Found problems: 25757
1974 Putnam, A2
A circle stands in a plane perpendicular to the ground and a point $A$ lies in this plane exterior to the circle and higher than its bottom. A particle starting from rest at $A$ slides without friction down an inclined straight line until it reaches the circle. Which straight line allows descent in the shortest time?
1968 Miklós Schweitzer, 10
Let $ h$ be a triangle of perimeter $ 1$, and let $ H$ be a triangle of perimeter $ \lambda$ homothetic to $ h$. Let $ h_1,h_2,...$ be translates of $ h$ such that , for all $ i$, $ h_i$ is different from $ h_{i\plus{}2}$ and touches $ H$ and $ h_{i\plus{}1}$ (that is, intersects without overlapping). For which values of $ \lambda$ can these triangles be chosen so that the sequence $ h_1,h_2,...$ is periodic? If $ \lambda \geq 1$ is such a value, then determine the number of different triangles in a periodic
chain $ h_1,h_2,...$ and also the number of times such a chain goes around the triangle $ H$.
[i]L. Fejes-Toth[/i]
1983 Poland - Second Round, 3
The point $ P $ lies inside the triangle $ ABC $, with $ \measuredangle PAC = \measuredangle PBC $. The points $ L $ and $ M $ are the projections $ P $ onto the lines $ BC $ and $ CA $, respectively, $ D $ is the midpoint of the segment $ AB $. Prove that $ DL = DM $.
2021 CHMMC Winter (2021-22), 7
Let $ABC$ be a triangle with $AB = 5$, $BC = 6$, and $CA = 7$. Denote $\Gamma$ the incircle of $ABC$, let $I$ be the center of $\Gamma$ . The circumcircle of $BIC$ intersects $\Gamma$ at $X_1$ and $X_2$. The circumcircle of $CIA$ intersects $\Gamma$ at $Y_1$ and $Y_2$. The circumcircle of $AIB$ intersects $\Gamma$ at $Z_1$ and $Z_2$. The area of the triangle determined by $\overline{X_1X_2}$, $\overline{Y_1Y_2}$, and $\overline{Z_1Z_2}$ equals $\frac{m \sqrt{p}}{n}$ for positive integers $m, n$, and $p$, where $m$ and$ n$ are relatively prime and $p$ is squarefree.
Compute $m+n+ p$.
1999 IMO Shortlist, 8
Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.
1988 IMO Shortlist, 12
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that
\[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
\]
2015 JBMO Shortlist, 2
The point ${P}$ is outside the circle ${\Omega}$. Two tangent lines, passing from the point ${P}$ touch the circle ${\Omega}$ at the points ${A}$ and ${B}$. The median${AM \left(M\in BP\right)}$ intersects the circle ${\Omega}$ at the point ${C}$ and the line ${PC}$ intersects again the circle ${\Omega}$ at the point ${D}$. Prove that the lines ${AD}$ and ${BP}$ are parallel.
(Moldova)
1969 Yugoslav Team Selection Test, Problem 6
Let $E$ be the set of $n^2+1$ closed intervals on the real axis. Prove that there exists a subset of $n+1$ intervals that are monotonically increasing with respect to inclusion, or a subset of $n+1$ intervals none of which contains any other interval from the subset.
2020 Bulgaria National Olympiad, P1
On the sides of $\triangle{ABC}$ points $P,Q \in{AB}$ ($P$ is between $A$ and $Q$) and $R\in{BC}$ are chosen. The points $M$ and $N$ are defined as the intersection point of $AR$ with the segments $CP$ and $CQ$, respectively. If $BC=BQ$, $CP=AP$, $CR=CN$ and $\angle{BPC}=\angle{CRA}$, prove that $MP+NQ=BR$.
1977 IMO Shortlist, 12
In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.
1989 Romania Team Selection Test, 4
Let $A,B,C$ be variable points on edges $OX,OY,OZ$ of a trihedral angle $OXYZ$, respectively.
Let $OA = a, OB = b, OC = c$ and $R$ be the radius of the circumsphere $S$ of $OABC$.
Prove that if points $A,B,C$ vary so that $a+b+c = R+l$, then the sphere $S$ remains tangent to a fixed sphere.
2009 Moldova National Olympiad, 7.4
Triangle $ABC$ with $AB = 10$ cm ¸and $\angle C= 15^o$, is right at $B$. Point $D \in (AC)$ is the foot of the altitude taken from $B$. Find the distance from point $D$ to the line $AB$.
1989 Putnam, A5
Show that we can find $\alpha>0$ such that, given any point $P$ inside a regular $2n+1$-gon which is inscribed in a circle radius $1$, we can find two vertices of the polygon whose distance from $P$ differ by less than $\frac1n-\frac\alpha{n^3}$.
2023 Chile National Olympiad, 6
Let $\vartriangle ABC$ be a triangle such that $\angle ABC = 30^o$, $\angle ACB = 15^o$. Let $M$ be midpoint of segment $BC$ and point $N$ lies on segment $MC$, such that the length of $NC$ is equal to length of $AB$. Proce that $AN$ is the bisector of the angle $\angle MAC$.
[img]https://cdn.artofproblemsolving.com/attachments/2/7/4c554b53f03288ee69931fdd2c6fbd3e27ab13.png[/img]
2017 India PRMO, 13
In a rectangle $ABCD, E$ is the midpoint of $AB, F$ is a point on $AC$ such that $BF$ is perpendicular to $AC$, and $FE$ perpendicular to $BD$. Suppose $BC = 8\sqrt3$. Find $AB$.
2013 Sharygin Geometry Olympiad, 4
Given a square cardboard of area $\frac{1}{4}$, and a paper triangle of area $\frac{1}{2}$ such that the square of its sidelength is a positive integer. Prove that the triangle can be folded in some ways such that the squace can be placed inside the folded figure so that both of its faces are completely covered with paper.
[i]Proposed by N.Beluhov, Bulgaria[/i]
XMO (China) 2-15 - geometry, 9.2
Given a $\triangle ABC$ with circumcenter $O$ and orthocenter $H(O\ne H)$. Denote the midpoints of $BC, AC$ as $D, E$ and let $D', E'$ be the reflections of $D, E$ w.r.t. point $H$, respectively. If lines $AD'$ and $BE'$ meet at $K$, compute $\frac{KO}{KH}$.
2007 Tournament Of Towns, 6
Let $P$ and $Q$ be two convex polygons. Let $h$ be the length of the projection of $Q$ onto a line perpendicular to a side of $P$ which is of length $p$. Define $f(P,Q)$ to be the sum of the products $hp$ over all sides of $P$. Prove that $f(P,Q) = f(Q, P)$.
1991 Brazil National Olympiad, 2
$P$ is a point inside the triangle $ABC$. The line through $P$ parallel to $AB$ meets $AC$ $A_0$ and $BC$ at $B_0$. Similarly, the line through $P$ parallel to $CA$ meets $AB$ at $A_1$ and $BC$ at $C_1$, and the line through P parallel to BC meets $AB$ at $B_2$ and $AC$ at $C_2$. Find the point $P$ such that $A_0B_0 = A_1B_1 = A_2C_2$.
2011 Belarus Team Selection Test, 2
The external angle bisector of the angle $A$ of an acute-angled triangle $ABC$ meets the circumcircle of $\vartriangle ABC$ at point $T$. The perpendicular from the orthocenter $H$ of $\vartriangle ABC$ to the line $TA$ meets the line $BC$ at point $P$. The line $TP$ meets the circumcircce of $\vartriangle ABC$ at point $D$. Prove that $AB^2+DC^2=AC^2+BD^2$
A. Voidelevich
2013 Oral Moscow Geometry Olympiad, 6
The trapezoid $ABCD$ is inscribed in the circle $w$ ($AD // BC$). The circles inscribed in the triangles $ABC$ and $ABD$ touch the base of the trapezoid $BC$ and $AD$ at points $P$ and $Q$ respectively. Points $X$ and $Y$ are the midpoints of the arcs $BC$ and $AD$ of circle $w$ that do not contain points $A$ and $B$ respectively. Prove that lines $XP$ and $YQ$ intersect on the circle $w$.
1998 All-Russian Olympiad Regional Round, 9.8
The endpoints of a compass are at two lattice points of an infinite unit square
grid. It is allowed to rotate the compass around one of its endpoints, not varying
its radius, and thus move the other endpoint to another lattice point. Can the
endpoints of the compass change places after several such steps?
1994 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 4
Two circles with radii 1 and 2 touch each other and a line as in the figure. In the region between the circles and the line, there is a circle with radius $ r$ which touches the two circles and the line. What is $ r$?
[img]http://i250.photobucket.com/albums/gg265/geometry101/GeometryImage2.jpg[/img]
A. 1/3
B. $ \frac {1}{\sqrt {5}}$
C. $ \sqrt {3} \minus{} \sqrt {2}$
D. $ 6 \minus{} 4 \sqrt {2}$
E. None of these
VMEO III 2006, 12.1
Given a circle $(O)$ and a point $P$ outside that circle. $M$ is a point running on the circle $(O)$. The circle with center $I$ and diameter $PM$ intersects circle $(O)$ again at $N$. The tangent of $(I)$ at $P$ intersects $MN$ at $Q$. The line through $Q$ perpendicular to $PO$ intersects $PM$ at $ A$. $AN$ intersects $(O)$ further at $ B$. $BM$ intersects $PO$ at $C$. Prove that $AC$ is perpendicular to $OQ$.
2016 Sharygin Geometry Olympiad, P15
Let $O, M, N$ be the circumcenter, the centroid and the Nagel point of a triangle. Prove that angle $MON$ is right if and only if one of the triangle’s angles is equal to $60^o$.