Found problems: 2023
2024 Czech-Polish-Slovak Junior Match, 3
Determine the possible interior angles of isosceles triangles that can be subdivided in two isosceles triangles with disjoint interior.
2001 China National Olympiad, 1
Let $a$ be real number with $\sqrt{2}<a<2$, and let $ABCD$ be a convex cyclic quadrilateral whose circumcentre $O$ lies in its interior. The quadrilateral's circumcircle $\omega$ has radius $1$, and the longest and shortest sides of the quadrilateral have length $a$ and $\sqrt{4-a^2}$, respectively. Lines $L_A,L_B,L_C,L_D$ are tangent to $\omega$ at $A,B,C,D$, respectively.
Let lines $L_A$ and $L_B$, $L_B$ and $L_C$,$L_C$ and $L_D$,$L_D$ and $L_A$ intersect at $A',B',C',D'$ respectively. Determine the minimum value of $\frac{S_{A'B'C'D'}}{S_{ABCD}}$.
2006 Mediterranean Mathematics Olympiad, 2
Let $P$ be a point inside a triangle $ABC$, and $A_1B_2,B_1C_2,C_1A_2$ be segments passing through $P$ and parallel to $AB, BC, CA$ respectively, where points $A_1, A_2$ lie on $BC, B_1, B_2$ on $CA$, and $C_1,C_2$ on $AB$. Prove that
\[ \text{Area}(A_1A_2B_1B_2C_1C_2) \ge \frac{1}{2}\text{Area}(ABC)\]
2002 Tournament Of Towns, 5
Does there exist a regular triangular prism that can be covered (without overlapping) by different equilateral triangles? (One is allowed to bend the triangles around the edges of the prism.)
2012 Regional Competition For Advanced Students, 4
In a triangle $ABC$, let $H_a$, $H_b$ and $H_c$ denote the base points of the altitudes on the sides $BC$, $CA$ and $AB$, respectively.
Determine for which triangles $ABC$ two of the lengths $H_aH_b$, $H_bH_c$ and $H_aH_c$ are equal.
2021 Taiwan TST Round 3, 6
Let $ ABCD $ be a rhombus with center $ O. $ $ P $ is a point lying on the side $ AB. $ Let $ I, $ $ J, $ and $ L $ be the incenters of triangles $ PCD, $ $ PAD, $ and $PBC, $ respectively. Let $ H $ and $ K $ be orthocenters of triangles $ PLB $ and $ PJA, $ respectively.
Prove that $ OI \perp HK. $
[i]Proposed by buratinogigle[/i]
1990 Federal Competition For Advanced Students, P2, 6
A convex pentagon $ ABCDE$ is inscribed in a circle. The distances of $ A$ from the lines $ BC,CD,DE$ are $ a,b,c,$ respectively. Compute the distance of $ A$ from the line $ BE$.
2006 Iran Team Selection Test, 3
Suppose $ABC$ is a triangle with $M$ the midpoint of $BC$.
Suppose that $AM$ intersects the incircle at $K,L$.
We draw parallel line from $K$ and $L$ to $BC$ and name their second intersection point with incircle $X$ and $Y$. Suppose that $AX$ and $AY$ intersect $BC$ at $P$ and $Q$.
Prove that $BP=CQ$.
2013 Sharygin Geometry Olympiad, 4
Given a square cardboard of area $\frac{1}{4}$, and a paper triangle of area $\frac{1}{2}$ such that the square of its sidelength is a positive integer. Prove that the triangle can be folded in some ways such that the squace can be placed inside the folded figure so that both of its faces are completely covered with paper.
[i]Proposed by N.Beluhov, Bulgaria[/i]
2007 Italy TST, 1
Let $ABC$ an acute triangle.
(a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$;
(b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.
2012 European Mathematical Cup, 2
Let $ABC$ be an acute triangle with orthocenter $H$. Segments $AH$ and $CH$ intersect segments $BC$ and $AB$ in points $A_1$ and $C_1$ respectively. The segments $BH$ and $A_1C_1$ meet at point $D$. Let $P$ be the midpoint of the segment $BH$. Let $D'$ be the reflection of the point $D$ in $AC$. Prove that quadrilateral $APCD'$ is cyclic.
[i]Proposed by Matko Ljulj.[/i]
2014 Korea - Final Round, 4
Let $ ABC $ be a isosceles triangle with $ AC=BC$. Let $ D $ a point on a line $ BA $ such that $ A $ lies between $ B, D $. Let $O_1 $ be the circumcircle of triangle $ DAC $. $ O_1 $ meets $ BC $ at point $ E $. Let $ F $ be the point on $ BC $ such that $ FD $ is tangent to circle $O_1 $, and let $O_2 $ be the circumcircle of $ DBF$. Two circles $O_1 , O_2 $ meet at point $ G ( \ne D) $. Let $ O $ be the circumcenter of triangle $ BEG$. Prove that the line $FG$ is tangent to circle $O$ if and only if $ DG \bot FO$.
2018 Taiwan TST Round 2, 2
Let $ABC$ be a triangle with circumcircle $\Omega$, circumcenter $O$ and orthocenter $H$. Let $S$ lie on $\Omega$ and $P$ lie on $BC$ such that $\angle ASP=90^\circ$, line $SH$ intersects the circumcircle of $\triangle APS$ at $X\neq S$. Suppose $OP$ intersects $CA,AB$ at $Q,R$, respectively, $QY,RZ$ are the altitude of $\triangle AQR$. Prove that $X,Y,Z$ are collinear.
[i]Proposed by Shuang-Yen Lee[/i]
Cono Sur Shortlist - geometry, 2012.G3
Let $ABC$ be a triangle, and $M$, $N$, and $P$ be the midpoints of $AB$, $BC$, and $CA$ respectively, such that $MBNP$ is a parallelogram. Let $R$ and $S$ be the points in which the line $MN$ intersects the circumcircle of $ABC$. Prove that $AC$ is tangent to the circumcircle of triangle $RPS$.
1990 Vietnam National Olympiad, 1
A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' \equal{} MB \cdot MB' \equal{} MC \cdot MC'$.
2013 Turkey Team Selection Test, 1
Let $E$ be intersection of the diagonals of convex quadrilateral $ABCD$. It is given that $m(\widehat{EDC}) = m(\widehat{DEC})=m(\widehat{BAD})$. If $F$ is a point on $[BC]$ such that $m(\widehat{BAF}) + m(\widehat{EBF})=m(\widehat{BFE})$, show that $A$, $B$, $F$, $D$ are concyclic.
1988 Federal Competition For Advanced Students, P2, 5
The bisectors of angles $ B$ and $ C$ of triangle $ ABC$ intersect the opposite sides in points $ B'$ and $ C'$ respectively. Show that the line $ B'C'$ intersects the incircle of the triangle.
2013 Romania National Olympiad, 3
Given $P$ a point m inside a triangle acute-angled $ABC$ and $DEF$ intersections of lines with that $AP$, $BP$, $CP$ with$\left[ BC \right],\left[ CA \right],$respective $\left[ AB \right]$
a) Show that the area of the triangle $DEF$ is at most a quarter of the area of the triangle $ABC$
b) Show that the radius of the circle inscribed in the triangle $DEF$ is at most a quarter of the radius of the circle circumscribed on triangle $4ABC.$
1995 Italy TST, 4
In a triangle $ABC$, $P$ and $Q$ are the feet of the altitudes from $B$ and $A$ respectively. Find the locus of the circumcentre of triangle $PQC$, when point $C$ varies (with $A$ and $B$ fixed) in such a way that $\angle ACB$ is equal to $60^{\circ}$.
1974 IMO Longlists, 27
Let $C_1$ and $C_2$ be circles in the same plane, $P_1$ and $P_2$ arbitrary points on $C_1$ and $C_2$ respectively, and $Q$ the midpoint of segment $P_1P_2.$ Find the locus of points $Q$ as $P_1$ and $P_2$ go through all possible positions.
[i]Alternative version[/i]. Let $C_1, C_2, C_3$ be three circles in the same plane. Find the locus of the centroid of triangle $P_1P_2P_3$ as $P_1, P_2,$ and $P_3$ go through all possible positions on $C_1, C_2$, and $C_3$ respectively.
1990 Dutch Mathematical Olympiad, 4
If $ ABCDEFG$ is a regular $ 7$-gon with side $ 1$, show that: $ \frac{1}{AC}\plus{}\frac{1}{AD}\equal{}1$.
1990 IMO Longlists, 69
Consider the set of cuboids: the three edges $a, b, c$ from a common vertex satisfy the condition
\[\frac ab = \frac{a^2}{c^5}\]
(i) Prove that there are $100$ pairs of cuboids in this set with equal volumes in each pair.
(ii) For each pair of the above cuboids, find the ratio of the sum of their edges.
2013 Turkey Team Selection Test, 2
Let the incircle of the triangle $ABC$ touch $[BC]$ at $D$ and $I$ be the incenter of the triangle. Let $T$ be midpoint of $[ID]$. Let the perpendicular from $I$ to $AD$ meet $AB$ and $AC$ at $K$ and $L$, respectively. Let the perpendicular from $T$ to $AD$ meet $AB$ and $AC$ at $M$ and $N$, respectively. Show that $|KM|\cdot |LN|=|BM|\cdot|CN|$.
2008 All-Russian Olympiad, 6
In a scalene triangle $ ABC$ the altitudes $ AA_{1}$ and $ CC_{1}$ intersect at $ H, O$ is the circumcenter, and $ B_{0}$ the midpoint of side $ AC$. The line $ BO$ intersects side $ AC$ at $ P$, while the lines $ BH$ and $ A_{1}C_{1}$ meet at $ Q$. Prove that the lines $ HB_{0}$ and $ PQ$ are parallel.
2007 South africa National Olympiad, 3
In acute-angled triangle $ ABC$, the points $ D,E,F$ are on sides $ BC,CA,AB$, respectively such that $ \angle AFE \equal{} \angle BFD, \angle FDB \equal{} \angle EDC, \angle DEC \equal{} \angle FEA$. Prove that $ AD$ is perpendicular to $ BC$.