Found problems: 25757
2023 APMO, 3
Let $ABCD$ be a parallelogram. Let $W, X, Y,$ and $Z$ be points on sides $AB, BC, CD,$ and $DA$, respectively, such that the incenters of triangles $AWZ, BXW, CYX,$ and $DZY$ form a parallelogram. Prove that $WXYZ$ is a parallelogram.
Ukrainian TYM Qualifying - geometry, 2020.13
In the triangle $ABC$ on the side $BC$, the points$ D$ and $E$ are chosen so that the angle $BAD$ is equal to the angle $EAC$. Let $I$ and $J$ be the centers of the inscribed circles of triangles $ABD$ and $AEC$ respectively, $F$ be the point of intersection of $BI$ and $EJ$, $G$ be the point of intersection of $DI$ and $CJ$. Prove that the points $I, J, F, G$ lie on one circle, the center of which belongs to the line $I_bI_c$, where $I_b$ and $I_c$ are the centers of the exscribed circles of the triangle $ABC$, which touch respectively sides $AC$ and $AB$.
Swiss NMO - geometry, 2013.10
Let $ABCD$ be a tangential quadrilateral with $BC> BA$. The point $P$ is on the segment $BC$, such that $BP = BA$ . Show that the bisector of $\angle BCD$, the perpendicular on line $BC$ through $P$ and the perpendicular on $BD$ through $A$, intersect at one point.
2023 ELMO Shortlist, G5
Let \(ABC\) be an acute triangle with circumcircle \(\omega\). Let \(P\) be a variable point on the arc \(BC\) of \(\omega\) not containing \(A\). Squares \(BPDE\) and \(PCFG\) are constructed such that \(A\), \(D\), \(E\) lie on the same side of line \(BP\) and \(A\), \(F\), \(G\) lie on the same side of line \(CP\). Let \(H\) be the intersection of lines \(DE\) and \(FG\). Show that as \(P\) varies, \(H\) lies on a fixed circle.
[i]Proposed by Karthik Vedula[/i]
2005 Kyiv Mathematical Festival, 4
Let $ M$ be the intersection point of medians of a triangle $ \triangle ABC.$ It is known that $ AC \equal{} 2BC$ and $ \angle ACM \equal{} \angle CBM.$ Find $ \angle ACB.$
2009 ISI B.Stat Entrance Exam, 9
Consider $6$ points located at $P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)$. Let $R$ be the region consisting of [i]all[/i] points in the plane whose distance from $P_0$ is smaller than that from any other $P_i$, $i=1,2,3,4,5$. Find the perimeter of the region $R$.
Azerbaijan Al-Khwarizmi IJMO TST 2025, 1
In isosceles triangle, the condition $AB=AC>BC$ is satisfied. Point $D$ is taken on the circumcircle of $ABC$ such that $\angle CAD=90^{\circ}$.A line parallel to $AC$ which passes from $D$ intersects $AB$ and $BC$ respectively at $E$ and $F$.Show that circumcircle of $ADE$ passes from circumcenter of $DFC$.
2020 USA IMO Team Selection Test, 6
Let $P_1P_2\dotsb P_{100}$ be a cyclic $100$-gon and let $P_i = P_{i+100}$ for all $i$. Define $Q_i$ as the intersection of diagonals $\overline{P_{i-2}P_{i+1}}$ and $\overline{P_{i-1}P_{i+2}}$ for all integers $i$.
Suppose there exists a point $P$ satisfying $\overline{PP_i}\perp\overline{P_{i-1}P_{i+1}}$ for all integers $i$. Prove that the points $Q_1,Q_2,\dots, Q_{100}$ are concyclic.
[i]Michael Ren[/i]
2017 AMC 12/AHSME, 24
Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$. What is $\tfrac{AB}{BC}$?
$\textbf{(A) \ } 1+\sqrt{2} \qquad \textbf{(B) \ } 2+\sqrt{2}\qquad \textbf{(C) \ } \sqrt{17}\qquad \textbf{(D) \ } 2+\sqrt{5} \qquad \textbf{(E) \ } 1+2\sqrt{3}$
1986 IMO Shortlist, 17
Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.
2001 Kurschak Competition, 3
In a square lattice let us take a lattice triangle that has the smallest area among all the lattice triangles similar to it. Prove that the circumcenter of this triangle is not a lattice point.
1974 Dutch Mathematical Olympiad, 1
A convex quadrilateral with area $1$ is divided into four quadrilaterals divided by connecting the midpoints of the opposite sides. Prove that each of those four quadrilaterals has area $< \frac38$.
2021 Stanford Mathematics Tournament, 8
In triangle $\vartriangle ABC$, $AB = 5$, $BC = 7$, and $CA = 8$. Let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively, and let $M$ be the midpoint of $BC$. The area of triangle $MEF$ can be expressed as $\frac{a \sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ such that the greatest common divisor of $a$ and $c$ is $1$ and $b$ is not divisible by the square of any prime. Compute $a + b + c$.
2018 PUMaC Geometry A, 8
Let $\omega$ be a circle. Let $E$ be on $\omega$ and $S$ outside $\omega$ such that line segment $SE$ is tangent to $\omega$. Let $R$ be on $\omega$. Let line $SR$ intersect $\omega$ at $B$ other than $R$, such that $R$ is between $S$ and $B$. Let $I$ be the intersection of the bisector of $\angle ESR$ with the line tangent to $\omega$ at $R$; let $A$ be the intersection of the bisector of $\angle ESR$ with $ER$. If the radius of the circumcircle of $\triangle EIA$ is $10$, the radius of the circumcircle of $\triangle SAB$ is $14$, and $SA = 18$, then $IA$ can be expressed in simplest form as $\frac{m}{n}$. Find $m + n$.
2005 MOP Homework, 6
Given a convex quadrilateral $ABCD$. The points $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$ respectively. The line $PQ$ intersects the lines $AB$ and $CD$ at $N$ and $M$ respectively. Prove that the circumcircles of triangles $NAP$, $NBQ$, $MQD$, and $MPC$ have a common point.
1949-56 Chisinau City MO, 33
Construct a triangle, the base of which lies on the given line, and the feet of the altitudes, drawn on the sides, coincide with the given points.
2010 All-Russian Olympiad Regional Round, 9.4
In triangle $ABC$, $\angle A =60^o$. Let $BB_1$ and $CC_1$ be angle bisectors of this triangle. Prove that the point symmetrical to vertex $A$ with respect to line $B_1C_1$ lies on side $BC$.
1963 AMC 12/AHSME, 25
Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches and the area of triangle $CEF$ is $200$ square inches. Then the number of inches in $BE$ is:
[asy]
size(6cm);
pair A = (0, 0), B = (1, 0), C = (1, 1), D = (0, 1), E = (1.3, 0), F = (0, 0.7);
draw(A--B--C--D--cycle);
draw(F--C--E--B);
label("$A$", A, SW);
label("$B$", B, S);
label("$C$", C, N);
label("$D$", D, NW);
label("$E$", E, SE);
label("$F$", F, W);
//Credit to MSTang for the asymptote
[/asy]
$\textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 14 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 20$
2004 Postal Coaching, 11
Three circles touch each other externally and all these cirlces also touch a fixed straight line. Let $A,B,C$ be the mutual points of contact of these circles. If $\omega$ denotes the Brocard angle of the triangle $ABC$, prove that $\cot{\omega}$ = 2.
2022 Kyiv City MO Round 1, Problem 2
There are $n$ sticks which have distinct integer length. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What's the largest possible value of $n$ under such conditions?
[i](Proposed by Bogdan Rublov)[/i]
2014-2015 SDML (Middle School), 1
Given that each unit square in the grid below is a $1\times1$ square, find the area of the shaded region in square units.
[asy]
fill((3,0)--(4,0)--(6,3)--(4,4)--(4,3)--(0,2)--(2,2)--cycle, grey);
draw((0,0)--(6,0));
draw((0,1)--(6,1));
draw((0,2)--(6,2));
draw((0,3)--(6,3));
draw((0,4)--(6,4));
draw((0,0)--(0,4));
draw((1,0)--(1,4));
draw((2,0)--(2,4));
draw((3,0)--(3,4));
draw((4,0)--(4,4));
draw((5,0)--(5,4));
draw((6,0)--(6,4));
[/asy]
$\text{(A) }8\qquad\text{(B) }9\qquad\text{(C) }10\qquad\text{(D) }11\qquad\text{(E) }12$
2010 AIME Problems, 15
In $ \triangle{ABC}$ with $ AB = 12$, $ BC = 13$, and $ AC = 15$, let $ M$ be a point on $ \overline{AC}$ such that the incircles of $ \triangle{ABM}$ and $ \triangle{BCM}$ have equal radii. Let $ p$ and $ q$ be positive relatively prime integers such that $ \tfrac{AM}{CM} = \tfrac{p}{q}$. Find $ p + q$.
2007 All-Russian Olympiad Regional Round, 8.1
In a convex quadrilateral. eight segments are drawn, each of them connects a vertex with the midpoint of some opposite side. Seven of these segments have the same length $ a$. Prove that the eight one is also of length $ a$.
2009 Today's Calculation Of Integral, 506
Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$. Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$.
1984 Tournament Of Towns, (061) O2
Six altitudes are constructed from the three vertices of the base of a tetrahedron to the opposite sides of the three lateral faces. Prove that all three straight lines joining two base points of the altitudes in each lateral face are parallel to a certain plane.
(IF Sharygin, Moscow)