Found problems: 25757
2019 Argentina National Olympiad, 3
In triangle $ABC$ it is known that $\angle ACB = 2\angle ABC$. Furthermore $P$ is an interior point of the triangle $ABC$ such that $AP = AC$ and $PB = PC$. Prove that $\angle BAC = 3 \angle BAP$.
2014 Math Prize for Girls Olympiad, 3
Say that a positive integer is [i]sweet[/i] if it uses only the digits 0, 1, 2, 4, and 8. For instance, 2014 is sweet. There are sweet integers whose squares are sweet: some examples (not necessarily the smallest) are 1, 2, 11, 12, 20, 100, 202, and 210. There are sweet integers whose cubes are sweet: some examples (not necessarily the smallest) are 1, 2, 10, 20, 200, 202, 281, and 2424. Prove that there exists a sweet positive integer $n$ whose square and cube are both sweet, such that the sum of all the digits of $n$ is 2014.
1997 Tournament Of Towns, (560) 1
$M$ and $N$ are the midpoints of the sides $AB$ and $AC$ of a triangle ABC respectively. $P$ and $Q$ are points on the sides $AB$ and $AC$ respectively such that the bisector of the angle $ACB$ also bisects the angle $MCP$, and the bisector of the angle $ABC$ also bisects the angle $NBQ$. If $AP = AQ$, does it follow that $ABC$ is isosceles?
(V Senderov)
Croatia MO (HMO) - geometry, 2018.7
Given an acute-angled triangle $ABC$ in which $|AB| <|AC|$. Point $D$ is the midpoint of the shorter arc $BC$ of its circumcircle. The point $I$ is the center of its incircle, and the point $J$ is symmetric point of $I$ wrt line $BC$. The line $DJ$ intersects the circumcircle of the triangle $ABC$ at the point $E$ belonging to the arc $AB$. Prove that $|AI |= |IE|$.
2022 Sharygin Geometry Olympiad, 5
Let the diagonals of cyclic quadrilateral $ABCD$ meet at point $P$. The line passing through $P$ and perpendicular to $PD$ meets $AD$ at point $D_1$, a point $A_1$ is defined similarly. Prove that the tangent at $P$ to the circumcircle of triangle $D_1PA_1$ is parallel to $BC$.
2022 AMC 12/AHSME, 2
In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$?
[asy]
import olympiad;
size(180);
real r = 3, s = 5, t = sqrt(r*r+s*s);
defaultpen(linewidth(0.6) + fontsize(10));
pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0);
draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D));
label("$A$",A,SW);
label("$B$", B, NW);
label("$C$",C,NE);
label("$D$",D,SE);
label("$P$",P,S);
[/asy]
$\textbf{(A) }3\sqrt 5 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }6\sqrt 5 \qquad
\textbf{(D) }20\qquad
\textbf{(E) }25$
2020 Thailand TST, 6
Let $\mathcal L$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell\in\mathcal L$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_1,\ell_2,\ell_3$ passing through $X$, the points $f(\ell_1),f(\ell_2),f(\ell_3)$, and $X$ lie on a circle.
Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$.
[i]Australia[/i]
2024 CMIMC Geometry, 4
Let $ABC$ be an equilateral triangle with side length $1$. Points $D$ and $E$ lie on $\overline{BC}$ and $\overline{AC}$ respectively such that $\triangle BDE$ is right isosceles, while points $F$ and $G$ lie on $\overline{BC}$ and $\overline{AB}$ respectively such that $\triangle CFG$ is right isosceles. Find the area of the intersection of $\triangle BDE$ and $\triangle CFG$.
[i]Proposed by Ishin Shah[/i]
2003 All-Russian Olympiad Regional Round, 11.2
On the diagonal $AC$ of a convex quadrilateral $ABCD$ is chosen such a point $K$ such that $KD = DC$, $\angle BAC = \frac12 \angle KDC$, $\angle DAC = \frac12 \angle KBC$. Prove that $\angle KDA = \angle BCA$ or $\angle KDA = \angle KBA$.
Today's calculation of integrals, 897
Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.
2018 Junior Balkan Team Selection Tests - Romania, 3
Let $ABCD$ be a cyclic quadrilateral. The line parallel to $BD$ passing through $A$ meets the line parallel to $AC$ passing through $B$ at $E$. The circumcircle of triangle $ABE$ meets the lines $EC$ and $ED$, again, at $F$ and $G$, respectively. Prove that the lines $AB, CD$ and $FG$ are either parallel or concurrent.
2022 JHMT HS, 3
Triangle $WSE$ has side lengths $WS=13$, $SE=15$, and $WE=14$. Points $J$ and $H$ lie on $\overline{SE}$ such that $SJ=JH=HE=5$. Let the angle bisector of $\angle{WES}$ intersect $\overline{WH}$ and $\overline{WJ}$ at points $M$ and $T$, respectively. Find the area of quadrilateral $JHMT$.
2013-2014 SDML (High School), 4
$ABCD$ is a rectangle. Segment $BA$ is extended through $A$ to a point $E$. Let the intersection of $EC$ and $AD$ be point $F$. Suppose that [the] measure of $\angle{ACD}$ is $60$ degrees, and that the length of segment $EF$ is twice the length of diagonal $AC$. What is the measure of $\angle{ECD}$?
2022 JHMT HS, 7
Two rays emanate from the origin $O$ and form a $45^\circ$ angle in the first quadrant of the Cartesian coordinate plane. For some positive numbers $X$, $Y$, and $S$, the ray with the larger slope passes through point $A = (X, S)$, and the ray with the smaller slope passes through point $B = (S, Y)$. If $6X + 6Y + 5S = 600$, then determine the maximum possible area of $\triangle OAB$.
2000 239 Open Mathematical Olympiad, 3
Let $ AA_1 $ and $ CC_1 $ be the altitudes of the acute-angled triangle $ ABC $. A line passing through the centers of the inscribed circles the triangles $ AA_1C $ and $ CC_1A $ intersect the sides of $ AB $ and $ BC $ triangle $ ABC $ at points $ X $ and $ Y $. Prove that $ BX = BY $.
2007 Indonesia TST, 1
Given triangle $ ABC$ and its circumcircle $ \Gamma$, let $ M$ and $ N$ be the midpoints of arcs $ BC$ (that does not contain $ A$) and $ CA$ (that does not contain $ B$), repsectively. Let $ X$ be a variable point on arc $ AB$ that does not contain $ C$. Let $ O_1$ and $ O_2$ be the incenter of triangle $ XAC$ and $ XBC$, respectively. Let the circumcircle of triangle $ XO_1O_2$ meets $ \Gamma$ at $ Q$.
(a) Prove that $ QNO_1$ and $ QMO_2$ are similar.
(b) Find the locus of $ Q$ as $ X$ varies.
2023 Bangladesh Mathematical Olympiad, P6
Let $\triangle ABC$ be an acute angle triangle and $\omega$ be its circumcircle. Let $N$ be a point on arc $AC$ not containing $B$ and $S$ be a point on line $AB$. The line tangent to $\omega$ at $N$ intersects $BC$ at $T$, $NS$ intersects $\omega$ at $K$. Assume that $\angle NTC = \angle KSB$. Prove that $CK\parallel AN \parallel TS$.
Denmark (Mohr) - geometry, 1992.2
In a right-angled triangle, $a$ and $b$ denote the lengths of the two catheti. A circle with radius $r$ has the center on the hypotenuse and touches both catheti. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$.
2005 Oral Moscow Geometry Olympiad, 3
In triangle $ABC$, points $K ,P$ are chosen on the side $AB$ so that $AK = BL$, and points $M,N$ are chosen on the side $BC$ so that $CN = BM$. Prove that $KN + LM \ge AC$.
(I. Bogdanov)
2014 NIMO Problems, 8
The side lengths of $\triangle ABC$ are integers with no common factor greater than $1$. Given that $\angle B = 2 \angle C$ and $AB < 600$, compute the sum of all possible values of $AB$.
[i]Proposed by Eugene Chen[/i]
2014 Saudi Arabia Pre-TST, 4.4
Let $\vartriangle ABC$ be an acute triangle, with $\angle A> \angle B \ge \angle C$. Let $D, E$ and $F$ be the tangency points between the incircle of triangle and sides $BC, CA, AB$, respectively. Let $J$ be a point on $(BD)$, $K$ a point on $(DC)$, $L$ a point on $(EC)$ and $M$ a point on $(FB)$, such that $$AF = FM = JD = DK = LE = EA.$$Let $P$ be the intersection point between $AJ$ and $KM$ and let $Q$ be the intersection point between $AK$ and $JL$. Prove that $PJKQ$ is cyclic.
2018 Moscow Mathematical Olympiad, 10
$ABC$ is acute-angled triangle, $AA_1,CC_1$ are altitudes. $M$ is centroid. $M$ lies on circumcircle of $A_1BC_1$. Find all values of $\angle B$
1994 Bundeswettbewerb Mathematik, 3
Let $A$ and $B$ be two spheres of different radii, both inscribed in a cone $K$. There are $m$ other, congruent spheres arranged in a ring such that each of them touches $A, B, K$ and two of the other spheres. Prove that this is possible for at most three values of $m.$
2007 National Olympiad First Round, 25
Let $A, B, C$ be points on a unit circle such that $|AB|=|BC|$ and $m(\widehat{ABC})=72^\circ$. Let $D$ be a point such that $\triangle BCD$ is equilateral. If $AD$ meets the circle at $D$, what is $|DE|$?
$
\textbf{(A)}\ \dfrac 12
\qquad\textbf{(B)}\ \dfrac {\sqrt 3}2
\qquad\textbf{(C)}\ \dfrac {\sqrt 2}2
\qquad\textbf{(D)}\ \sqrt 3 -1
\qquad\textbf{(E)}\ \text{None of the above}
$
1968 All Soviet Union Mathematical Olympiad, 114
Given a quadrangle $ABCD$. The lengths of all its sides and diagonals are the rational numbers. Let $O$ be the point of its diagonals intersection. Prove that $|AO|$ - the length of the $[AO]$ segment is also rational.