Found problems: 25757
2016 Sharygin Geometry Olympiad, P19
Let $ABCDEF$ be a regular hexagon. Points $P$ and $Q$ on tangents to its circumcircle at $A$ and $D$ respectively are such that $PQ$ touches the minor arc $EF$ of this circle. Find the angle between $PB$ and $QC$.
2024-IMOC, G7
Triangle $ABC$ has circumcenter $O$ and incenter $I$. The incircle is tangent to $AC, AB$ at $E, F$, respectively. $H$ is the orthocenter of $\triangle BIC$. $\odot(AEF)$ and $\odot(ABC)$ intersects again at $S$. $BC, AH$ intersects $OI$ again at $J, K$, respectively. Prove that $H, K, J, S$ are concyclic.
[i]Proposed by chengbilly[/i]
2017 Hanoi Open Mathematics Competitions, 11
Let $S$ denote a square of the side-length $7$, and let eight squares of the side-length $3$ be given. Show that $S$ can be covered by those eight small squares.
Novosibirsk Oral Geo Oly VIII, 2023.4
An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.
2024 All-Russian Olympiad Regional Round, 9.2
On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the diagonals of all such trapezoids share a common point.
2013 National Olympiad First Round, 9
Let $ABC$ be a triangle with $|AB|=18$, $|AC|=24$, and $m(\widehat{BAC}) = 150^\circ$. Let $D$, $E$, $F$ be points on sides $[AB]$, $[AC]$, $[BC]$, respectively, such that $|BD|=6$, $|CE|=8$, and $|CF|=2|BF|$. Let $H_1$, $H_2$, $H_3$ be the reflections of the orthocenter of triangle $ABC$ over the points $D$, $E$, $F$, respectively. What is the area of triangle $H_1H_2H_3$?
$
\textbf{(A)}\ 70
\qquad\textbf{(B)}\ 72
\qquad\textbf{(C)}\ 84
\qquad\textbf{(D)}\ 96
\qquad\textbf{(E)}\ 108
$
2022 Tuymaada Olympiad, 7
$M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and parallel to $MD$ there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150.$ Find the $\angle MT D.$
[i](K. Ivanov )[/i]
2007 Ukraine Team Selection Test, 6
Find all primes $ p$ for that there is an integer $ n$ such that there are no integers $ x,y$ with $ x^3 \plus{} y^3 \equiv n \mod p$ (so not all residues are the sum of two cubes).
E.g. for $ p \equal{} 7$, one could set $ n \equal{} \pm 3$ since $ x^3,y^3 \equiv 0 , \pm 1 \mod 7$, thus $ x^3 \plus{} y^3 \equiv 0 , \pm 1 , \pm 2 \mod 7$ only.
2014 NIMO Summer Contest, 14
Let $ABC$ be a triangle with circumcenter $O$ and let $X$, $Y$, $Z$ be the midpoints of arcs $BAC$, $ABC$, $ACB$ on its circumcircle. Let $G$ and $I$ denote the centroid of $\triangle XYZ$ and the incenter of $\triangle ABC$.
Given that $AB = 13$, $BC = 14$, $CA = 15$, and $\frac {GO}{GI} = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Evan Chen[/i]
2020 AMC 10, 12
Triangle $AMC$ is isoceles with $AM = AC$. Medians $\overline{MV}$ and $\overline{CU}$ are perpendicular to each other, and $MV=CU=12$. What is the area of $\triangle AMC?$
[asy]
draw((-4,0)--(4,0)--(0,12)--cycle);
draw((-2,6)--(4,0));
draw((2,6)--(-4,0));
draw((-2,6)--(2,6));
label("M", (-4,0), W);
label("C", (4,0), E);
label("A", (0, 12), N);
label("V", (2, 6), NE);
label("U", (-2, 6), NW);
draw(rightanglemark((-2,6),(0,4),(-4,0),17));
[/asy]
$\textbf{(A) } 48 \qquad \textbf{(B) } 72 \qquad \textbf{(C) } 96 \qquad \textbf{(D) } 144 \qquad \textbf{(E) } 192$
2020 Thailand TST, 1
Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$.
(Vietnam)
1996 AMC 8, 22
The horizontal and vertical distances between adjacent points equal $1$ unit. The area of triangle $ABC$ is
[asy]
for (int a = 0; a < 5; ++a)
{
for (int b = 0; b < 4; ++b)
{
dot((a,b));
}
}
draw((0,0)--(3,2)--(4,3)--cycle);
label("$A$",(0,0),SW);
label("$B$",(3,2),SE);
label("$C$",(4,3),NE);
[/asy]
$\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/2 \qquad \text{(C)}\ 3/4 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 5/4$
2000 IMO Shortlist, 5
The tangents at $B$ and $A$ to the circumcircle of an acute angled triangle $ABC$ meet the tangent at $C$ at $T$ and $U$ respectively. $AT$ meets $BC$ at $P$, and $Q$ is the midpoint of $AP$; $BU$ meets $CA$ at $R$, and $S$ is the midpoint of $BR$. Prove that $\angle ABQ=\angle BAS$. Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.
1997 Pre-Preparation Course Examination, 5
Let $H$ be the orthocenter of the triangle $ABC$ and $P$ an arbitrary point on circumcircle of triangle. $BH$ meets $AC$ at $E$. $PAQB$ and $PARC$ are two parallelograms and $AQ$ meets $HR$ at $X$. Show that $EX \parallel AP$.
2004 USA Team Selection Test, 1
Suppose $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ are real numbers such that \[ (a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 -1)(b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 - 1) > (a_1 b_1 + a_2 b_2 + \cdots + a_n b_n - 1)^2. \] Prove that $a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 > 1$ and $b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 > 1$.
2014 Stanford Mathematics Tournament, 9
In cyclic quadrilateral $ABCD$, $AB= AD$. If $AC = 6$ and $\frac{AB}{BD} =\frac35$ , find the maximum possible area of $ABCD$.
2005 China Team Selection Test, 3
Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.
2013 BMT Spring, 5
Circle $C_1$ has center $O$ and radius $OA$, and circle $C_2$ has diameter $OA$. $AB$ is a chord of circle $C_1$ and $BD$ may be constructed with $D$ on $OA$ such that $BD$ and $OA$ are perpendicular. Let $C$ be the point where $C_2$ and $BD$ intersect. If $AC = 1$, find $AB$.
2012 German National Olympiad, 3
Let $ABC$ a triangle and $k$ a circle such that:
(1) The circle $k$ passes through $A$ and $B$ and touches the line $AC.$
(2) The tangent to $k$ at $B$ intersects the line $AC$ in a point $X\ne C.$
(3) The circumcircle $\omega$ of $BXC$ intersects $k$ in a point $Q\ne B.$
(4) The tangent to $\omega$ at $X$ intersects the line $AB$ in a point $Y.$
Prove that the line $XY$ is tangent to the circumcircle of $BQY.$
2019 Iran MO (3rd Round), 1
Let $A_1,A_2, \dots A_k$ be points on the unit circle.Prove that:
$\sum\limits_{1\le i<j \le k} d(A_i,A_j)^2 \le k^2 $
Where $d(A_i,A_j)$ denotes the distance between $A_i,A_j$.
Kyiv City MO 1984-93 - geometry, 1993.10.4
Prove theat for an arbitrary triangle holds the inequality $$a \cos A+ b \cos B + c \cos C \le p ,$$ where $a, b, c$ are the sides of the triangle, $A, B, C$ are the angles, $p$ is the semiperimeter.
II Soros Olympiad 1995 - 96 (Russia), 10.10
The Order "For Faithful Service" of the $7$th degree in shape is a combination of a semicircle with a diameter $AB = 2$ and a triangle $AM B$. The sides$ AM$ and $BM$ intersect the semicircle (the border of the semicircle). The part of the circle outside the triangle and the part of the triangle outside the circle are made of pure copper. What should the side of the triangle be equal to in order for the area of the copper part to be the smallest?
1994 Abels Math Contest (Norwegian MO), 1b
Let $C$ be a point on the extension of the diameter $AB$ of a circle. A line through $C$ is tangent to the circle at point $N$. The bisector of $\angle ACN$ meets the lines $AN$ and $BN$ at $P$ and $Q$ respectively. Prove that $PN = QN$.
2000 Czech And Slovak Olympiad IIIA, 3
In the plane are given $2000$ congruent triangles of area $1$, which are all images of one triangle under translations. Each of these triangles contains the centroid of every other triangle. Prove that the union of these triangles has area less than $22/9$.
2013 Harvard-MIT Mathematics Tournament, 26
Triangle $ABC$ has perimeter $1$. Its three altitudes form the side lengths of a triangle. Find the set of all possible values of $\min(AB,BC,CA)$.