by $17$ rectangle. What is this square's perimeter? $\textbf{(A) }20\qquad\textbf{(B) }32\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad\textbf{(E) }56$

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and $$\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ$$ Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$ Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$ and $CC_1C_2$ all pass through two common points. (Note: a scalene triangle is one where no two sides have equal length.) [i]Proposed by Ankan Bhattacharya, USA[/i]

Let $\vartriangle ABC$ be an equilateral triangle with side $1$. $1011$ points $P_1$, $P_2$, $P_3$, $...$, $P_{1011}$ on the side $AC$ and $1011$ points $Q_1$, $Q_2$, $Q_3$, $...$ ,$ Q_{1011}$ on side AB (see figure) in such a way as to generate $2023$ triangles of equal area. Find the length of the segment $AP_{1011}$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/fea495c16a0b626e0c3882df66d66011a1a3af.png[/img] PS. Harder version of [url=https://artofproblemsolving.com/community/c4h3323135p30741470]2023 Chile NMO L1 P3[/url]

Let $ABCD$ be a convex quadrilateral and let $P$ be the intersection of the diagonals $AC$ and $BD$. The radii of the circles inscribed in the triangles $\vartriangle ABP$, $\vartriangle BCP$, $\vartriangle CDP$ and $\vartriangle DAP$ are the same. Prove that $ABCD$ is a rhombus,

Given two points $P,Q$ of the plane, denote $P+Q$ the midpoint of (possibly degenerate) segment $PQ$ and $P\cdot Q$ the image of $P$ in rotation around the origin $Q$ under $+90^\circ.$ a) Are these operations commutative? b) Given two distinct points $A,B$ the equation \[Y\cdot X=(A\cdot X)+B\] defines a map $X\mapsto Y.$ Determine what the mapping is. c) Construct all fixed points of the map from b).

Let $\omega$ be the circumcircle of triangle $ABC$ with $AB<AC$. Let $I$ be its incenter and let $M$ be the midpoint of $BC$. The foot of the perpendicular from $M$ to $AI$ is $H$. The lines $MH, BI, AB$ form a triangle $T_b$ and the lines $MH, CI, AC$ form a triangle $T_c$. The circumcircle of $T_b$ meets $\omega$ at $B'$ and the circumcircle of $T_c$ meets $\omega$ at $C'$. Prove that $B', H, C'$ are collinear.

Given a square $ABCD$, points $E$ and $F$ are taken inside the segments $BC$ and $CD$ so that $\angle EAF = 45^o$. The lines $AE$ and $AF$ intersect the circle circumscribed to the square at points $G$ and $H$ respectively. Prove that lines $EF$ and $GH$ are parallel.

In a convex quadrilateral $ABCD$, ${\angle}ABD=65^\circ$,${\angle}CBD=35^\circ$, ${\angle}ADC=130^\circ$ and $BC=AB$.Find the angles of $ABCD$.

Let $\Omega$ be the circumcircle of triangle $ABC$. Suppose that $X$ is a point on the segment $AB$ with $XB=XC$, and the angle bisector of $\angle BAC$ intersects $BC$ and $\Omega$ at $D$, $M$, respectively. If $P$ is a point on $BC$ such that $AP$ is tangent to $\Omega$ and $Q$ is a point on $DX$ such that $CQ$ is tangent to $\Omega$, show that $AB$, $CM$, $PQ$ are concurrent.

At what $n$ can a regular $n$-gon be cut by disjoint diagonals into $n- 2$ isosceles (including equilateral) triangles?

On the side $CD$ of the square $ABCD$, the point $F$ is chosen and the equal squares $DGFE$ and $AKEH$ are constructed ($E$ and $H$ lie inside the square). Let $M$ be the midpoint of $DF$, $J$ is the incenter of the triangle $CFH$. Prove that: a) the points $D, K, H, J, F$ lie on the same circle; b) the circles inscribed in triangles $CFH$ and $GMF$ have the same radii.

The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily similar?

A triangle has area 114 and sides of integer length. What is the perimeter of the triangle?