Found problems: 5
2005 BAMO, 2
Prove that if two medians in a triangle are equal in length, then the triangle is isosceles.
(Note: A median in a triangle is a segment which connects a vertex of the triangle to the midpoint of the opposite side of the triangle.)
2005 Sharygin Geometry Olympiad, 23
Envelop the cube in one layer with five convex pentagons of equal areas.
2009 Greece JBMO TST, 2
Given convex quadrilateral $ABCD$ inscribed in circle $(O,R)$ (with center $O$ and radius $R$). With centers the vertices of the quadrilateral and radii $R$, we consider the circles $C_A(A,R), C_B(B,R), C_C(C,R), C_D(D,R)$. Circles $C_A$ and $C_B$ intersect at point $K$, circles $C_B$ and $C_C$ intersect at point $L$, circles $C_C$ and $C_D$ intersect at point $M$ and circles $C_D$ and $C_A$ intersect at point $N$ (points $K,L,M,N$ are the second common points of the circles given they all pass through point $O$). Prove that quadrilateral $KLMN$ is a parallelogram.
2022 Indonesia TST, C
Five numbers are chosen from $\{1, 2, \ldots, n\}$. Determine the largest $n$ such that we can always pick some of the 5 chosen numbers so that they can be made into two groups whose numbers have the same sum (a group may contain only one number).
1969 Vietnam National Olympiad, 4
Two circles centers $O$ and $O'$, radii $R$ and $R'$, meet at two points. A variable line $L$ meets the circles at $A, C, B, D$ in that order and $\frac{AC}{AD} = \frac{CB}{BD}$. The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$.
Find the locus of $H$ and $H'$.
If $OO'^2 < R^2 + R'^2$, find a point $P$ on $L$ such that $PO + PO'$ has the smallest possible value.
Show that this value does not depend on the position of $L$.
Comment on the case $OO'^2 > R^2 + R'^2$.