Found problems: 25757
1978 IMO Shortlist, 13
We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.
2008 Sharygin Geometry Olympiad, 23
(V.Protasov, 10--11) In the space, given two intersecting spheres of different radii and a point $ A$ belonging to both spheres. Prove that there is a point $ B$ in the space with the following property:
if an arbitrary circle passes through points $ A$ and $ B$ then the second points of its meet with the given spheres are equidistant from $ B$.
2018 239 Open Mathematical Olympiad, 8-9.4
In a triangle, each median forms an angle with the side it is drawn to, which is less than $\alpha$. Prove that one of the angles of the triangle is greater than $180^\circ-\frac{3}{2}\alpha$.
[i]Proposed by Sergey Berlov[/i]
2020 MIG, 6
The top vertex of this equilateral triangle is folded over the shown dashed line. Which of the 5 points
will the vertex lie closest to after this fold?
[asy]
size(110);
draw((0,0)--(1,0)--(0.5,sqrt(3)/2)--cycle);
dot((0.5,sqrt(3)/2));
pair A_1=(0,0);label("$A_1$",A_1,S);dot(A_1);
pair A_2=(0.25,0);label("$A_2$",A_2,S);dot(A_2);
pair A_3=(0.5,0);label("$A_3$",A_3,S);dot(A_3);
pair A_4=(0.75,0);label("$A_4$",A_4,S);dot(A_4);
pair A_5=(1,0);label("$A_5$",A_5,S);dot(A_5);
draw((0.23,0.38)--(0.86,0.22),dashed);
[/asy]
$\textbf{(A) }A_1\qquad\textbf{(B) }A_2\qquad\textbf{(C) }A_3\qquad\textbf{(D) }A_4\qquad\textbf{(E) }A_5$
2017 All-Russian Olympiad, 2
$ABCD$ is an isosceles trapezoid with $BC || AD$. A circle $\omega$ passing through $B$ and $C$ intersects the side $AB$ and the diagonal $BD$ at points $X$ and $Y$ respectively. Tangent to $\omega$ at $C$ intersects the line $AD$ at $Z$. Prove that the points $X$, $Y$, and $Z$ are collinear.
2021 Yasinsky Geometry Olympiad, 2
Given a rectangle $ABCD$, which is located on the line $\ell$ They want it "turn over" by first turning around the vertex $D$, and then as point $C$ appears on the line $\ell$ - by making a turn around the vertex $C$ (see figure). What is the length of the curve along which the vertex $A$ is moving , at such movement, if $AB = 30$ cm, $BC = 40$ cm?
(Alexey Panasenko)
[img]https://cdn.artofproblemsolving.com/attachments/d/9/3cca36b08771b1897e385d43399022049bbcde.png[/img]
2008 Sharygin Geometry Olympiad, 4
(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A'$, $ B'$; $ C_1$ is the meet of lines $ AA'$ and $ BB'$. Prove that $ \angle C_1CA \equal{} \angle C_0CB$.
2006 Macedonia National Olympiad, 4
Let $M$ be a point on the smaller arc $A_1A_n$ of the circumcircle of a regular $n$-gon $A_1A_2\ldots A_n$ .
$(a)$ If $n$ is even, prove that $\sum_{i=1}^n(-1)^iMA_i^2=0$.
$(b)$ If $n$ is odd, prove that $\sum_{i=1}^n(-1)^iMA_i=0$.
2019 Yasinsky Geometry Olympiad, p5
In the triangle $ABC$ it is known that $BC = 5, AC - AB = 3$. Prove that $r <2 <r_a$ .
(here $r$ is the radius of the circle inscribed in the triangle $ABC$, $r_a$ is the radius of an exscribed circle that touches the sides of $BC$).
(Mykola Moroz)
2015 CCA Math Bonanza, L4.4
Sierpinski's triangle is formed by taking a triangle, and drawing an upside down triangle inside each upright triangle that appears. A snake sees the fractal, but decides that the triangles need circles inside them. Therefore, she draws a circle inscribed in every upside down triangle she sees (assume that the snake can do an infinite amount of work). If the original triangle had side length $1$, what is the total area of all the individual circles?
[i]2015 CCA Math Bonanza Lightning Round #4.4[/i]
2007 Vietnam Team Selection Test, 2
Let $ABC$ be an acute triangle with incricle $(I)$. $(K_{A})$ is the cricle such that $A\in (K_{A})$ and $AK_{A}\perp BC$ and it in-tangent for $(I)$ at $A_{1}$, similary we have $B_{1},C_{1}$.
a) Prove that $AA_{1},BB_{1},CC_{1}$ are concurrent, called point-concurrent is $P$.
b) Assume circles $(J_{A}),(J_{B}),(J_{C})$ are symmetry for excircles $(I_{A}),(I_{B}),(I_{C})$ across midpoints of $BC,CA,AB$ ,resp. Prove that $P_{P/(J_{A})}=P_{P/(J_{B})}=P_{P/(J_{C})}$.
Note. If $(O;R)$ is a circle and $M$ is a point then $P_{M/(O)}=OM^{2}-R^{2}$.
Kyiv City MO Juniors 2003+ geometry, 2009.89.5
A chord $AB$ is drawn in the circle, on which the point $P$ is selected in such a way that $AP = 2PB$. The chord $DE$ is perpendicular to the chord $AB $ and passes through the point $P$. Prove that the midpoint of the segment $AP$ is the orthocener of the triangle $AED$.
2013 Olympic Revenge, 2
Let $ABC$ to be an acute triangle. Also, let $K$ and $L$ to be the two intersections of the perpendicular from $B$ with respect to side $AC$ with the circle of diameter $AC$, with $K$ closer to $B$ than $L$. Analogously, $X$ and $Y$ are the two intersections of the perpendicular from $C$ with respect to side $AB$ with the circle of diamter $AB$, with $X$ closer to $C$ than $Y$. Prove that the intersection of $XL$ and $KY$ lies on $BC$.
1961 AMC 12/AHSME, 11
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of triangle $APR$ is
${{ \textbf{(A)}\ 42\qquad\textbf{(B)}\ 40.5 \qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 39\frac{7}{8} }\qquad\textbf{(E)}\ \text{not determined by the given information} } $
2004 Flanders Math Olympiad, 1
[u][b]The author of this posting is : Peter VDD[/b][/u]
____________________________________________________________________
most of us didn't really expect to get this, but here it goes (flanders mathematical olympiad 2004, today)
triangle with sides 501m, 668m, 835m
how many lines can be draws so that the line halves both area and circumference?
2017 Adygea Teachers' Geometry Olympiad, 3
Jack has a quadrilateral that consists of four sticks. It turned out that Jack can form three different triangles from those sticks. Prove that he can form a fourth triangle that is different from the others.
2007 Hungary-Israel Binational, 2
Given is an ellipse $ e$ in the plane. Find the locus of all points $ P$ in space such that the cone of apex $ P$ and directrix $ e$ is a right circular cone.
2007 Iran Team Selection Test, 2
Triangle $ABC$ is isosceles ($AB=AC$). From $A$, we draw a line $\ell$ parallel to $BC$. $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$. $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$. Prove that
\[\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}\]
2010 Dutch Mathematical Olympiad, 1
Consider a triangle $ABC$ such that $\angle A = 90^o, \angle C =60^o$ and $|AC|= 6$. Three circles with centers $A, B$ and $C$ are pairwise tangent in points on the three sides of the triangle.
Determine the area of the region enclosed by the three circles (the grey area in the figure).
[asy]
unitsize(0.2 cm);
pair A, B, C;
real[] r;
A = (6,0);
B = (6,6*sqrt(3));
C = (0,0);
r[1] = 3*sqrt(3) - 3;
r[2] = 3*sqrt(3) + 3;
r[3] = 9 - 3*sqrt(3);
fill(arc(A,r[1],180,90)--arc(B,r[2],270,240)--arc(C,r[3],60,0)--cycle, gray(0.7));
draw(A--B--C--cycle);
draw(Circle(A,r[1]));
draw(Circle(B,r[2]));
draw(Circle(C,r[3]));
dot("$A$", A, SE);
dot("$B$", B, NE);
dot("$C$", C, SW);
[/asy]
2023 Sharygin Geometry Olympiad, 9.8
Let $ABC$ be a triangle with $\angle A = 120^\circ$, $I$ be the incenter, and $M$ be the midpoint of $BC$. The line passing through $M$ and parallel to $AI$ meets the circle with diameter $BC$ at points $E$ and $F$ ($A$ and $E$ lie on the same semiplane with respect to $BC$). The line passing through $E$ and perpendicular to $FI$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. Find the value of $\angle PIQ$.
2022 VN Math Olympiad For High School Students, Problem 2
Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$, $T$ is its [i]Fermat-Torricelli[/i] point.
Construct 3 equilateral triangles $BCD, CAE, ABF$ outside $\triangle ABC$
Prove that: $AD, BE, CF$ are concurrent at $T$.
2012 Hanoi Open Mathematics Competitions, 14
[b]Q14.[/b] Let be given a trinagle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$. Suppose that $IH$ is perpendicular to $BC$ ($H$ belongs to $BC$). If $HB=5 \text{cm}, \; HC=8 \text{cm}$, compute the area of $\triangle ABC$.
2021 HMNT, 8
Let $n$ be the answer to this problem. Find the number of distinct (i.e. non-congruent), non-degenerate triangles with integer side lengths and perimeter $n$.
2009 All-Russian Olympiad Regional Round, 9.3
In an acute triangle $ABC$ the altitudes $AA_1$, $BB_1$, $CC_1$ are drawn. A line perpendicular to side $AC$ and passing through a point $A_1$, intersects the line $B_1C_1$ at point $D$. Prove that angle $ADC$ is right.
1978 IMO Longlists, 48
Prove that it is possible to place $2n(2n + 1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times 2 \times (n + 1)$ in a cubic box with edge $2n + 1$ if and only if $n$ is even or $n = 1$.
[i]Remark[/i]. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.