Found problems: 25757
1964 Poland - Second Round, 6
Prove that from any five points in the plane it is possible to choose three points that are not vertices of an acute triangle.
1969 IMO Longlists, 12
$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.
2019 Saudi Arabia Pre-TST + Training Tests, 5.2
Let the bisector of the outside angle of $A$ of triangle $ABC$ and the circumcircle of triangle $ABC$ meet at point $P$. The circle passing through points $A$ and $P$ intersects segments $BP$ and $CP$ at points $E$ and $F$ respectively. Let $AD$ is the angle bisector of triangle $ABC$. Prove that $\angle PED = \angle PFD$.
[img]https://cdn.artofproblemsolving.com/attachments/0/3/0638429a220f07227703a682479ed150302aae.png[/img]
2013 Princeton University Math Competition, 4
Let $f(x)=1-|x|$. Let \begin{align*}f_n(x)&=(\overbrace{f\circ \cdots\circ f}^{n\text{ copies}})(x)\\g_n(x)&=|n-|x| |\end{align*} Determine the area of the region bounded by the $x$-axis and the graph of the function $\textstyle\sum_{n=1}^{10}f(x)+\textstyle\sum_{n=1}^{10}g(x).$
2024 China Team Selection Test, 17
$ABCDE$ is a convex pentagon with $BD=CD=AC$, and $B$, $C$, $D$, $E$ are concyclic. If $\angle BAC+\angle AED=180^{\circ}$ and $\angle DCA=\angle BDE$, prove that $AB=DE$ or $AB=2AE$.
2009 Iran Team Selection Test, 10
Let $ ABC$ be a triangle and $ AB\ne AC$ . $ D$ is a point on $ BC$ such that $ BA \equal{} BD$ and $ B$ is between $ C$ and $ D$ . Let $ I_{c}$ be center of the circle which touches $ AB$ and the extensions of $ AC$ and $ BC$ . $ CI_{c}$ intersect the circumcircle of $ ABC$ again at $ T$ .
If $ \angle TDI_{c} \equal{} \frac {\angle B \plus{} \angle C}{4}$ then find $ \angle A$
2019 India IMO Training Camp, P1
Let the points $O$ and $H$ be the circumcenter and orthocenter of an acute angled triangle $ABC.$ Let $D$ be the midpoint of $BC.$ Let $E$ be the point on the angle bisector of $\angle BAC$ such that $AE\perp HE.$ Let $F$ be the point such that $AEHF$ is a rectangle. Prove that $D,E,F$ are collinear.
2013 Canada National Olympiad, 3
Let $G$ be the centroid of a right-angled triangle $ABC$ with $\angle BCA = 90^\circ$. Let $P$ be the point on ray $AG$ such that $\angle CPA = \angle CAB$, and let $Q$ be the point on ray $BG$ such that $\angle CQB = \angle ABC$. Prove that the circumcircles of triangles $AQG$ and $BPG$ meet at a point on side $AB$.
2017 IMAR Test, 1
Let $P$ be a point in the interior $\triangle ABC$, and $AD,BE,CF$ 3 concurrent cevians through $P$, with $D,E,F$ on $BC,CA,AB$. The circle with the diameter $BC$ intersects the circle with the diameter $AD$ in $D_1,D_2$. Analogously we define $E_1,E_2$ and $F_1,F_2$. Prove that $D_1,D_2,E_1,E_2,F_1,F_2$ are concylic.
Estonia Open Senior - geometry, 2014.2.3
The angles of a triangle are $22.5^o, 45^o$ and $112.5^o$. Prove that inside this triangle there exists a point that is located on the median through one vertex, the angle bisector through another vertex and the altitude through the third vertex.
1997 AMC 8, 10
What fraction of this square region is shaded? Stripes are equal in width, and the figure is drawn to scale.
[asy]
unitsize(8);
fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,black);
fill((0,0)--(5,0)--(5,5)--(0,5)--cycle,white);
fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,black);
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,white);
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,black);
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,white);
draw((0,6)--(0,0)--(6,0));
[/asy]
$\textbf{(A)}\ \dfrac{5}{12} \qquad \textbf{(B)}\ \dfrac{1}{2} \qquad \textbf{(C)}\ \dfrac{7}{12} \qquad \textbf{(D)}\ \dfrac{2}{3} \qquad \textbf{(E)}\ \dfrac{5}{6}$
2000 JBMO ShortLists, 19
Let $ABC$ be a triangle. Find all the triangles $XYZ$ with vertices inside triangle $ABC$ such that $XY,YZ,ZX$ and six non-intersecting segments from the following $AX, AY, AZ, BX, BY, BZ, CX, CY, CZ$ divide the triangle $ABC$ into seven regions with equal areas.
2021 Kurschak Competition, 3
Let $A_1B_3A_2B_1A_3B_2$ be a cyclic hexagon such that $A_1B_1,A_2B_2,A_3B_3$ intersect at one point. Let $C_1=A_1B_1\cap A_2A_3,C_2=A_2B_2\cap A_1A_3,C_3=A_3B_3\cap A_1A_2$. Let $D_1$ be the point on the circumcircle of the hexagon such that $C_1B_1D_1$ touches $A_2A_3$. Define $D_2,D_3$ analogously. Show that $A_1D_1,A_2D_2,A_3D_3$ meet at one point.
2022 Yasinsky Geometry Olympiad, 5
In an acute-angled triangle $ABC$, point $I$ is the incenter, $H$ is the orthocenter, $O$ is the center of the circumscribed circle, $T$ and $K$ are the touchpoints of the $A$-excircle and incircle with side $BC$ respectively. It turned out that the segment $TI$ is passing through the point $O$. Prove that $HK$ is the angle bisector of $\angle BHC$.
(Matvii Kurskyi)
1976 Czech and Slovak Olympiad III A, 3
Consider a half-plane with the boundary line $p$ and two points $M,N$ in it such that the distances $Mp$ and $Np$ are different. Construct a trapezoid $MNPQ$ with area $MN^2$ such that $P,Q\in p.$ Discuss conditions of solvability.
2021 Sharygin Geometry Olympiad, 8
Let $ABC$ be an isosceles triangle ($AB=BC$) and $\ell$ be a ray from $B$. Points $P$ and $Q$ of $\ell$ lie inside the triangle in such a way that $\angle BAP=\angle QCA$. Prove that $\angle PAQ=\angle PCQ$.
2008 Germany Team Selection Test, 2
Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear.
[i]Author: Waldemar Pompe, Poland[/i]
2015 Singapore Junior Math Olympiad, 2
In a convex hexagon $ABCDEF, AB$ is parallel to $DE, BC$ is parallel to $EF$ and $CD$ is parallel to $FA$. Prove that the triangles $ACE$ and $BDF$ have the same area.
2008 Sharygin Geometry Olympiad, 7
(A.Zaslavsky, 8--9) Given a circle and a point $ O$ on it. Another circle with center $ O$ meets the first one at points $ P$ and $ Q$. The point $ C$ lies on the first circle, and the lines $ CP$, $ CQ$ meet the second circle for the second time at points $ A$ and $ B$. Prove that $ AB\equal{}PQ$.
2010 Irish Math Olympiad, 5
Suppose $a,b,c$ are the side lengths of a triangle $ABC$. Show that $$x=\sqrt{a(b+c-a)}, y=\sqrt{b(c+a-b)}, z=\sqrt{c(a+b-c)}$$ are the side lengths of an acute-angled triangle $XYZ$, with the same area as $ABC$, but with a smaller perimeter, unless $ABC$ is equilateral.
1953 Moscow Mathematical Olympiad, 242
Let $A$ be a vertex of a regular star-shaped pentagon, the angle at $A$ being less than $180^o$ and the broken line $AA_1BB_1CC_1DD_1EE_1$ being its contour. Lines $AB$ and $DE$ meet at $F$. Prove that polygon $ABB_1CC_1DED_1$ has the same area as the quadrilateral $AD_1EF$.
Note: A regular star pentagon is a figure formed along the diagonals of a regular pentagon.
2001 Croatia National Olympiad, Problem 1
On the unit circle $k$ with center $O$, points $A$ and $B$ with $AB=1$ are chosen and unit circles $k_1$ and $k_2$ with centers $A$ and $B$ are drawn. A sequence of circles $(l_n)$ is defined as follows: circle $l_1$ is tangent to $k$ internally at $D_1$ and to $k_1,k_2$ externally, and for $n>1$ circle $l_n$ is tangent to $k_1$ and $k_2$ and to $l_{n-1}$ at $D_n$. For each $n$, compute $d_n=OD_n$ and the radius $r_n$ of $l_n$.
2023 Irish Math Olympiad, P1
We are given a triangle $ABC$ such that $\angle BAC < 90^{\circ}$. The point $D$ is on the opposite side of the line $AB$ to $C$ such that $|AD| = |BD|$ and $\angle ADB = 90^{\circ}$. Similarly, the point $E$ is on the opposite side of $AC$ to $B$ such that $|AE| = |CE|$ and $\angle AEC = 90^{\circ}$. The point $X$ is such that $ADXE$ is a parallelogram.
Prove that $|BX| = |CX|$.
1969 IMO Longlists, 44
$(MON 5)$ Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation $x^2 - ax + b = 0$.
2018 JBMO Shortlist, G6
Let $XY$ be a chord of a circle $\Omega$, with center $O$, which is not a diameter. Let $P, Q$ be two distinct points inside the segment $XY$, where $Q$ lies between $P$ and $X$. Let $\ell$ the perpendicular line drawn from $P$ to the diameter which passes through $Q$. Let $M$ be the intersection point of $\ell$ and $\Omega$, which is closer to $P$. Prove that $$ MP \cdot XY \ge 2 \cdot QX \cdot PY$$