Found problems: 25757
1962 IMO, 6
Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]
Ukrainian TYM Qualifying - geometry, I.17
A right triangle when rotating around a large leg forms a cone with a volume of $100\pi$. Calculate the length of the path that passes through each vertex of the triangle at rotation of $180^o$ around the point of intersection of its bisectors, if the sum of the diameters of the circles, inscribed in the triangle and circumscribed around it, are equal to $17$.
2019 Yasinsky Geometry Olympiad, p1
It is known that in the triangle $ABC$ the distance from the intersection point of the angle bisector to each of the vertices of the triangle does not exceed the diameter of the circle inscribed in this triangle. Find the angles of the triangle $ABC$.
(Grigory Filippovsky)
2017 Saint Petersburg Mathematical Olympiad, 6
In acute-angled triangle $ABC$, the height $AH$ and median $BM$ were drawn. Point $D$ lies on the circumcircle of triangle $BHM$ such that $AD \parallel BM$ and $B, D$ are on opposite sides of line $AC$. Prove that $BC=BD$.
1978 IMO Longlists, 50
A variable tetrahedron $ABCD$ has the following properties:
Its edge lengths can change as well as its vertices, but the opposite edges remain equal $(BC = DA, CA = DB, AB = DC)$; and the vertices $A,B,C$ lie respectively on three fixed spheres with the same center $P$ and radii $3, 4, 12$. What is the maximal length of $PD$?
2000 Saint Petersburg Mathematical Olympiad, 10.2
Let $AA_1$ and $BB_1$ be the altitudes of acute angled triangle $ABC$. Points $K$ and $M$ are midpoints of $AB$ and $A_1B_1$ respectively. Segments $AA_1$ and $KM$ intersect at point $L$. Prove that points $A$, $K$, $L$ and $B_1$ are noncyclic.
[I]Proposed by S. Berlov[/i]
1992 AIME Problems, 15
Define a positive integer $ n$ to be a factorial tail if there is some positive integer $ m$ such that the decimal representation of $ m!$ ends with exactly $ n$ zeroes. How many positive integers less than $ 1992$ are not factorial tails?
2005 IberoAmerican, 2
A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$. Afterwards, if its last jump was from $A$ to $B$, then the next jump is from $B$ to one of the points $B + (B - A) - 1$, $B + (B - A)$, $B + (B-A) + 1$.
Prove that if the flea arrived twice at the point $n$, $n$ positive integer, then it performed at least $\lceil 2\sqrt n\rceil$ jumps.
2014-2015 SDML (Middle School), 8
Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?
Kvant 2020, M2615
In the triangle $ABC$, the inscribed circle touches the sides $CA{}$ and $AB{}$ at the points $B_1{}$ and $C_1{}$, respectively. An arbitrary point $D{}$ is selected on the side $AB{}$. The point $L{}$ is the center of the inscribed circle of the triangle $BCD$. The bisector of the angle $ACD$ intersects the line $B_1C_1$ at the point $M{}$. Prove that $\angle CML=90^\circ$.
[i]Proposed by Chan Quang Heung (Vietnam)[/i]
2019 Bosnia and Herzegovina EGMO TST, 3
The circle inscribed in the triangle $ABC$ touches the sides $AB$ and $AC$ at the points $K$ and $L$ , respectively. The angle bisectors from $B$ and $C$ intersect the altitude of the triangle from the vertex $A$ at the points $Q$ and $R$ , respectively. Prove that one of the points of intersection of the circles circumscribed around the triangles $BKQ$ and $CPL$ lies on $BC$.
2014 AMC 12/AHSME, 20
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?
$\textbf{(A) }6\sqrt 3+3\qquad
\textbf{(B) }\dfrac{27}2\qquad
\textbf{(C) }8\sqrt 3\qquad
\textbf{(D) }14\qquad
\textbf{(E) }3\sqrt 3+9\qquad$
2025 International Zhautykov Olympiad, 5
Let $A_1C_2B_1B_2C_1A_2$ be a cyclic convex hexagon inscribed in circle $\Omega$, centered at $O$. Let $\{ P \} = A_2B_2 \cap A_1B_1$ and $\{ Q \} = A_2C_2 \cap A_1C_1$. Let $\Gamma_1$ be a circle tangent to $OB_1$ and $OC_1$ at $B_1,C_1$ respectively. Similarly, define $\Gamma_2$ to be the circle tangent to $OB_2,OC_2$ at $B_2, C_2$ respectively. Prove that there is a homothety that sends $\Gamma_1$ to $\Gamma_2$, whose center lies on $PQ$
2023 Stanford Mathematics Tournament, 6
Let ABC be a triangle and $\omega_1$ its incircle. Let points $D$ and $E$ be on segments $AB$, $AC$ respectively such that $DE$ is parallel to $BC$ and tangent to $\omega_1$ . Now let $\omega_2$ be the incircle of $\vartriangle ADE$ and let points $F$ and $G$ be on segments $AD,$ $AE$ respectively such that F G is parallel to $DE$ and tangent to $\omega_2$. Given that $\omega_2$ is tangent to line $AF$ at point X and line $AG$ at point $Y$ , the radius of $\omega_1$ is $60$, and $$4(AX) = 5(F G) = 4(AY),$$ compute the radius of $\omega_2$.
2001 Flanders Math Olympiad, 2
Consider a triangle and 2 lines that each go through a corner and intersects the opposing segment, such that the areas are as on the attachment.
Find the "?"
2014 May Olympiad, 2
In a convex quadrilateral $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. If $MP$ and $NQ$ divide $ABCD$ in four quadrilaterals with the same area, prove that $ABCD$ is a parallelogram.
1983 AIME Problems, 12
Diameter $AB$ of a circle has length a 2-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$.
2013 Iran Team Selection Test, 12
Let $ABCD$ be a cyclic quadrilateral that inscribed in the circle $\omega$.Let $I_{1},I_{2}$ and $r_{1},r_{2}$ be incenters and radii of incircles of triangles $ACD$ and $ABC$,respectively.assume that $r_{1}=r_{2}$. let $\omega'$ be a circle that touches $AB,AD$ and touches $\omega$ at $T$. tangents from $A,T$ to $\omega$ meet at the point $K$.prove that $I_{1},I_{2},K$ lie on a line.
Kyiv City MO Juniors 2003+ geometry, 2015.8.3
In the isosceles triangle $ABC$, $ (AB = BC)$ the bisector $AD$ was drawn, and in the triangle $ABD$ the bisector $DE$ was drawn. Find the values of the angles of the triangle $ABC$, if it is known that the bisectors of the angles $ABD$ and $AED$ intersect on the line $AD$.
(Fedak Ivan)
2015 Paraguayan Mathematical Olympiad, Problem 3
A cube is divided into $8$ smaller cubes of the same size, as shown in the figure. Then, each of these small cubes is divided again into $8$ smaller cubes of the same size. This process is done $4$ more times to each resulting cube. What is the ratio between the sum of the total areas of all the small cubes resulting from the last division and the total area of the initial cube?
2005 Estonia National Olympiad, 1
The height drawn on the hypotenuse of a right triangle divides the hypotenuse into two sections with a length ratio of $9: 1$ and two triangles of the starting triangle with a difference of areas of $48$ cm$^2$. Find the original triangle sidelengths.
2005 Spain Mathematical Olympiad, 3
In a triangle with sides $a, b, c$ the side $a$ is the arithmetic mean of $b$ and $c$. Prove that:
a) $0^o \le A \le 60^o$.
b) The height relative to side $a$ is three times the inradius $r$.
c) The distance from the circumcenter to side $a$ is $R - r$, where $R$ is the circumradius.
2004 Czech and Slovak Olympiad III A, 5
Let $L$ be an arbitrary point on the minor arc $CD$ of the circumcircle of square $ABCD$. Let $K,M,N$ be the intersection points of $AL,CD$; $CL,AD$; and $MK,BC$ respectively. Prove that $B,M,L,N$ are concyclic.
1999 USAMTS Problems, 5
In $\triangle ABC$, $AC>BC$, $CM$ is the median, and $CH$ is the altitude emanating from $C$, as shown in the figure on the right. Determine the measure of $\angle MCH$ if $\angle ACM$ and $\angle BCH$ each have measure $17^\circ$.
[asy]
size(200);
defaultpen(linewidth(0.8));
pair A=origin,B=(10,0),C=(7,5),M=(5,0),H=(7,0);
draw(A--C--B--cycle^^H--C--M);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,NE);
label("$M$",M,NW);
label("$H$",H,NE);
[/asy]
2009 India IMO Training Camp, 4
Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly.
Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.