Found problems: 25757
1988 Polish MO Finals, 3
Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.
2021 Science ON grade IX, 2
Consider the acute-angled triangle $ABC$, with orthocentre $H$ and circumcentre $O$. $D$ is the intersection point of lines $AH$ and $BC$ and $E$ lies on $\overline{AH}$ such that $AE=DH$.
Suppose $EO$ and $BC$ meet at $F$. Prove that $BD=CF$.
[i] (Călin Pop & Vlad Robu) [/i]
2021 Korea Winter Program Practice Test, 2
Let $ABC$ be a triangle with $\angle A=60^{\circ}$. Point $D, E$ in lines $\overrightarrow{AB}, \overrightarrow{AC}$ respectively satisfies $DB=BC=CE$. ($A,B,D$ lies on this order, and $A,C,E$ likewise) Circle with diameter $BC$ and circle with diameter $DE$ meets at two points $X, Y$. Prove that $\angle XAY\ge 60^{\circ}$
1990 IMO Shortlist, 17
Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $ A$ be the beginning vertex and $ B$ be the end vertex. Let there be $ p \times q \times r$ cubes on the string $ (p, q, r \geq 1).$
[i](a)[/i] Determine for which values of $ p, q,$ and $ r$ it is possible to build a block with dimensions $ p, q,$ and $ r.$ Give reasons for your answers.
[i](b)[/i] The same question as (a) with the extra condition that $ A \equal{} B.$
2020 Purple Comet Problems, 15
Daniel had a string that formed the perimeter of a square with area $98$. Daniel cut the string into two pieces. With one piece he formed the perimeter of a rectangle whose width and length are in the ratio $2 : 3$. With the other piece he formed the perimeter of a rectangle whose width and length are in the ratio $3 : 8$. The two rectangles that Daniel formed have the same area, and each of those areas is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2007 Nicolae Coculescu, 3
Let $ M,N $ be points on the segments $ AB,AC, $ respectively, of the triangle $ ABC. $ Also, let $ P,Q, $ be the midpoints of the segments $ MN,BC, $ respectively. Knowing that $ PQ $ is parallel to the bisector of $ \angle BAC , $ show that $
BM=CN. $
[i]Gheorghe Duță[/i]
2022 Middle European Mathematical Olympiad, 5
Let $\Omega$ be the circumcircle of a triangle $ABC$ with $\angle CAB = 90$. The medians through $B$ and $C$ meet $\Omega$ again at $D$ and $E$, respectively. The tangent to $\Omega$ at $D$ intersects the line $AC$ at $X$ and the tangent to $\Omega$ at $E$ intersects the line $AB$ at $Y$ . Prove that the line $XY$ is tangent to $\Omega$.
2006 IMO Shortlist, 9
Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.
1997 IMO Shortlist, 18
The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$
1996 Israel National Olympiad, 3
The angles of an acute triangle $ABC$ at $\alpha , \beta, \gamma$. Let $AD$ be a height, $CF$ a median, and $BE$ the bisector of $\angle B$. Show that $AD,CF$ and $BE$ are concurrent if and only if $\cos \gamma \tan\beta = \sin \alpha$ .
1992 IMTS, 5
In $\triangle ABC$, shown on the right, let $r$ denote the radius of the inscribed circle, and let $r_A$, $r_B$, and $r_C$ denote the radii of the smaller circles tangent to the inscribed circle and to the sides emanating from $A$, $B$, and $C$, respectively. Prove that
$r \leq r_A + r_B + r_C$
2007 Germany Team Selection Test, 3
Let $ ABC$ be a triangle and $ P$ an arbitrary point in the plane. Let $ \alpha, \beta, \gamma$ be interior angles of the triangle and its area is denoted by $ F.$ Prove:
\[ \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F
\]
When does equality occur?
2008 Silk Road, 2
In a triangle $ABC$ $A_0$,$B_0$ and $C_0$ are the midpoints of the sides $BC$,$CA$ and $AB$.$A_1$,$B_1$,$C_1$ are the midpoints of the broken lines $BAC,CAB,ABC$.Show that $A_0A_1,B_0B_1,C_0C_1$ are concurrent.
1986 IMO Longlists, 70
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that
\[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\]
where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.
2024 Benelux, 3
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$ such that $\left|AC\right|\neq\left|BC\right|$. The internal angle bisector of $\angle CAB$ intersects side $BC$ at $D$ and the external angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $\Omega$ at $E$ and $F$ respectively. Let $G$ be the intersection of lines $AE$ and $FI$ and let $\Gamma$ be the circumcircle of triangle $BDI$. Show that $E$ lies on $\Gamma$ if and only if $G$ lies on $\Gamma$.
2009 Stanford Mathematics Tournament, 2
The pattern in the fi gure below continues inward in finitely. The base of the biggest triangle is 1. All triangles are equilateral. Find the shaded area.
[asy]
defaultpen(linewidth(0.8));
pen blu = rgb(0,112,191);
real r=sqrt(3);
fill((8,0)--(0,8r)--(-8,0)--cycle, blu);
fill(origin--(4,4r)--(-4,4r)--cycle, white);
fill((2,2r)--(0,4r)--(-2,2r)--cycle, blu);
fill((0,2r)--(1,3r)--(-1,3r)--cycle, white);[/asy]
2016 NIMO Problems, 4
In rhombus $ABCD$, let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. If $CN = 7$ and $DM = 24$, compute $AB^2$.
[i]Proposed by Andy Liu[/i]
2005 Belarusian National Olympiad, 2
A line parallel to the side $AC$ of a triangle $ABC$ with $\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$, so that $CN/BN = AC/BC = 2/1$. The segments $CM$ and $AN$ meet at $O$. Let $K$ be a point on the segment $ON$ such that $MO+OK = KN$. The bisector of $\angle ABC$ meets the line through $K$ perpendicular to $AN$ at point $T$.
Determine $\angle MTB$.
2021 Vietnam National Olympiad, 3
Let $\bigtriangleup ABC$ is not an isosceles triangle and is an acute triangle, $AD,BE,CF$ be the altitudes and $H$ is the orthocenter .Let $I$ is the circumcenter of $\bigtriangleup HEF$ and let $K,J$ is the midpoint of $BC,EF$ respectively.Let $HJ$ intersects $(I)$ again at $G$ and $GK$ intersects $(I)$ at $L\neq G$.
a) Prove that $AL$ is perpendicular to $EF$.
b) Let $AL$ intersects $EF$ at $M$, the line $IM$ intersects the circumcircle $\bigtriangleup IEF$ again at $N$, $DN$ intersects $AB,AC$ at $P$ and $Q$ respectively then prove that $PE,QF,AK$ are concurrent.
1993 Poland - First Round, 11
A triangle with perimeter $2p$ is inscribed in a circle of radius $R$ and also circumscribed on a circle of radius $r$. Prove that $p < 2(R+r)$.
2015 AIME Problems, 14
For each integer $n \ge 2$, let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\le x \le n$ and $0\le y \le x \left\lfloor \sqrt x \right\rfloor$, where $\left\lfloor \sqrt x \right\rfloor$ is the greatest integer not exceeding $\sqrt x$. Find the number of values of $n$ with $2\le n \le 1000$ for which $A(n)$ is an integer.
2019 ELMO Shortlist, G5
Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$, let $B_1, C_1$ be variable points on $AB,AC$, respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$. Denote the circumcircle of $AB_1C_1$ by $\omega$. Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$, then $\omega$ passes through a fixed point other than $A$.
[i]Proposed by Max Jiang[/i]
2011 IMAC Arhimede, 2
Let $ABCD$ be a cyclic quadrilatetral inscribed in a circle $k$. Let $M$ and $N$ be the midpoints of the arcs $AB$ and $CD$ which do not contain $C$ and $A$ respectively. If $MN$ meets side $AB$ at $P$, then show that
$\frac{AP}{BP}=\frac{AC+AD}{BC+BD}$
2024 Yasinsky Geometry Olympiad, 3
Let \( W \) be the midpoint of the arc \( BC \) of the circumcircle of triangle \( ABC \), such that \( W \) and \( A \) lie on opposite sides of line \( BC \). On sides \( AB \) and \( AC \), points \( P \) and \( Q \) are chosen respectively so that \( APWQ \) is a parallelogram, and on side \( BC \), points \( K \) and \( L \) are chosen such that \( BK = KW \) and \( CL = LW \). Prove that the lines \( AW \), \( KQ \), and \( LP \) are concurrent.
[i]Proposed by Matthew Kurskyi[/i]
2010 USA Team Selection Test, 4
Let $ABC$ be a triangle. Point $M$ and $N$ lie on sides $AC$ and $BC$ respectively such that $MN || AB$. Points $P$ and $Q$ lie on sides $AB$ and $CB$ respectively such that $PQ || AC$. The incircle of triangle $CMN$ touches segment $AC$ at $E$. The incircle of triangle $BPQ$ touches segment $AB$ at $F$. Line $EN$ and $AB$ meet at $R$, and lines $FQ$ and $AC$ meet at $S$. Given that $AE = AF$, prove that the incenter of triangle $AEF$ lies on the incircle of triangle $ARS$.