Found problems: 25757
2018 Sharygin Geometry Olympiad, 3
Let $AL$ be the bisector of triangle $ABC$, $D$ be its midpoint, and $E$ be the projection of $D$ to $AB$. It is known that $AC = 3AE$. Prove that $CEL$ is an isosceles triangle.
2000 Korea Junior Math Olympiad, 3
Acute triangle $ABC$ is inscribed in circle $O$. $P$ is the foot of altitude from $A$ to $BC$, and $D$ is the intersection of $O$ and line $AP$. $M, N$ are midpoint of $AB, AC$ respectively. $MP$ and $CD$ intersects at $Q$, and $NP$ and $BD$ intersects at $R$. Show that $AD, BQ, CR$ meet at one point if and only if $AB=AC$.
1993 Iran MO (2nd round), 2
Let $ABC$ be an acute triangle with sides and area equal to $a, b, c$ and $S$ respectively. [color=#FF0000]Prove or disprove[/color] that a necessary and sufficient condition for existence of a point $P$ inside the triangle $ABC$ such that the distance between $P$ and the vertices of $ABC$ be equal to $x, y$ and $z$ respectively is that there be a triangle with sides $a, y, z$ and area $S_1$, a triangle with sides $b, z, x$ and area $S_2$ and a triangle with sides $c, x, y$ and area $S_3$ where $S_1 + S_2 + S_3 = S.$
Kvant 2023, M2735
Let $AB$ be a diameter of the circle $\Omega$ with center $O{}$. The points $C, D, X$ and $Y{}$ are chosen on $\Omega$ so that the segments $CX$ and $DX$ intersect the segment $AB$ at points symmetric with respect to $O{}$, and $XY\parallel AB$. Let the lines $AB{}$ and $CD{}$ intersect at the point $E$. Prove that the tangent to $\Omega$ through $Y{}$ passes through $E{}$.
2000 Bosnia and Herzegovina Team Selection Test, 2
Let $S$ be a point inside triangle $ABC$ and let lines $AS$, $BS$ and $CS$ intersect sides $BC$, $CA$ and $AB$ in points $X$, $Y$ and $Z$, respectively. Prove that $$\frac{BX\cdot CX}{AX^2}+\frac{CY\cdot AY}{BY^2}+\frac{AZ\cdot BZ}{CZ^2}=\frac{R}{r}-1$$ iff $S$ is incenter of $ABC$
1997 Kurschak Competition, 1
Let $p>2$ be a prime number and let $L=\{0,1,\dots,p-1\}^2$. Prove that we can find $p$ points in $L$ with no three of them collinear.
2013 Saudi Arabia GMO TST, 3
$ABC$ is a triangle, $H$ its orthocenter, $I$ its incenter, $O$ its circumcenter and $\omega$ its circumcircle. Line $CI$ intersects circle $\omega$ at point $D$ different from $C$. Assume that $AB = ID$ and $AH = OH$. Find the angles of triangle $ABC$.
1998 National High School Mathematics League, 11
If ellipse $x^2+4(y-a)^2=4$ and parabola $x^2=2y$ have intersections, then the range value of $a$ is________.
1994 Denmark MO - Mohr Contest, 1
A wine glass with a cross section as shown has the property of an orange in shape as a sphere with a radius of $3$ cm just can be placed in the glass without protruding above glass. Determine the height $h$ of the glass.
[img]https://1.bp.blogspot.com/-IuLm_IPTvTs/XzcH4FAjq5I/AAAAAAAAMYY/qMi4ng91us8XsFUtnwS-hb6PqLwAON_jwCLcBGAsYHQ/s0/1994%2BMohr%2Bp1.png[/img]
1996 IberoAmerican, 1
Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$.
2014 Balkan MO Shortlist, N3
$\boxed{N3}$Prove that there exist infinitely many non isosceles triangles with rational side lengths$,$rational lentghs of altitudes and$,$ perimeter equal to $3.$
2017 Vietnamese Southern Summer School contest, Problem 4
Let $ABC$ be a triangle. A point $P$ varies inside $BC$. Let $Q, R$ be the points on $AC, AB$ in that order, such that $PQ\parallel AB, PR\parallel AC$.
1. Prove that, when $P$ varies, the circumcircle of triangle $AQR$ always passes through a fixed point $X$ other than $A$.
2. Extend $AX$ so that it cuts the circumcircle of $ABC$ a second time at point $K$. Prove that $AX=XK$.
2015 CHMMC (Fall), 1
$3$ players take turns drawing lines that connect vertices of a regular $n$-gon. No player may draw a line that intersects another line at a point other than a vertex of the $n-$gon. The last player able to draw a line wins. For how many $n$ in the range $4\le n \le 100$ does the first player have a winning strategy?
2023 Estonia Team Selection Test, 6
Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$.
Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$
1993 National High School Mathematics League, 3
Horizontal line $m$ passes the center of circle $\odot O$. Line $l\perp m$, $l$ and $m$ intersect at $M$, and $M$ is on the right side of $O$. Three points $A,B,C$ ($B$ is in the middle) lie on line $l$, which are outside the circle, above line $m$. $AP,BQ,CR$ are tangent to $\odot O$ at $P,Q,R$. Prove:
[b](a)[/b] If $l$ is tangent to $\odot O$, then $AB\cdot CR+BC\cdot AP=AC\cdot BQ$.
[b](b)[/b] If $l$ and $\odot O$ intersect, then $AB\cdot CR+BC\cdot AP<AC\cdot BQ$.
[b](c)[/b] If $l$ and $\odot O$ are apart, then $AB\cdot CR+BC\cdot AP>AC\cdot BQ$.
2003 India National Olympiad, 5
Let a, b, c be the sidelengths and S the area of a triangle ABC. Denote $x=a+\frac{b}{2}$, $y=b+\frac{c}{2}$ and $z=c+\frac{a}{2}$. Prove that there exists a triangle with sidelengths x, y, z, and the area of this triangle is $\geq\frac94 S$.
2001 National Olympiad First Round, 13
Let $ABC$ be a triangle such that $|BC|=7$ and $|AB|=9$. If $m(\widehat{ABC}) = 2m(\widehat{BCA})$, then what is the area of the triangle?
$
\textbf{(A)}\ 14\sqrt 5
\qquad\textbf{(B)}\ 30
\qquad\textbf{(C)}\ 10\sqrt 6
\qquad\textbf{(D)}\ 20 \sqrt 2
\qquad\textbf{(E)}\ 12 \sqrt 3
$
1981 USAMO, 4
The sum of the measures of all the face angles of a given complex polyhedral angle is equal to the sum of all its dihedral angles. Prove that the polyhedral angle is a trihedral angle.
$\mathbf{Note:}$ A convex polyhedral angle may be formed by drawing rays from an exterior point to all points of a convex polygon.
2023 Princeton University Math Competition, A5 / B7
Let $\vartriangle ABC$ have $AB = 15$, $AC = 20$, and $BC = 21$. Suppose $\omega$ is a circle passing through $A$ that is tangent to segment $BC$. Let point $D\ne A$ be the second intersection of AB with $\omega$, and let point $E \ne A$ be the second intersection of $AC$ with $\omega$. Suppose $DE$ is parallel to $BC$. If $DE = \frac{a}{b}$ , where $a$, $b$ are relatively prime positive integers, find $a + b$.
1986 All Soviet Union Mathematical Olympiad, 428
A line is drawn through the $A$ vertex of triangle $ABC$ with $|AB|\ne|AC|$. Prove that the line can not contain more than one point $M$ such, that $M$ is not a triangle vertex, and $\angle ABM = \angle ACM$. What lines do not contain such a point $M$ at all?
2007 District Olympiad, 3
Let $ABC$ be a triangle with $BC=a$ $AC=b$ $AB=c$. For each line $\Delta$ we denote $d_{A}, d_{B}, d_{C}$ the distances from $A,B, C$ to $\Delta$ and we consider the expresion $E(\Delta)=ad_{A}^{2}+bd_{B}^{2}+cd_{C}^{2}$. Prove that if $E(\Delta)$ is minimum, then $\Delta$ passes through the incenter of $\Delta ABC$.
2006 China Northern MO, 3
$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$.
[i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.
2010 AMC 10, 5
The area of a circle whose circumference is $ 24\pi$ is $ k\pi$. What is the value of $ k$?
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 12 \qquad
\textbf{(C)}\ 24 \qquad
\textbf{(D)}\ 36 \qquad
\textbf{(E)}\ 144$
1974 IMO Longlists, 17
Show that there exists a set $S$ of $15$ distinct circles on the surface of a sphere, all having the same radius and such that $5$ touch exactly $5$ others, $5$ touch exactly $4$ others, and $5$ touch exactly $3$ others.
[i][General Problem: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=384764][/i]
Durer Math Competition CD 1st Round - geometry, 2021.D4
In the triangle $ABC$ we have $30^o$ at the vertex $A$, and $50^o$ at the vertex $B$. Let $O$ be the center of inscribed circle. Show that $AC + OC = AB$.