Found problems: 25757
1982 USAMO, 3
If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\triangle{A_1BC}$, prove that \[\operatorname{I.Q.} (A_1BC) > \operatorname{I.Q.} (A_2BC),\] where the [i]isoperrimetric quotient[/i] of a figure $F$ is defined by \[\operatorname{I.Q.}(F) = \frac{\operatorname{Area}(F)}{[\operatorname{Perimeter}(F)]^2}.\]
2002 AMC 12/AHSME, 18
Let $ C_1$ and $ C_2$ be circles defined by
\[ (x \minus{} 10)^2 \plus{} y^2 \equal{} 36\]and
\[ (x \plus{} 15)^2 \plus{} y^2 \equal{} 81,\]respectively. What is the length of the shortest line segment $ \overline{PQ}$ that is tangent to $ C_1$ at $ P$ and to $ C_2$ at $ Q$?
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$
Geometry Mathley 2011-12, 7.1
Let $ABCD$ be a cyclic quadrilateral. Suppose that $E$ is the intersection of $AB$ and $CD, F$ is the intersection of $AD$ and $CB, I$ is the intersection of $AC$ and $BD$. The circumcircles $(FAB), (FCD)$ meet $FI$ at $K, L$. Prove that $EK = EL$
Nguyễn Minh Hà
2023 Sharygin Geometry Olympiad, 10
Altitudes $BE$ and $CF$ of an acute-angled triangle $ABC$ meet at point $H$. The perpendicular from $H$ to $EF$ meets the line $\ell$ passing through $A$ and parallel to $BC$ at point $P$. The bisectors of two angles between $\ell$ and $HP$ meet $BC$ at points $S$ and $T$. Prove that the circumcircles of triangles $ABC$ and $PST$ are tangent.
2022 Assara - South Russian Girl's MO, 7
In a $7\times 7\times 7$ cube, the unit cubes are colored white, black and gray colors so that for any two colors the number of cubes of these two colors are different. In this case, $N$ parallel rows of $7$ cubes were found, each of which there are more white cubes than gray and than black. Likewise, there were $N$ parallel rows of $7$ cubes, each of which contained gray there are more cubes than white and than black, and there are also N parallel rows of $7$ cubes, each of which contains more black cubes than white ones and than gray ones. What is the largest $N$ for which this is possible?
2004 Iran MO (2nd round), 1
$ABC$ is a triangle and $\angle A=90^{\circ}$. Let $D$ be the meet point of the interior bisector of $\angle A$ and $BC$. And let $I_a$ be the $A-$excenter of $\triangle ABC$. Prove that:
\[\frac{AD}{DI_a}\leq\sqrt{2}-1.\]
1952 Czech and Slovak Olympiad III A, 4
Let $p,q$ be positive integers. Consider a rectangle $ABCD$ with lengths of sides $p$ and $q$ that consists of $pq$ unital squares. How many of these squares are crossed by diagonal $AC$?
2019 USA TSTST, 9
Let $ABC$ be a triangle with incenter $I$. Points $K$ and $L$ are chosen on segment $BC$ such that the incircles of $\triangle ABK$ and $\triangle ABL$ are tangent at $P$, and the incircles of $\triangle ACK$ and $\triangle ACL$ are tangent at $Q$. Prove that $IP=IQ$.
[i]Ankan Bhattacharya[/i]
1975 All Soviet Union Mathematical Olympiad, 206
Given a triangle $ABC$ with the unit area. The first player chooses a point $X$ on the side $[AB]$, than the second -- $Y$ on $[BC]$ side, and, finally, the first chooses a point $Z$ on $[AC]$ side. The first tries to obtain the greatest possible area of the $XYZ$ triangle, the second -- the smallest. What area can obtain the first for sure and how?
2012 ELMO Shortlist, 1
In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$.
[i]Ray Li.[/i]
2023 JBMO TST - Turkey, 2
Let $ABC$ is acute angled triangle and $K,L$ is points on $AC,BC$ respectively such that $\angle{AKB}=\angle{ALB}$. $P$ is intersection of $AL$ and $BK$ and $Q$ is the midpoint of segment $KL$. Let $T,S$ are the intersection $AL,BK$ with $(ABC)$ respectively. Prove that $TK,SL,PQ$ are concurrent.
1991 Bundeswettbewerb Mathematik, 4
A strip of width $1$ is to be divided by rectangular panels of common width $1$ and denominations long $a_1$, $a_2$, $a_3$, $. . .$ be paved without gaps ($a_1 \ne 1$). From the second panel on, each panel is similar but not congruent to the already paved part of the strip. When the first $n$ slabs are laid, the length of the paved part of the strip is $sn$. Given $a_1$, is there a number that is not surpassed by any $s_n$? The accuracy answer has to be proven.
2014 NIMO Problems, 3
Let $ABCD$ be a square with side length $2$. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{CD}$ respectively, and let $X$ and $Y$ be the feet of the perpendiculars from $A$ to $\overline{MD}$ and $\overline{NB}$, also respectively. The square of the length of segment $\overline{XY}$ can be written in the form $\tfrac pq$ where $p$ and $q$ are positive relatively prime integers. What is $100p+q$?
[i]Proposed by David Altizio[/i]
2011 Pre-Preparation Course Examination, 1
[b]a)[/b] prove that for every compressed set $K$ in the space $\mathbb R^3$, the function $f:\mathbb R^3 \longrightarrow \mathbb R$ that $f(p)=inf\{|p-k|,k\in K\}$ is continuous.
[b]b)[/b] prove that we cannot cover the sphere $S^2\subseteq \mathbb R^3$ with it's three closed sets, such that none of them contain two antipodal points.
2006 Sharygin Geometry Olympiad, 12
In the triangle $ABC$, the bisector of angle $A$ is equal to the half-sum of the height and median drawn from vertex $A$. Prove that if $\angle A$ is obtuse, then $AB = AC$.
2009 USAMTS Problems, 5
Let $ABC$ be a triangle with $AB = 3, AC = 4,$ and $BC = 5$, let $P$ be a point on $BC$, and let $Q$ be the point (other than $A$) where the line through $A$ and $P$ intersects the circumcircle of $ABC$. Prove that
\[PQ\le \frac{25}{4\sqrt{6}}.\]
2015 Auckland Mathematical Olympiad, 4
The bisector of angle $A$ in parallelogram $ABCD$ intersects side $BC$ at $M$ and the bisector of $\angle AMC$ passes through point $D$. Find angles of the parallelogram if it is known that $\angle MDC = 45^o$.
[img]https://cdn.artofproblemsolving.com/attachments/e/7/7cfb22f0c26fe39aa3da3898e181ae013a0586.png[/img]
2014 BMT Spring, 17
A convex solid is formed in four-dimensional Euclidean space with vertices at the $24$ possible permutations of $\{1, 2, 3, 4\}$ (so $(1, 2, 3, 4)$, $(1, 2, 4, 3)$, etc.). What is the product of the number of faces and edges of this solid?
2020 IOM, 6
In convex pentagon $ABCDE$ points $A_1$, $B_1$, $C_1$, $D_1$, $E_1$ are intersections of pairs of diagonals $(BD, CE)$, $(CE, DA)$, $(DA, EB)$, $(EB, AC)$ and $(AC, BD)$ respectively. Prove that if four of quadrilaterals $AB_{1}A_{1}B$, $BC_{1}B_{1}C$, $CD_{1}C_{1}D$, $DE_{1}D_{1}E$ and $EA_{1}E_{1}A$ are cyclic then the fifth one is also cyclic.
1988 Greece National Olympiad, 2
Let $ABC$ be a triangle inscribed in circle $C(O,R)$. Let $M$ ba apoint on the arc $BC$ . Let $D,E,Z$ be the feet of the perpendiculars drawn from $M$ on lines $AB,AC,BC$ respectively. Prove that $\frac{(BC)^2}{(MZ)^2} \ge 8\frac{R U_a}{(MD)\cdot(ME)}$ where $U_a$ is the altitude drawn on $BC$.
Kvant 2019, M2553
A circle centred at $I$ is tangent to the sides $BC, CA$, and $AB$ of an acute-angled triangle $ABC$ at $A_1, B_1$, and $C_1$, respectively. Let $K$ and $L$ be the incenters of the quadrilaterals $AB_1IC_1$ and $BA_1IC_1$, respectively. Let $CH$ be an altitude of triangle $ABC$. Let the internal angle bisectors of angles $AHC$ and $BHC$ meet the lines $A_1C_1$ and $B_1C_1$ at $P$ and $Q$, respectively. Prove that $Q$ is the orthocenter of the triangle $KLP$.
Kolmogorov Cup 2018, Major League, Day 3, Problem 1; A. Zaslavsky
2015 Czech-Polish-Slovak Junior Match, 1
Let $I$ be the center of the circle of the inscribed triangle $ABC$ and $M$ be the center of its side $BC$.
If $|AI| = |MI|$, prove that there are two of the sides of triangle $ABC$, of which one is twice of the other.
2021 Yasinsky Geometry Olympiad, 6
Three lines were drawn through the point $X$ in space. These lines crossed some sphere at six points. It turned out that the distances from point $X$ to some five of them are equal to $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm. What can be the distance from point $X$ to the sixth point?
(Alexey Panasenko)
2022 Thailand TSTST, 3
An acute scalene triangle $ABC$ with circumcircle $\Omega$ is given. The altitude from $B$ intersects side $AC$ at $B_1$ and circle $\Omega$ at $B_2$. The circle with diameter $B_1B_2$ intersects circle $\Omega$ again at $B_3$. Similarly, the altitude from $C$ intersects side $AB$ at $C_1$ and circle $\Omega$ at $C_2$. The circle with diameter $C_1C_2$ intersects circle $\Omega$ again at $C_3$. Let $X$ be the intersection of lines $B_1B_3$ and $C_1C_3$, and let $Y$ be the intersection of lines $B_3C$ and $C_3B$. Prove that line $XY$ bisects side $BC$.
May Olympiad L2 - geometry, 2003.5
An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.