Found problems: 25757
2022 Purple Comet Problems, 18
In $\vartriangle ABC$, let $D$ be on $BC$ such that $\overline{AD} \perp \overline{BC}$. Suppose also that $\tan B = 4 \sin C$, $AB^2 +CD^2 = 17$, and $AC^2 + BC^2 = 21$. Find the measure of $\angle C$ in degrees between $0^o$ and $180^o$ .
1999 Belarusian National Olympiad, 7
Let [i]O[/i] be the center of circle[i] W[/i]. Two equal chords [i]AB[/i] and [i]CD [/i]of[i] W [/i]intersect at [i]L [/i]such that [i]AL>LB [/i]and [i]DL>LC[/i]. Let [i]M [/i]and[i] N [/i]be points on [i]AL[/i] and [i]DL[/i] respectively such that ([i]ALC[/i])=2*([i]MON[/i]). Prove that the chord of [i]W[/i] passing through [i]M [/i]and [i]N[/i] is equal to [i]AB[/i] and [i]CD[/i].
2008 Portugal MO, 2
Let $AEBC$ be a cyclic quadrilateral. Let $D$ be a point on the ray $AE$ which is outside the circumscribed circumference of $AEBC$. Suppose that $\angle CAB=\angle BAE$. Prove that $AB=BD$ if and only if $DE=AC$.
1996 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 3
Let $ ABCD$ be a trapezoid with $ AB$ and $ CD$ parallel, $ \angle D \equal{} 2 \angle B, AD \equal{} 5,$ and $ CD \equal{} 2.$ Then $ AB$ equals
A. 7
B. 8
C. 13/2
D. 27/4
E. $ 5 \plus{} \frac{3 \sqrt{2}}{2}$
JBMO Geometry Collection, 2000
A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$.
[i]Albania[/i]
2001 Brazil Team Selection Test, Problem 4
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ and $Q$ be points on the segments $AB$ and $AC$, respectively, such that $BP : PQ : QC = AC : CB : BA$.
Prove that the points $A$, $P$, $Q$ and $O$ lie on one circle.
[i]Alternative formulation.[/i] Let $O$ be the center of the circumcircle of a triangle $ABC$. If $P$ and $Q$ are points on the sides $AB$ and $AC$, respectively, satisfying $\frac{BP}{PQ}=\frac{CA}{BC}$ and $\frac{CQ}{PQ}=\frac{AB}{BC}$, then show that the points $A$, $P$, $Q$ and $O$ lie on one circle.
2015 Taiwan TST Round 3, 2
In a scalene triangle $ABC$ with incenter $I$, the incircle is tangent to sides $CA$ and $AB$ at points $E$ and $F$. The tangents to the circumcircle of triangle $AEF$ at $E$ and $F$ meet at $S$. Lines $EF$ and $BC$ intersect at $T$. Prove that the circle with diameter $ST$ is orthogonal to the nine-point circle of triangle $BIC$.
[i]Proposed by Evan Chen[/i]
2014 Estonia Team Selection Test, 4
In an acute triangle the feet of altitudes drawn from vertices $A$ and $B$ are $D$ and $E$, respectively. Let $M$ be the midpoint of side $AB$. Line $CM$ intersects the circumcircle of $CDE$ again in point $P$ and the circumcircle of $CAB$ again in point $Q$. Prove that $|MP| = |MQ|$.
2008 Junior Balkan Team Selection Tests - Romania, 1
Let $ ABCD$ be a convex quadrilateral with opposite side not parallel. The line through $ A$ parallel to $ BD$ intersect line $ CD$ in $ F$, but parallel through $ D$ to $ AC$ intersect line $ AB$ at $ E$. Denote by $ M,N,P,Q$ midpoints of the segments $ AC,BD,AF,DE$. Prove that lines $ MN,PQ$ and $ AD$ are concurrent.
2016 AMC 10, 18
Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
$\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$
2004 Purple Comet Problems, 3
In $\triangle ABC$, three lines are drawn parallel to side $BC$ dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$, what is the area of the whole $\triangle ABC$?
[asy]
defaultpen(linewidth(0.7)); size(120);
pair B = (0,0), C = (1,0), A = (0.7,1); pair[] AB, AC;
draw(A--B--C--cycle);
for(int i = 1; i < 4; ++i) {
AB.push((i*A + (4-i)*B)/4); AC.push((i*A + (4-i)*C)/4);
draw(AB[i-1] -- AC[i-1]);
}
filldraw(AB[1]--AB[0]--AC[0]--AC[1]--cycle, gray(0.7));
label("$A$",A,N); label("$B$",B,S); label("$C$",C,S);[/asy]
2013 All-Russian Olympiad, 4
Let $ \omega $ be the incircle of the triangle $ABC$ and with centre $I$. Let $\Gamma $ be the circumcircle of the triangle $AIB$. Circles $ \omega $ and $ \Gamma $ intersect at the point $X$ and $Y$. Let $Z$ be the intersection of the common tangents of the circles $\omega$ and $\Gamma$. Show that the circumcircle of the triangle $XYZ$ is tangent to the circumcircle of the triangle $ABC$.
2019 Cono Sur Olympiad, 6
Let $ABC$ be an acute-angled triangle with $AB< AC$, and let $H$ be its orthocenter. The circumference with diameter $AH$ meets the circumscribed circumference of $ABC$ at $P\neq A$. The tangent to the circumscribed circumference of $ABC$ through $P$ intersects line $BC$ at $Q$. Show that $QP=QH$.
1987 Greece Junior Math Olympiad, 1
We color all the points of the plane with two colors. Prove that there are (at least) two points of the plane having the same color and at distance $1$ among them.
2024 All-Russian Olympiad Regional Round, 9.2
On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the diagonals of all such trapezoids share a common point.
2018 Harvard-MIT Mathematics Tournament, 8
Let $ABC$ be an equilateral triangle with side length $8.$ Let $X$ be on side $AB$ so that $AX=5$ and $Y$ be on side $AC$ so that $AY=3.$ Let $Z$ be on side $BC$ so that $AZ,BY,CX$ are concurrent. Let $ZX,ZY$ intersect the circumcircle of $AXY$ again at $P,Q$ respectively. Let $XQ$ and $YP$ intersect at $K.$ Compute $KX\cdot KQ.$
2016 HMIC, 2
Let $ABC$ be an acute triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\Omega$. Let $M$ be the midpoint of $AH$ and $N$ the midpoint of $BH$. Assume the points $M$, $N$, $O$, $H$ are distinct and lie on a circle $\omega$. Prove that the circles $\omega$ and $\Omega$ are internally tangent to each other.
[i]Dhroova Aiylam and Evan Chen[/i]
2015 IFYM, Sozopol, 3
The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.
2018 BMT Spring, 5
A point is picked uniformly at random inside of a square. Four segments are then drawn in connecting the point to each of the vertices of the square, cutting the square into four triangles. What is the probability that at least two of the resulting triangles are obtuse?
LMT Speed Rounds, 19
Evin picks distinct points $A, B, C, D, E$, and $F$ on a circle. What is the probability that there are exactly two intersections among the line segments $AB$, $CD$, and $EF$?
[i]Proposed by Evin Liang[/i]
2022 Cyprus JBMO TST, 3
Let $ABC$ be an acute-angled triangle, and let $D, E$ and $K$ be the midpoints of its sides $AB, AC$ and $BC$ respectively. Let $O$ be the circumcentre of triangle $ABC$, and let $M$ be the foot of the perpendicular from $A$ on the line $BC$. From the midpoint $P$ of $OM$ we draw a line parallel to $AM$, which meets the lines $DE$ and $OA$ at the points $T$ and $Z$ respectively. Prove that:
(a) the triangle $DZE$ is isosceles
(b) the area of the triangle $DZE$ is given by the formula
\[E_{DZE}=\frac{BC\cdot OK}{8}\]
2014 BMT Spring, 13
Let $ABC$ be a triangle with $AB = 16$, $AC = 10$, $BC = 18$. Let $D$ be a point on $AB$ such that $4AD = AB$ and let E be the foot of the angle bisector from $B$ onto $AC$. Let $P$ be the intersection of $CD$ and $BE$. Find the area of the quadrilateral $ADPE$.
2024 Belarusian National Olympiad, 10.6
Let $\omega$ be the circumcircle of triangle $ABC$. Tangent lines to $\omega$ at points $A$ and $C$ intersect at $K$. Line $BK$ intersects $\omega$ for the second time at $M$. On the line $BC$ point $N$ is chosen such that $\angle BAN = 90$. Line $MN$ intersects $\omega$ for the second time at $D$.
Prove that $BD=BC$
[i]P. Chernikova[/i]
1993 Austrian-Polish Competition, 9
Point $P$ is taken on the extension of side $AB$ of an equilateral triangle $ABC$ so that $A$ is between $B$ and $P$. Denote by $a$ the side length of triangle $ABC$, by $r_1$ the inradius of triangle $PAC$, and by $r_2$ the exradius of triangle $PBC$ opposite $P$. Find the sum $r_1+r_2$ as a function in $a$.
2017-IMOC, C7
There are $12$ monsters in a plane. Each monster is capable of spraying fire in a $30$-degree cone. Prove that monsters can destroy the plane.