Found problems: 25757
2018 Sharygin Geometry Olympiad, 9
A square is inscribed into an acute-angled triangle: two vertices of this square lie on the same side of the triangle and two remaining vertices lies on two remaining sides. Two similar squares are constructed for the remaining sides. Prove that three segments congruent to the sides of these squares can be the sides of an acute-angled triangle.
Estonia Open Senior - geometry, 1997.1.4
Let $H, K, L$ be the feet from the altitudes from vertices $A, B, C$ of the triangle $ABC$, respectively. Prove that
$| AK | \cdot | BL | \cdot| CH | = | HK | \cdot | KL | \cdot | LH | = | AL | \cdot | BH | \cdot | CK | $.
May Olympiad L2 - geometry, 2015.3
Let $ABCDEFGHI$ be a regular polygon of $9$ sides. The segments $AE$ and $DF$ intersect at $P$. Prove that $PG$ and $AF$ are perpendicular.
2015 Balkan MO Shortlist, G7
Let scalene triangle $ABC$ have orthocentre $H$ and circumcircle $\Gamma$. $AH$ meets $\Gamma$ at $D$ distinct from $A$. $BH$ and $CH$ meet $CA$ and $AB$ at $E$ and $F$ respectively, and $EF$ meets $BC$ at $P$. The tangents to $\Gamma$ at $B$ and $C$ meet at $T$. Show that $AP$ and $DT$ are concurrent on the circumcircle of $AFE$.
Ukrainian TYM Qualifying - geometry, XII.2
The figure shows a triangle, a circle circumscribed around it and the center of its inscribed circle. Using only one ruler (one-sided, without divisions), construct the center of the circumscribed circle.
2019 Iranian Geometry Olympiad, 5
Let $ABC$ be a triangle with $\angle A = 60^\circ$. Points $E$ and $F$ are the foot of angle bisectors of vertices $B$ and $C$ respectively. Points $P$ and $Q$ are considered such that quadrilaterals $BFPE$ and $CEQF$ are parallelograms. Prove that $\angle PAQ > 150^\circ$. (Consider the angle $PAQ$ that does not contain side $AB$ of the triangle.)
[i]Proposed by Alireza Dadgarnia[/i]
2013 Balkan MO, 1
In a triangle $ABC$, the excircle $\omega_a$ opposite $A$ touches $AB$ at $P$ and $AC$ at $Q$, while the excircle $\omega_b$ opposite $B$ touches $BA$ at $M$ and $BC$ at $N$. Let $K$ be the projection of $C$ onto $MN$ and let $L$ be the projection of $C$ onto $PQ$. Show that the quadrilateral $MKLP$ is cyclic.
([i]Bulgaria[/i])
PEN R Problems, 2
Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.
2019 IMO Shortlist, G7
Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$.
Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.
[i]Proposed by Anant Mudgal, India[/i]
2021 Science ON grade VI, 2
Is it possible for an isosceles triangle with all its sides of positive integer lengths to have an angle of $36^o$?
[i] (Adapted from Archimedes 2011, Traian Preda)[/i]
1988 Romania Team Selection Test, 12
The four vertices of a square are the centers of four circles such that the sum of theirs areas equals the square's area. Take an arbitrary point in the interior of each circle. Prove that the four arbitrary points are the vertices of a convex quadrilateral.
[i]Laurentiu Panaitopol[/i]
1989 All Soviet Union Mathematical Olympiad, 500
An insect is on a square ceiling side $1$. The insect can jump to the midpoint of the segment joining it to any of the four corners of the ceiling. Show that in $8$ jumps it can get to within $1/100$ of any chosen point on the ceiling
2007 Today's Calculation Of Integral, 191
(1) For integer $n=0,\ 1,\ 2,\ \cdots$ and positive number $a_{n},$ let $f_{n}(x)=a_{n}(x-n)(n+1-x).$ Find $a_{n}$ such that the curve $y=f_{n}(x)$ touches to the curve $y=e^{-x}.$
(2) For $f_{n}(x)$ defined in (1), denote the area of the figure bounded by $y=f_{0}(x), y=e^{-x}$ and the $y$-axis by $S_{0},$ for $n\geq 1,$ the area of the figure bounded by $y=f_{n-1}(x),\ y=f_{n}(x)$ and $y=e^{-x}$ by $S_{n}.$ Find $\lim_{n\to\infty}(S_{0}+S_{1}+\cdots+S_{n}).$
2023 Lusophon Mathematical Olympiad, 5
Let $ABCDEF$ be a regular hexagon with side 1. Point $X, Y$ are on sides $CD$ and $DE$ respectively, such that the perimeter of $DXY$ is $2$. Determine $\angle XAY$.
2007 Singapore Team Selection Test, 1
Two circles $ (O_1)$ and $ (O_2)$ touch externally at the point $C$ and internally at the points $A$ and $B$ respectively with another circle $(O)$. Suppose that the common tangent of $ (O_1)$ and $ (O_2)$ at $C$ meets $(O)$ at $P$ such that $PA=PB$. Prove that $PO$ is perpendicular to $AB$.
2016 PUMaC Team, 7
In triangle $ABC$, let $S$ be on $BC$ and $T$ be on $AC$ so that $AS \perp BC$ and $BT \perp AC$, and let $AS$ and $BT$ intersect at $H$. Let $O$ be the center of the circumcircle of $\vartriangle AHT, P$ be the center of the circumcircle of $\vartriangle BHS$, and $G$ be the other point of intersection (besides $H$) of the two circles. Let $GH$ and $OP$ intersect at $X$. If $AB = 14, BH = 6$, and HA = 11, then $XO - XP$ can be written in simplest form as $\frac{m}{n}$ . Find $m + n$.
2011 Saudi Arabia IMO TST, 1
Let $I$ be the incenter of a triangle $ABC$ and let $A', B', C'$ be midpoints of sides $BC$, $CA$, $AB$, respectively. If $IA'= IB'= IC'$ , then prove that triangle $ABC$ is equilateral.
2022 China Team Selection Test, 2
Given a non-right triangle $ABC$ with $BC>AC>AB$. Two points $P_1 \neq P_2$ on the plane satisfy that, for $i=1,2$, if $AP_i, BP_i$ and $CP_i$ intersect the circumcircle of the triangle $ABC$ at $D_i, E_i$, and $F_i$, respectively, then $D_iE_i \perp D_iF_i$ and $D_iE_i = D_iF_i \neq 0$. Let the line $P_1P_2$ intersects the circumcircle of $ABC$ at $Q_1$ and $Q_2$. The Simson lines of $Q_1$, $Q_2$ with respect to $ABC$ intersect at $W$.
Prove that $W$ lies on the nine-point circle of $ABC$.
2006 Bosnia and Herzegovina Team Selection Test, 5
Triangle $ABC$ is inscribed in circle with center $O$. Let $P$ be a point on arc $AB$ which does not contain point $C$. Perpendicular from point $P$ on line $BO$ intersects side $AB$ in point $S$, and side $BC$ in $T$. Perpendicular from point $P$ on line $AO$ intersects side $AB$ in point $Q$, and side $AC$ in $R$.
(i) Prove that triangle $PQS$ is isosceles
(ii) Prove that $\frac{PQ}{QR}=\frac{ST}{PQ}$
1996 Akdeniz University MO, 4
$25$ point in a plane and for all $3$ points, we find $2$ points such that this $2$ points' distance less than $1$ $cm$ . Prove that at least $13$ points in a circle of radius $1$ $cm$.
2008 Brazil Team Selection Test, 3
Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$.
Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$.
[i]Author: Farzan Barekat, Canada[/i]
2006 Harvard-MIT Mathematics Tournament, 2
Suppose $ABC$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{AC}$ such that $\angle ABP=45^\circ$. Given that $AP=1$ and $CP=2$, compute the area of $ABC$.
2010 Princeton University Math Competition, 6
In the following diagram, a semicircle is folded along a chord $AN$ and intersects its diameter $MN$ at $B$. Given that $MB : BN = 2 : 3$ and $MN = 10$. If $AN = x$, find $x^2$.
[asy]
size(120); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
real r = sqrt(80)/5;
pair M=(-1,0), N=(1,0), A=intersectionpoints(arc((M+N)/2, 1, 0, 180),circle(N,r))[0], C=intersectionpoints(circle(A,1),circle(N,1))[0], B=intersectionpoints(circle(C,1),M--N)[0];
draw(arc((M+N)/2, 1, 0, 180)--cycle); draw(A--N); draw(arc(C,1,180,180+2*aSin(r/2)));
label("$A$",D2(A),NW);
label("$B$",D2(B),SW);
label("$M$",D2(M),S);
label("$N$",D2(N),SE);
[/asy]
2023 Stanford Mathematics Tournament, 7
Triangle $ABC$ has $AC = 5$. $D$ and $E$ are on side $BC$ such that $AD$ and $AE$ trisect $\angle BAC$, with $D$ closer to $B$ and $DE =\frac32$, $EC =\frac52$ . From $B$ and $E$, altitudes $BF$ and $EG$ are drawn onto side $AC$. Compute $\frac{CF}{CG}-\frac{AF}{AG}$ .
2011 IFYM, Sozopol, 2
Five distinct points $A,B,C,D$ and $E$ lie on a line with $|AB|=|BC|=|CD|=|DE|$. The point $F$ lies outside the line. Let $G$ be the circumcentre of the triangle $ADF$ and $H$ the circumcentre of the triangle $BEF$. Show that the lines $GH$ and $FC$ are perpendicular.