Found problems: 25757
2005 Postal Coaching, 6
Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, and let $E$ be the midpoint of its side $BC$. Suppose we can inscribe a circle into the quadrilateral $ABED$, and that we can inscribe a circle into the quadrilateral $AECD$. Denote $|AB|=a$, $|BC|=b$, $|CD|=c$, $|DA|=d$. Prove that \[a+c=\frac{b}{3}+d;\] \[\frac{1}{a}+\frac{1}{c}=\frac{3}{b}\]
1985 Tournament Of Towns, (080) T1
A median , a bisector and an altitude of a certain triangle intersect at an inner point $O$ . The segment of the bisector from the vertex to $O$ is equal to the segment of the altitude from the vertex to $O$ . Prove that the triangle is equilateral .
2011 Uzbekistan National Olympiad, 3
Given an acute triangle $ABC$ with altituties AD and BE. O circumcinter of $ABC$.If o lies on the segment DE then find the value of $sinAsinBcosC$
2004 Oral Moscow Geometry Olympiad, 5
Trapezoid $ABCD$ with bases $AB$ and $CD$ is inscribed in a circle. Prove that the quadrilateral formed by orthogonal projections of any point of this circle onto lines $AC, BC, AD$ and $BD$ is inscribed.
2010 Chile National Olympiad, 6
Prove that in the interior of an equilateral triangle with side $a$ you can put a finite number of equal circles that do not overlap, with radius $r = \frac{a}{2010}$, so that the sum of their areas is greater than $\frac{17\sqrt3}{80}$ a$^2$.
2016 Sharygin Geometry Olympiad, P13
$L$ is a Line that intersect with the side $AB,BC,AC$ of triangle $ABC$ at $F,D,E$
The line perpendicular to $BC$ from $D$ intersect $AB,AC$ at $A_{1},A_{2}$ respectively
Name $B_{1},B_{2},C_{1},C_{2}$ similarly
Prove that the circumcenters of $AA_{1}A_{2},BB_{1}B_{2},CC_{1}C_{2}$ are collinear
2019 India IMO Training Camp, P2
Let $ABC$ be a triangle with $\angle A=\angle C=30^{\circ}.$ Points $D,E,F$ are chosen on the sides $AB,BC,CA$ respectively so that $\angle BFD=\angle BFE=60^{\circ}.$ Let $p$ and $p_1$ be the perimeters of the triangles $ABC$ and $DEF$, respectively. Prove that $p\le 2p_1.$
1973 Poland - Second Round, 1
Prove that if positive numbers $ x, y, z $ satisfy the inequality
$$
\frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1,$$
then they are the lengths of the sides of a certain triangle.
2018 Brazil Undergrad MO, 6
Given an equilateral triangle $ABC$ in the plane, how many points $P$ in the plane are such that the three triangles $AP B, BP C $ and $CP A$ are isosceles and not degenerate?
2010 Kazakhstan National Olympiad, 4
It is given that for some $n \in \mathbb{N}$ there exists a natural number $a$, such that $a^{n-1} \equiv 1 \pmod{n}$ and that for any prime divisor $p$ of $n-1$ we have $a^{\frac{n-1}{p}} \not \equiv 1 \pmod{n}$.
Prove that $n$ is a prime.
2019 India IMO Training Camp, P3
Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.
Swiss NMO - geometry, 2006.7
Let $ABCD$ be a cyclic quadrilateral with $\angle ABC = 60^o$ and $| BC | = | CD |$. Prove that $|CD| + |DA| = |AB|$
Durer Math Competition CD Finals - geometry, 2013.C5
The points $A, B, C, D, P$ lie on an circle as shown in the figure such that $\angle AP B = \angle BPC = \angle CPD$. Prove that the lengths of the segments are denoted by $a, b, c, d$ by $\frac{a + c}{b + d} =\frac{b}{c}$.
[img]https://cdn.artofproblemsolving.com/attachments/a/2/ba8965f5d7d180426db26e8f7dd5c7ad02c440.png[/img]
1998 Romania National Olympiad, 4
Let $ABCD$ be an arbitrary tetrahedron. The bisectors of the angles $\angle BDC$, $\angle CDA$ and $\angle ADB$ intersect $BC$, $CA$ and $AB$, in the points $M$, $N$, $P$, respectively.
a) Show that the planes $(ADM)$, $(BDN)$ and $(CDP)$ have a common line $d$.
b) Let the points $A' \in (AD)$, $B' \in (BD)$ and $C' \in (CD)$ be such that $(AA') = (BB') = (CC')$ ; show that if $G$ and $G'$ are the centroids of $ABC$ and $A'B'C'$ then the lines $GG'$ and $d$ are either parallel or identical.
1937 Moscow Mathematical Olympiad, 033
* On a plane two points $A$ and $B$ are on the same side of a line. Find point $M$ on the line such that $MA +MB$ is equal to a given length.
1981 Czech and Slovak Olympiad III A, 2
Let $n$ be a positive integer. Consider $n^2+1$ (closed, i.e. including endpoints) segments on a single line. Show that at least one of the following statements holds:
a) there are $n+1$ segments with non-empty intersection,
b) there are $n+1$ segments among which two of them are disjoint.
1960 IMO, 3
In a given right triangle $ABC$, the hypotenuse $BC$, of length $a$, is divided into $n$ equal parts ($n$ and odd integer). Let $\alpha$ be the acute angel subtending, from $A$, that segment which contains the mdipoint of the hypotenuse. Let $h$ be the length of the altitude to the hypotenuse fo the triangle. Prove that: \[ \tan{\alpha}=\dfrac{4nh}{(n^2-1)a}. \]
2018 Polish Junior MO Finals, 5
Point $M$ is middle of side $AB$ of equilateral triangle $ABC$. Points $D$ and $E$ lie on segments $AC$ and $BC$, respectively and $\angle DME = 60 ^{\circ}$. Prove that, $AD + BE = DE + \frac{1}{2}AB$.
2010 Iran MO (3rd Round), 6
[b]polyhedral[/b]
we call a $12$-gon in plane good whenever:
first, it should be regular, second, it's inner plane must be filled!!, third, it's center must be the origin of the coordinates, forth, it's vertices must have points $(0,1)$,$(1,0)$,$(-1,0)$ and $(0,-1)$.
find the faces of the [u]massivest[/u] polyhedral that it's image on every three plane $xy$,$yz$ and $zx$ is a good $12$-gon.
(it's obvios that centers of these three $12$-gons are the origin of coordinates for three dimensions.)
time allowed for this question is 1 hour.
2004 Junior Balkan MO, 2
Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.
2017 Harvard-MIT Mathematics Tournament, 2
Let $A$, $B$, $C$, $D$, $E$, $F$ be $6$ points on a circle in that order. Let $X$ be the intersection of $AD$ and $BE$, $Y$ is the intersection of $AD$ and $CF$, and $Z$ is the intersection of $CF$ and $BE$. $X$ lies on segments $BZ$ and $AY$ and $Y$ lies on segment $CZ$. Given that $AX = 3$, $BX = 2$, $CY = 4$, $DY = 10$, $EZ = 16$, and $FZ = 12$, find the perimeter of triangle $XYZ$.
1985 IMO Longlists, 39
Given a triangle $ABC$ and external points $X, Y$ , and $Z$ such that $\angle BAZ = \angle CAY , \angle CBX = \angle ABZ$, and $\angle ACY = \angle BCX$, prove that $AX,BY$ , and $CZ$ are concurrent.
1946 Moscow Mathematical Olympiad, 111
Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.
2024 Turkey Olympic Revenge, 4
Let the circumcircle of a triangle $ABC$ be $\Gamma$. The tangents to $\Gamma$ at $B,C$ meet at point $E$. For a point $F$ on line $BC$ which is not on the segment $BC$, let the midpoint of $EF$ be $G$. Lines $GB,GC$ meet $\Gamma$ again at points $I,H$ respectively. Let $M$ be the midpoint of $BC$. Prove that the points $F,I,H,M$ lie on a circle.
Proposed by [i]Mehmet Can Baştemir[/i]
1955 AMC 12/AHSME, 14
The length of rectangle R is $ 10$ percent more than the side of square S. The width of the rectangle is $ 10$ percent less than the side of the square. The ratio of the areas, R:S, is:
$ \textbf{(A)}\ 99: 100 \qquad
\textbf{(B)}\ 101: 100 \qquad
\textbf{(C)}\ 1: 1 \qquad
\textbf{(D)}\ 199: 200 \qquad
\textbf{(E)}\ 201: 200$