Found problems: 25757
2009 Stanford Mathematics Tournament, 5
In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has two
lanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane,
and Sammy on the outer. They will race for one complete lap, measured by the inner track.
What is the square of the distance between Willy and Sammy's starting positions so that they will both race
the same distance? Assume that they are of point size and ride perfectly along their respective lanes
2024 BMT, 8
Points $A, B, C, D,$ and $F$ lie on a sphere with radius $\sqrt{10}$ such that lines $AD, BE,$ and $CF$ are concurrent at point $P$ inside the sphere and are pairwise perpendicular. If $PA=\sqrt{6}, PB=\sqrt{10},$ and $PC=\sqrt{15},$ what is the volume of tetrahedron $DEFP$?
1982 IMO Longlists, 5
Among all triangles with a given perimeter, find the one with the maximal radius of its incircle.
2016 Kosovo National Mathematical Olympiad, 5
In trapezoid $ABCD$ with $AB$ parallel to $CD$ show that :
$\frac{|AB|^2-|BC|^2+|AC|^2}{|CD|^2-|AD|^2+|AC|^2}=\frac{|AB|}{|CD|}=\frac{|AB|^2-|AD|^2+|BD|^2}{|CD|^2-|BC|^2+|BD|^2}$
2021 Latvia Baltic Way TST, P12
Five points $A,B,C,P,Q$ are chosen so that $A,B,C$ aren't collinear. The following length conditions hold: $\frac{AP}{BP}=\frac{AQ}{BQ}=\frac{21}{20}$ and $\frac{BP}{CP}=\frac{BQ}{CQ}=\frac{20}{19}$. Prove that line $PQ$ goes through the circumcentre of $\triangle ABC$.
2010 Sharygin Geometry Olympiad, 20
The incircle of an acute-angled triangle $ABC$ touches $AB, BC, CA$ at points $C_1, A_1, B_1$ respectively. Points $A_2, B_2$ are the midpoints of the segments $B_1C_1, A_1C_1$ respectively. Let $P$ be a common point of the incircle and the line $CO$, where $O$ is the circumcenter of triangle $ABC.$ Let also $A'$ and $B'$ be the second common points of $PA_2$ and $PB_2$ with the incircle. Prove that a common point of $AA'$ and $BB'$ lies on the altitude of the triangle dropped from the vertex $C.$
1965 German National Olympiad, 6
Let $\alpha,\beta, \gamma$ be the angles of a triangle. Prove that $\cos\alpha, + \cos\beta + \cos\gamma \le \frac{3}{2} $ and find the cases of equality.
2004 Estonia National Olympiad, 5
Three different circles of equal radii intersect in point $Q$. The circle $C$ touches all of them. Prove that $Q$ is the center of $C$.
2014 Contests, 3
Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to $AC$ if $DE$ is parallel to $AC$.
2005 Federal Math Competition of S&M, Problem 2
Let $ABC$ be an acute triangle. Circle $k$ with diameter $AB$ intersects $AC$ and $BC$ again at $M$ and $N$ respectively. The tangents to $k$ at $M$ and $N$ meet at point $P$. Given that $CP=MN$, determine $\angle ACB$.
Geometry Mathley 2011-12, 1.4
Given are three circles $(O_1), (O_2), (O_3)$, pairwise intersecting each other, that is, every single circle meets the other two circles at two distinct points. Let $(X_1)$ be the circle externally tangent to $(O_1)$ and internally tangent to the circles $(O_2), (O_3),$ circles $(X_2), (X_3)$ are defined in the same manner. Let $(Y_1)$ be the circle internally tangent to $(O_1)$ and externally tangent to the circles $(O_2), (O_3)$, the circles $(Y_2), (Y_3)$ are defined in the same way. Let $(Z_1), (Z_2)$ be two circles internally tangent to all three circles $(O_1), (O_2), (O_3)$. Prove that the four lines $X_1Y_1, X_2Y_2, X_3Y_3, Z_1Z_2$ are concurrent.
Nguyễn Văn Linh
1978 IMO Shortlist, 4
Let $T_1$ be a triangle having $a, b, c$ as lengths of its sides and let $T_2$ be another triangle having $u, v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that
\[16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).\]
When does equality hold?
2023 Iran MO (3rd Round), 3
In triangle $\triangle ABC$ points $M,N$ lie on $BC$ st : $\angle BAM= \angle MAN= \angle NAC$ . Points $P,Q$ are on the angle bisector of $BAC$, on the same side of $BC$ as A , st :
$$\frac{1}{3} \angle BAC = \frac{1}{2} \angle BPC = \angle BQC$$
Let $E = AM \cap CQ$ and $F = AN \cap BQ$ . Prove that the common tangents to $(EPF), (EQF)$ and the circumcircle of $\triangle ABC$ , are concurrent.
2004 Bulgaria Team Selection Test, 1
The points $P$ and $Q$ lie on the diagonals $AC$ and $BD$, respectively, of a quadrilateral $ABCD$ such that $\frac{AP}{AC} + \frac{BQ}{BD} =1$. The line $PQ$ meets the sides $AD$ and $BC$ at points $M$ and $N$. Prove that the circumcircles of the triangles $AMP$, $BNQ$, $DMQ$, and $CNP$ are concurrent.
2022-IMOC, G4
Let $\vartriangle ABC$ be an acute triangle with circumcircle $\Omega$. A line passing through $A$ perpendicular to $BC$ meets $\Omega$ again at $D$. Draw two circles $\omega_b$, $\omega_c$ with $B, C$ as centers and $BD$, $CD$ as radii, respectively, and they intersect $AB$, $AC$ at $E, F,$ respectively. Let $K\ne A$ be the second intersection of $(AEF)$ and $\Omega$, and let $\omega_b$, $\omega_c$ intersect $KB$, $KC$ at $P, Q$, respectively. The circumcenter of triangle $DP Q$ is $O$, prove that $K, O, D$ are collinear.
[i]proposed by Li4[/i]
2002 Czech and Slovak Olympiad III A, 4
Find all pairs of real numbers $a, b$ for which the equation in the domain of the real numbers
\[\frac{ax^2-24x+b}{x^2-1}=x\]
has two solutions and the sum of them equals $12$.
2015 Princeton University Math Competition, A3
Let $I$ be the incenter of a triangle $ABC$ with $AB = 20$, $BC = 15$, and $BI = 12$. Let $CI$ intersect the circumcircle $\omega_1$ of $ABC$ at $D \neq C $. Alice draws a line $l$ through $D$ that intersects $\omega_1$ on the minor arc $AC$ at $X$ and the circumcircle $\omega_2$ of $AIC$ at $Y$ outside $\omega_1$. She notices that she can construct a right triangle with side lengths $ID$, $DX$, and $XY$. Determine, with proof, the length of $IY$.
1996 Czech and Slovak Match, 3
The base of a regular quadrilateral pyramid $\pi$ is a square with side length $2a$ and its lateral edge has length a$\sqrt{17}$. Let $M$ be a point inside the pyramid. Consider the five pyramids which are similar to $\pi$ , whose top vertex is at $M$ and whose bases lie in the planes of the faces of $\pi$ . Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of $\pi$ , and find for which $M$ equality holds.
2017 Kosovo Team Selection Test, 5
Given triangle $ABC$. Let $P$, $Q$, $R$, be the tangency points of inscribed circle of $\triangle ABC$ on sides $AB$, $BC$, $AC$ respectively. We take the reflection of these points with respect to midpoints of the sides they lie on, and denote them as $P',Q'$ and $R'$. Prove that $AP'$, $BQ'$, and $CR'$ are concurrent.
2010 Argentina Team Selection Test, 4
Two players, $A$ and $B$, play a game on a board which is a rhombus of side $n$ and angles of $60^{\circ}$ and $120^{\circ}$, divided into $2n^2$ equilateral triangles, as shown in the diagram for $n=4$.
$A$ uses a red token and $B$ uses a blue token, which are initially placed in cells containing opposite corners of the board (the $60^{\circ}$ ones). In turns, players move their token to a neighboring cell (sharing a side with the previous one). To win the game, a player must either place his token on the cell containing the other player's token, or get to the opposite corner to the one where he started.
If $A$ starts the game, determine which player has a winning strategy.
2010 Today's Calculation Of Integral, 560
Let $ K$ be the figure bounded by the graph of function $ y \equal{} \frac {x}{\sqrt {1 \minus{} x^2}}$, $ x$ axis and the line $ x \equal{} \frac {1}{2}$.
(1) Find the volume $ V_1$ of the solid generated by rotation of $ K$ around $ x$ axis.
(2) Find the volume $ V_2$ of the solid generated by rotation of $ K$ around $ y$ axis.
Please solve question (2) without using the shell method for Japanese High School Students those who don't learn it.
2024 Sharygin Geometry Olympiad, 10.4
Let $I$ be the incenter of a triangle $ABC$. The lines passing through $A$ and parallel to $BI, CI$ meet the perpendicular bisector to $AI$ at points $S, T$ respectively. Let $Y$ be the common point of $BT$ and $CS$, and $A^*$ be a point such that $BICA^*$ is a parallelogram. Prove that the midpoint of segment $YA^*$ lies on the excircle of the triangle touching the side $BC$.
2018 Moscow Mathematical Olympiad, 4
$ABCD$ is convex and $AB\not \parallel CD,BC \not \parallel DA$. $P$ is variable point on $AD$. Circumcircles of $\triangle ABP$ and $\triangle CDP$ intersects at $Q$. Prove, that all lines $PQ$ goes through fixed point.
2011 HMNT, 7
Let $XY Z$ be a triangle with $\angle XY Z = 40^o$ and $\angle Y ZX = 60^o$. A circle $\Gamma$, centered at the point $I$, lies inside triangle $XY Z$ and is tangent to all three sides of the triangle. Let $A$ be the point of tangency of $\Gamma$ with $Y Z$, and let ray $\overrightarrow{XI}$ intersect side $Y Z$ at $B$. Determine the measure of $\angle AIB$.
2014 BAMO, 1
The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.