Found problems: 25757
2022 Novosibirsk Oral Olympiad in Geometry, 2
A quadrilateral is given, in which the lengths of some two sides are equal to $1$ and $4$. Also, the diagonal of length $2$ divides it into two isosceles triangles. Find the perimeter of this quadrilateral.
1992 Turkey Team Selection Test, 3
A circle with radius $4$ and $251$ distinct points inside the circle are given. Show that it is possible to draw a circle with radius $1$ and containing at least $11$ of these points.
2016 District Olympiad, 4
Let $ ABCDA’B’C’D’ $ a right parallelepiped and $ M,N $ the feet of the perpendiculars of $ BD $ through $ A’, $ respectively, $ C’. $ We know that $ AB=\sqrt 2, BC=\sqrt 3, AA’=\sqrt 2. $
[b]a)[/b] Prove that $ A’M\perp C’N. $
[b]b)[/b] Calculate the dihedral angle between the plane formed by $ A’MC $ and the plane formed by $ ANC’. $
2014 Saudi Arabia IMO TST, 3
Let $ABC$ be a triangle and let $P$ be a point on $BC$. Points $M$ and $N$ lie on $AB$ and $AC$, respectively such that $MN$ is not parallel to $BC$ and $AMP N$ is a parallelogram. Line $MN$ meets the circumcircle of $ABC$ at $R$ and $S$. Prove that the circumcircle of triangle $RP S$ is tangent to $BC$.
2004 Nicolae Păun, 2
The following geometry is embedded in the Cartesian plane.
[b]a)[/b] Prove that any line that passes through at least two lattice points, passes through at least three lattice points.
[b]b)[/b] Find a point on the plane which doesn't lie on any line that passes through at least two lattice points.
[b]c)[/b] Show that any point with rational coordinates lie on a line that passes through two lattice points.
[i]Lavinia Savu[/i]
1994 Denmark MO - Mohr Contest, 5
In a right-angled and isosceles triangle, the two catheti are both length $1$. Find the length of the shortest line segment dividing the triangle into two figures with the same area, and specify the location of this line segment
2018 BMT Spring, 9
What is the least integer a greater than $14$ so that the triangle with side lengths $a - 1$, $a$, and $a + 1$ has integer area?
2018 Hanoi Open Mathematics Competitions, 1
How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle).
A. $6$ B. $8$ C. $12$ D. $18$ E. $16$
1992 Cono Sur Olympiad, 2
Let $P$ be a point outside the circle $C$. Find two points $Q$ and $R$ on the circle, such that $P,Q$ and $R$ are collinear and $Q$ is the midpopint of the segmenet $PR$. (Discuss the number of solutions).
2023 Sharygin Geometry Olympiad, 21
Let $ABCD$ be a cyclic quadrilateral; $M_{ac}$ be the midpoint of $AC$; $H_d,H_b$ be the orthocenters of $\triangle ABC,\triangle ADC$ respectively; $P_d,P_b$ be the projections of $H_d$ and $H_b$ to $BM_{ac}$ and $DM_{ac}$ respectively. Define similarly $P_a,P_c$ for the diagonal $BD$. Prove that $P_a,P_b,P_c,P_d$ are concyclic.
2019 India PRMO, 22
In parallelogram $ABCD$, $AC=10$ and $BD=28$. The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. What is the maximum walue of $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$ ?
2015 Sharygin Geometry Olympiad, P12
Find the maximal number of discs which can be disposed on the plane so that each two of them have a common point and no three have it
1998 German National Olympiad, 6b
Prove that the following statement holds for all odd integers $n \ge 3$:
If a quadrilateral $ABCD$ can be partitioned by lines into $n$ cyclic quadrilaterals, then $ABCD$ is itself cyclic.
2005 Oral Moscow Geometry Olympiad, 5
The triangle $ABC$ is inscribed in the circle. Construct a point $P$ such that the points of intersection of lines $AP, BP$ and $CP$ with this circle are the vertices of an equilateral triangle.
(A. Zaslavsky)
1946 Moscow Mathematical Olympiad, 122
On the sides $PQ, QR, RP$ of $\vartriangle PQR$ segments $AB, CD, EF$ are drawn. Given a point $S_0$ inside triangle $\vartriangle PQR$, find the locus of points $S$ for which the sum of the areas of triangles $\vartriangle SAB$, $\vartriangle SCD$ and $\vartriangle SEF$ is equal to the sum of the areas of triangles $\vartriangle S_0AB$, $\vartriangle S_0CD$, $\vartriangle S0EF$. Consider separately the case $$\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}.$$
2020 Sharygin Geometry Olympiad, 23
A non-self-intersecting polygon is nearly convex if precisely one of its interior angles is greater than $180^\circ$.
One million distinct points lie in the plane in such a way that no three of them are collinear. We would like to construct a nearly convex one-million-gon whose vertices are precisely the one million given points. Is it possible that there exist precisely ten such polygons?
2012 Junior Balkan Team Selection Tests - Romania, 3
Consider the triangle $ABC$ and the points $D \in (BC)$ and $M \in (AD)$. Lines $BM$ and $AC$ meet at $E$, lines $CM$ and $AB$ meet at $F$, and lines $EF$ and $AD$ meet at $N$. Prove that $$\frac{AN}{DN}=\frac{1}{2}\cdot \frac{AM}{DM}$$
2019 IMO Shortlist, G6
Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$.
(Slovakia)
1993 India National Olympiad, 9
Show that there exists a convex hexagon in the plane such that
(i) all its interior angles are equal;
(ii) its sides are $1,2,3,4,5,6$ in some order.
2014 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be an acute triangle and $D \in (BC) , E \in (AD)$ be mobile points. The circumcircle of triangle $CDE$ meets the median from $C$ of the triangle $ABC$ at $F$ Prove that the circumcenter of triangle $AEF$ lies on a fixed line.
1903 Eotvos Mathematical Competition, 3
Let $A,B,C,D$ be the vertices of a rhombus, let $k_1$ be the circle through $B,C$ and $D$, let $k_2$ be the circle through $A,C$ and $D$, let $k_3$ be the circle through $A,B$ and $D$, let $k_4$ be the circle through $A,B$ and $C$. Prove that the tangents to $k_1$ and $k_3$ at $B$ form the same angle as the tangents to $k_2$ and $k_4$ at $A$.
2006 France Team Selection Test, 2
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]
2015 BMT Spring, P1
Suppose that circles $C_1$ and $C_2$ intersect at $X$ and $Y$ . Let $A, B$ be on $C_1$, $C_2$, respectively, such that $A, X, B$ lie on a line in that order. Let $A, C$ be on $C_1$, $C_2$, respectively, such that $A, Y, C$ lie on a line in that order. Let $A', B', C'$ be another similarly defined triangle with $A \ne A'$. Prove that $BB' = CC'$.
1983 IMO Longlists, 29
Let $O$ be a point outside a given circle. Two lines $OAB, OCD$ through $O$ meet the circle at $A,B,C,D$, where $A,C$ are the midpoints of $OB,OD$, respectively. Additionally, the acute angle $\theta$ between the lines is equal to the acute angle at which each line cuts the circle. Find $\cos \theta$ and show that the tangents at $A,D$ to the circle meet on the line $BC.$
1977 IMO Longlists, 22
Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.