Found problems: 85335
2012 Sharygin Geometry Olympiad, 6
Let $ABC$ be an isosceles triangle with $BC = a$ and $AB = AC = b$. Segment $AC$ is the base of an isosceles triangle $ADC$ with $AD = DC = a$ such that points $D$ and $B$ share the opposite sides of AC. Let $CM$ and $CN$ be the bisectors in triangles $ABC$ and $ADC$ respectively. Determine the circumradius of triangle $CMN$.
(M.Rozhkova)
2016 Saudi Arabia Pre-TST, 2.3
Let $ABC$ be a non isosceles triangle with circumcircle $(O)$ and incircle $(I)$. Denote $(O_1)$ as the circle internal tangent to $(O)$ at $A_1$ and also tangent to segments $AB,AC$ at $A_b,A_c$ respectively. Define the circles $(O_2), (O_3)$ and the points $B_1, C_1, B_c , B_a, C_a, C_b$ similarly.
1. Prove that $AA_1, BB_1, CC_1$ are concurrent at the point $M$ and $3$ points $I,M,O$ are collinear.
2. Prove that the circle $(I)$ is inscribed in the hexagon with $6$ vertices $A_b,A_c , B_c , B_a, C_a, C_b$.
2022 Centroamerican and Caribbean Math Olympiad, 5
Esteban the alchemist have $8088$ copper pieces, $6066$ bronze pieces, $4044$ silver pieces and $2022$ gold pieces. He can take two pieces of different metals and use a magic hammer to turn them into two pieces of different metals that he take and different each other. Find the largest number of gold pieces that Esteban can obtain after using the magic hammer a finite number of times.
$\textbf{Note:}$ [i]If Esteban takes a copper and bronze pieces, then he turn them into a silver and a gold pieces.[/i]
2005 Thailand Mathematical Olympiad, 21
Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$ ranges over the real numbers.
2021 Cono Sur Olympiad, 2
Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.
2020 Azerbaijan IZHO TST, 2
Consider two circles $k_1,k_2$ touching at point $T$.
A line touches $k_2$ at point $X$ and intersects $k_1$ at points $A,B$ where $B$ lies between $A$ and $X$.Let $S$ be the second intersection point of $k_1$ with $XT$. On the arc $\overarc{TS}$ not containing $A$ and $B$ , a point $C$ is choosen.
Let $CY$ be the tangent line to $k_2$ with $Y\in{k_2}$ , such that the segment $CY$ doesn't intersect the segment $ST$ .If $I=XY\cap{SC}$ , prove that :
$(a)$ the points $C,T,Y,I$ are concyclic.
$(b)$ $I$ is the $A-excenter$ of $\triangle ABC$
2022 Azerbaijan Junior National Olympiad, N2
If $x,y,z \in\mathbb{N}$ and $2x^2+3y^3=4z^4$, Prove that $6|x,y,z$
2020 Iran Team Selection Test, 4
Given a function $g:[0,1] \to \mathbb{R}$ satisfying the property that for every non empty dissection of the trivial $[0,1]$ to subsets $A,B$ we have either $\exists x \in A; g(x) \in B$ or $\exists x \in B; g(x) \in A$ and we have furthermore $g(x)>x$ for $x \in [0,1]$. Prove that there exist infinite $x \in [0,1]$ with $g(x)=1$.
[i]Proposed by Ali Zamani [/i]
2017 QEDMO 15th, 4
Let $a$ be a real number such that $\left(a + \frac{1}{a}\right)^2=11$. What possible values can $a^3 + \frac{1}{a^3}$ and $a^5 + \frac{1}{a^5}$ take?
1998 Gauss, 9
Two numbers have a sum of $32$. If one of the numbers is $ – 36$, what is the other number?
$\textbf{(A)}\ 68 \qquad \textbf{(B)}\ -4 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 72 \qquad \textbf{(E)}\ -68$
1987 Greece Junior Math Olympiad, 4
If $$x+y+z=x^2+y^2+z^2=x^3+y^3+z^3=1 \ \ with \ \ x,y,z\in \mathbb{R},$$ prove that at least one of $x,y,z$ is equal to zero.
2016 ELMO Problems, 6
Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$, respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$. Finally, let $\gamma$ be the circumcircle of $\triangle AST$.
(a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$.
(b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$.
[i]James Lin[/i]
2012 Thailand Mathematical Olympiad, 4
Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at most $1$.
1999 National Olympiad First Round, 17
In a regular pyramid with top point $ T$ and equilateral base $ ABC$, let $ P$, $ Q$, $ R$, $ S$ be the midpoints of $ \left[AB\right]$, $ \left[BC\right]$, $ \left[CT\right]$ and $ \left[TA\right]$, respectively. If $ \left|AB\right| \equal{} 6$ and the altitude of pyramid is equal to $ 2\sqrt {15}$, then area of $ PQRS$ will be
$\textbf{(A)}\ 4\sqrt {15} \qquad\textbf{(B)}\ 8\sqrt {2} \qquad\textbf{(C)}\ 8\sqrt {3} \qquad\textbf{(D)}\ 6\sqrt {5} \qquad\textbf{(E)}\ 9\sqrt {2}$
2004 Hong kong National Olympiad, 3
Points $P$ and $Q$ are taken sides $AB$ and $AC$ of a triangle $ABC$ respectively such that $\hat{APC}=\hat{AQB}=45^{0}$. The line through $P$ perpendicular to $AB$ intersects $BQ$ at $S$, and the line through $Q$ perpendicular to $AC$ intersects $CP$ at $R$. Let $D$ be the foot of the altitude of triangle $ABC$ from $A$. Prove that $SR\parallel BC$ and $PS,AD,QR$ are concurrent.
2001 Macedonia National Olympiad, 3
Let $ABC$ be a scalene triangle and $k$ be its circumcircle. Let $t_A,t_B,t_C$ be the tangents to $k$ at $A, B, C,$ respectively. Prove that points $AB\cap t_C$, $CA\cap t_B$, and $BC\cap t_A$ exist, and that they are collinear.
1995 May Olympiad, 4
We have four white equilateral triangles of $3$ cm on each side and join them by their sides to obtain a triangular base pyramid. At each edge of the pyramid we mark two red dots that divide it into three equal parts. Number the red dots, so that when you scroll them in the order they were numbered, result a path with the smallest possible perimeter. How much does that path measure?
2015 Dutch Mathematical Olympiad, 4
Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$
2025 Korea - Final Round, P3
An acute triangle $\bigtriangleup ABC$ is given which $BC>CA>AB$.
$I$ is the interior and the incircle of $\bigtriangleup ABC$ meets $BC, CA, AB$ at $D,E,F$. $AD$ and $BE$ meet at $P$. Let $l_{1}$ be a tangent from D to the circumcircle of $\bigtriangleup DIP$, and define $l_{2}$ and $l_{3}$ on $E$ and $F$, respectively.
Prove $l_{1},l_{2},l_{3}$ meet at one point.
2014 Dutch BxMO/EGMO TST, 3
In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent
to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more
in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$.
Prove that $|AR|\cdot |BQ|=|P I|^2$
2022 Moldova EGMO TST, 5
Solve the equation in $\mathbb{R}$ $$\left\{\left\{\frac{x^2-x}{2021}\right \}-\left\{\frac{x^2+x}{2022}\right \} \right \}=0.$$
the 7th XMO, 1
As shown in the figure, it is known that $BC = AC$ in $ABC$, $M$ is the midpoint of $AB$, points $D$ and $E$ lie on $AB$ satisfying $\angle DCE = \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F$ (different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$ and $O_2$ respectively. Prove that $O_1O_2\perp CF$.
[img]https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png[/img]
JBMO Geometry Collection, 2016
A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.
2014-2015 SDML (Middle School), 1
The sum of $10$ consecutive integers is $75$. What is the smallest of these $10$ integers?
2006 National Olympiad First Round, 1
Let $ABC$ be an equilateral triangle. $D$ and $E$ are midpoints of $[AB]$ and $[AC]$. The ray $[DE$ cuts the circumcircle of $\triangle ABC$ at $F$. What is $\frac {|DE|}{|DF|}$?
$
\textbf{(A)}\ \frac 12
\qquad\textbf{(B)}\ \frac {\sqrt 3}3
\qquad\textbf{(C)}\ \frac 23(\sqrt 3 - 1)
\qquad\textbf{(D)}\ \frac 23
\qquad\textbf{(E)}\ \frac {\sqrt 5 - 1}2
$