Found problems: 25757
1985 Tournament Of Towns, (085) 1
$a, b$ and $c$ are sides of a triangle, and $\gamma$ is its angle opposite $c$.
Prove that $c \ge (a + b) \sin \frac{\gamma}{2}$
(V. Prasolov )
2015 Oral Moscow Geometry Olympiad, 2
Line $\ell$ is perpendicular to one of the medians of the triangle. The perpendicular bisectors of the sides of this triangle intersect line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the remaining two.
1997 Greece Junior Math Olympiad, 1
Let $ABC$ be an equilateral triangle whose angle bisectors of $B$ and $C$ intersect at $D$. Perpendicular bisectors of $BD$ and $CD$ intersect $BC$ at points $E$ and $Z$ respectively.
a) Prove that $BE=EZ=ZC$.
b) Find the ratio of the areas of the triangles $BDE$ to $ABC$
2021/2022 Tournament of Towns, P2
A cube was split into 8 parallelepipeds by three planes parallel to its faces. The resulting parts were painted in a chessboard pattern. The volumes of the black parallelepipeds are 1, 6, 8, 12. Find the volumes of the white parallelepipeds.
[i]Oleg Smirnov[/i]
2023 Iranian Geometry Olympiad, 4
Let $ABCD$ be a convex quadrilateral. Let $E$ be the intersection of its diagonals. Suppose that $CD = BC = BE$. Prove that $AD + DC\ge AB$.
[i]Proposed by Dominik Burek - Poland[/i]
2006 Putnam, A1
Find the volume of the region of points $(x,y,z)$ such that
\[\left(x^{2}+y^{2}+z^{2}+8\right)^{2}\le 36\left(x^{2}+y^{2}\right). \]
2017 Iranian Geometry Olympiad, 1
In triangle $ABC$, the incircle, with center $I$, touches the sides $BC$ at point $D$. Line $DI$ meets $AC$ at $X$. The tangent line from $X$ to the incircle (different from $AC$) intersects $AB$ at $Y$. If $YI$ and $BC$ intersect at point $Z$, prove that $AB=BZ$.
[i]Proposed by Hooman Fattahimoghaddam[/i]
1976 Euclid, 3
Source: 1976 Euclid Part B Problem 3
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$I$ is the centre of the inscribed circle of $\triangle{ABC}$. $AI$ meets the circumcircle of $\triangle{ABC}$ at $D$. Prove that $D$ is equidistant from $I$, $B$, and $C$.
1999 National High School Mathematics League, 6
Points $A(1,2)$, a line that passes $(5,-2)$ intersects the parabola $y^2=4x$ at two points $B,C$. Then, $\triangle ABC$ is
$\text{(A)}$ an acute triangle
$\text{(B)}$ an obtuse triangle
$\text{(C)}$ a right triangle
$\text{(D)}$ not sure
2002 India IMO Training Camp, 13
Let $ABC$ and $PQR$ be two triangles such that
[list]
[b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.
[b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$
[/list]
Prove that $AB+AC=PQ+PR$.
2021 Kosovo National Mathematical Olympiad, 4
Let $ABC$ be a triangle with $AB<AC$. Let $D$ be the point where the bisector of angle $\angle BAC$ touches $BC$ and let $D'$ be the reflection of $D$ in the midpoint of $BC$. Let $X$ be the intersection of the bisector of angle $\angle BAC$ with the line parallel to $AB$ that passes through $D'$. Prove that the line $AC$ is tangent with the circumscribed circle of triangle $XCD'$
2018 Ramnicean Hope, 2
Let be the points $ M,N,P, $ on the sides $ BC,AC,AB $ (not on their endpoints), respectively, of a triangle $ ABC, $ such that $ \frac{BM}{MC} =\frac{CN}{NA} =\frac{AP}{PB} . $ Denote $ G_1,G_2,G_3 $ the centroids of $ APN,BMP,CNM, $ respectively. Show that the $ MNP $ has the same centroid as $ G_1G_2G_3. $
[i]Ovidiu Țâțan[/i]
1995 Tournament Of Towns, (446) 2
From a regular $10$-gon $ABCDEFGHIJ$ of side length $1$ a straight line cuts off a triangle $PAQ$ such that $PA +AQ = 1$. Find the sum of angles under which the segment $PQ$ is seen from the points $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$ and $J$.
(V Proizvolov)
2013 Oral Moscow Geometry Olympiad, 4
Let $ABC$ be a triangle. On the extensions of sides $AB$ and $CB$ towards $B$, points $C_1$ and $A_1$ are taken, respectively, so that $AC = A_1C = AC_1$. Prove that circumscribed circles of triangles $ABA_1$ and $CBC_1$ intersect on the bisector of angle $B$.
1980 Bulgaria National Olympiad, Problem 2
(a) Prove that the area of a given convex quadrilateral is at least twice the area of an arbitrary convex quadrilateral inscribed in it whose sides are parallel to the diagonals of the original one.
(b) A tetrahedron with surface area $S$ is intersected by a plane perpendicular to two opposite edges. If the area of the cross-section is $Q$, prove that $S>4Q$.
2013 All-Russian Olympiad, 4
A square with horizontal and vertical sides is drawn on the plane. It held several segments parallel to the sides, and there are no two segments which lie on one line or intersect at an interior point for both segments. It turned out that the segments cuts square into rectangles, and any vertical line intersecting the square and not containing segments of the partition intersects exactly $ k $ rectangles of the partition, and any horizontal line intersecting the square and not containing segments of the partition intersects exactly $\ell$ rectangles. How much the number of rectangles can be?
[i]I. Bogdanov, D. Fon-Der-Flaass[/i]
2002 Tournament Of Towns, 1
In a convex $2002\text{-gon}$ several diagonals are drawn so that they do not intersect inside the polygon. As a result the polygon splits into $2000$ triangles. Isit possible that exactly $1000$ triangles have diagonals for all their three sides?
2003 All-Russian Olympiad, 2
The diagonals of a cyclic quadrilateral $ABCD$ meet at $O$. Let $S_1, S_2$ be the circumcircles of triangles $ABO$ and $CDO$ respectively, and $O,K$ their intersection points. The lines through $O$ parallel to $AB$ and $CD$ meet $S_1$ and $S_2$ again at $L$ and $M$, respectively. Points $P$ and $Q$ on segments $OL$ and $OM$ respectively are taken such that $OP : PL = MQ : QO$. Prove that $O,K, P,Q$ lie on a circle.
2025 Alborz Mathematical Olympiad, P2
In the Jordan Building (the Olympiad building of High School Mandegar Alborz), Ali and Khosro are playing a game. First, Ali selects 2025 points on the plane such that no three points are collinear and no four points are concyclic. Then, Khosro selects a point, followed by Ali selecting another point, and then Khosro selects one more point. The circumcircle of these three points is drawn, and the number of points inside the circle is denoted by \( t \). If Khosro's goal is to maximize \( t \) and Ali's goal is to minimize \( t \), and both play optimally, determine the value of \( t \).
Proposed by Reza Tahernejad Karizi
1974 IMO Longlists, 43
An $(n^2+n+1) \times (n^2+n+1)$ matrix of zeros and ones is given. If no four ones are vertices of a rectangle, prove that the number of ones does not exceed $(n + 1)(n^2 + n + 1).$
2011 Morocco TST, 3
For a given triangle $ ABC$, let $ X$ be a variable point on the line $ BC$ such that $ C$ lies between $ B$ and $ X$ and the incircles of the triangles $ ABX$ and $ ACX$ intersect at two distinct points $ P$ and $ Q.$ Prove that the line $ PQ$ passes through a point independent of $ X$.
1979 IMO Longlists, 70
There are $1979$ equilateral triangles: $T_1,T_2, . . . ,T_{1979}$. A side of triangle $T_k$ is equal to $\frac{1}{k}$, $k = 1,2, . . . ,1979$. At what values of a number $a$ can one place all these triangles into the equilateral triangle with side length $a$ so that they don’t intersect (points of contact are allowed)?
2000 AMC 8, 22
A cube has edge length $2$. Suppose that we glue a cube of edge length $1$ on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to
[asy]
draw((0,0)--(2,0)--(3,1)--(3,3)--(2,2)--(0,2)--cycle);
draw((2,0)--(2,2));
draw((0,2)--(1,3));
draw((1,7/3)--(1,10/3)--(2,10/3)--(2,7/3)--cycle);
draw((2,7/3)--(5/2,17/6)--(5/2,23/6)--(3/2,23/6)--(1,10/3));
draw((2,10/3)--(5/2,23/6));
draw((3,3)--(5/2,3));
[/asy]
$\text{(A)}\ 10 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 21 \qquad \text{(E)}\ 25$
2006 Taiwan TST Round 1, 1
Circle $O$ is the incircle of the square $ABCD$. $O$ is tangent to $AB$ and $AD$ at $E$ and $F$, respectively. Let $K$ be a point on the minor arc $EF$, and let the tangent of $O$ at $K$ intersect $AB$, $AC$, $AD$ at $X$, $Y$, $Z$, respectively. Show that $\displaystyle \frac{AX}{XB} + \frac{AY}{YC} + \frac{AZ}{ZD} =1$.
2022 Kosovo & Albania Mathematical Olympiad, 4
Consider $n>9$ lines on the plane such that no two lines are parallel. Show that there exist at least $n/9$ lines such that the angle between any two of the lines is at most $20^\circ$.