Found problems: 25757
Denmark (Mohr) - geometry, 2023.4
In the $9$-gon $ABCDEFGHI$, all sides have equal lengths and all angles are equal. Prove that $|AB| + |AC| = |AE|$.
[img]https://cdn.artofproblemsolving.com/attachments/6/2/8c82e8a87bf8a557baaf6ac72b3d18d2ba3965.png[/img]
1995 Singapore MO Open, 2
Let $A_1A_2A_3$ be a triangle and $M$ an interior point. The straight lines $MA_1, MA_2, MA_3$ intersect the opposite sides at the points $B_1, B_2, B_3$ respectively (see Fig.). Show that if the areas of triangles $A_2B_1M, A_3B_2M$ and $A_1B_3M$ are equal, then $M$ coincides with the centroid of triangle $A_1A_2A_3$.
[img]https://cdn.artofproblemsolving.com/attachments/1/7/b29bdbb1f2b103be1f3cb2650b3bfff352024a.png[/img]
1950 Kurschak Competition, 2
Three circles $C_1$, $C_2$, $C_3$ in the plane touch each other (in three different points). Connect the common point of $C_1$ and $C_2$ with the other two common points by straight lines. Show that these lines meet $C_3$ in diametrically opposite points.
2001 AMC 10, 15
A street has parallel curbs $ 40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $ 15$ feet and each stripe is $ 50$ feet long. Find the distance, in feet, between the stripes.
$ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 25$
2013 Princeton University Math Competition, 4
An equilateral triangle is given. A point lies on the incircle of this triangle. If the smallest two distances from the point to the sides of the triangle is $1$ and $4$, the sidelength of this equilateral triangle can be expressed as $\tfrac{a\sqrt b}c$ where $(a,c)=1$ and $b$ is not divisible by the square of an integer greater than $1$. Find $a+b+c$.
1998 Bosnia and Herzegovina Team Selection Test, 4
Circle $k$ with radius $r$ touches the line $p$ in point $A$. Let $AB$ be a dimeter of circle and $C$ an arbitrary point of circle distinct from points $A$ and $B$. Let $D$ be a foot of perpendicular from point $C$ to line $AB$. Let $E$ be a point on extension of line $CD$, over point $D$, such that $ED=BC$. Let tangents on circle from point $E$ intersect line $p$ in points $K$ and $N$. Prove that length of $KN$ does not depend from $C$
2002 Germany Team Selection Test, 2
Prove: If $x, y, z$ are the lengths of the angle bisectors of a triangle with perimeter 6, than we have:
\[\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} \geq 1.\]
2017 Sharygin Geometry Olympiad, P8
Let $AD$ be the base of trapezoid $ABCD$. It is known that the circumcenter of triangle $ABC$ lies on $BD$. Prove that the circumcenter of triangle $ABD$ lies on $AC$.
[i]Proposed by Ye.Bakayev[/i]
Kharkiv City MO Seniors - geometry, 2019.11.5
In the acute-angled triangle $ABC$, let $CD, AE$ be the altitudes. Points $F$ and $G$ are the projections of $A$ and $C$ on the line $DE$, respectively, $H$ and $K$ are the projections of $D$ and $E$ on the line $AC$, respectively. The lines $HF$ and $KG$ intersect at point $P$. Prove that line $BP$ bisects the segment $DE$.
1987 Bulgaria National Olympiad, Problem 5
Let $E$ be a point on the median $AD$ of a triangle $ABC$, and $F$ be the projection of $E$ onto $BC$. From a point $M$ on $EF$ the perpendiculars $MN$ to $AC$ and $MP$ to $AB$ are drawn. Prove that if the points $N,E,P$ lie on a line, then $M$ lies on the bisector of $\angle BAC$.
1982 Poland - Second Round, 2
The plane is covered with circles in such a way that the center of each circle does not belong to any other circle. Prove that each point of the plane belongs to at most five circles.
2004 Silk Road, 3
In-circle of $ABC$ with center $I$ touch $AB$ and $AC$ at $P$ and $Q$ respectively. $BI$ and $CI$ intersect $PQ$ at $K$ and $L$ respectively. Prove, that circumcircle of $ILK$ touch incircle of $ABC$ iff $|AB|+|AC|=3|BC|$.
2014 China Girls Math Olympiad, 6
In acute triangle $ABC$, $AB > AC$.
$D$ and $E$ are the midpoints of $AB$, $AC$ respectively.
The circumcircle of $ADE$ intersects the circumcircle of $BCE$ again at $P$.
The circumcircle of $ADE$ intersects the circumcircle $BCD$ again at $Q$.
Prove that $AP = AQ$.
2004 AMC 12/AHSME, 24
A plane contains points $ A$ and $ B$ with $ AB \equal{} 1$. Let $ S$ be the union of all disks of radius $ 1$ in the plane that cover $ \overline{AB}$. What is the area of $ S$?
$ \textbf{(A)}\ 2\pi \plus{} \sqrt3 \qquad \textbf{(B)}\ \frac {8\pi}{3} \qquad \textbf{(C)}\ 3\pi \minus{} \frac {\sqrt3}{2} \qquad \textbf{(D)}\ \frac {10\pi}{3} \minus{} \sqrt3 \qquad \textbf{(E)}\ 4\pi \minus{} 2\sqrt3$
1998 AIME Problems, 12
Let $ABC$ be equilateral, and $D, E,$ and $F$ be the midpoints of $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. There exist points $P, Q,$ and $R$ on $\overline{DE}, \overline{EF},$ and $\overline{FD},$ respectively, with the property that $P$ is on $\overline{CQ}, Q$ is on $\overline{AR},$ and $R$ is on $\overline{BP}.$ The ratio of the area of triangle $ABC$ to the area of triangle $PQR$ is $a+b\sqrt{c},$ where $a, b$ and $c$ are integers, and $c$ is not divisible by the square of any prime. What is $a^{2}+b^{2}+c^{2}$?
2013 AIME Problems, 8
A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.
2015 Mathematical Talent Reward Programme, MCQ: P 8
In $\triangle A B C$, $A B=A C$ and $D$ is foot of the perpendicular from $C$ to $A B$ and $E$ the foot of the perpendicular from $B$ to $A C,$ then
[list=1]
[*] $BC^3>BD^3+BE^3$
[*] $BC^3 <BD^3+BE^3$
[*] $BC^3=BD^3+BE^3$
[*] None of these
[/list]
2019 District Olympiad, 4
Consider the isosceles right triangle$ ABC, \angle A = 90^o$, and point $D \in (AB)$ such that $AD = \frac13 AB$. In the half-plane determined by the line $AB$ and point $C$ , consider a point $E$ such that $\angle BDE = 60^o$ and $\angle DBE = 75^o$. Lines $BC$ and $DE$ intersect at point $G$, and the line passing through point $G$ parallel to the line $AC$ intersects the line $BE$ at point $H$. Prove that the triangle $CEH$ is equilateral.
1966 IMO Shortlist, 32
The side lengths $a,$ $b,$ $c$ of a triangle $ABC$ form an arithmetical progression (such that $b-a=c-b$). The side lengths $a_{1},$ $b_{1},$ $c_{1}$ of a triangle $A_{1}B_{1}C_{1}$ also form an arithmetical progression (with $b_{1}-a_{1}=c_{1}-b_{1}$). [Hereby, $a=BC,$ $b=CA,$ $c=AB, $ $a_{1}=B_{1}C_{1},$ $b_{1}=C_{1}A_{1},$ $c_{1}=A_{1}B_{1}.$] Moreover, we know that $\measuredangle CAB=\measuredangle C_{1}A_{1}B_{1}.$
Show that triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.
1956 Moscow Mathematical Olympiad, 330
A square of side $a$ is inscribed in a triangle so that two of the square’s vertices lie on the base, and the other two lie on the sides of the triangle. Prove that if $r$ is the radius of the circle inscribed in the triangle, then $r\sqrt2 < a < 2r$.
1989 ITAMO, 4
Points $A,M,B,C,D$ are given on a circle in this order such that $A$ and $B$ are equidistant from $M$. Lines $MD$ and $AC$ intersect at $E$ and lines $MC$ and $BD$ intersect at $F$. Prove that the quadrilateral $CDEF$ is inscridable in a circle.
2018 Argentina National Olympiad Level 2, 3
A geometry program on the computer allows the following operations to be performed:
[list]
[*]Mark points on segments, on lines or outside them.
[*]Draw the line that joins two points.
[*]Find the point of intersection of two lines.
[*]Given a point $P$ and a line $\ell$, trace the symmetric of $P$ with respect to $\ell$.
[/list]
Given an triangle $ABC$, using exclusively the allowed operations, construct the intersection point of the perpendicular bisectors of the triangle.
2012 Tuymaada Olympiad, 3
A circle is contained in a quadrilateral with successive sides of lengths $3,6,5$ and $8$. Prove that the length of its radius is less than $3$.
[i]Proposed by K. Kokhas[/i]
2001 Tournament Of Towns, 1
On the plane is a triangle with red vertices and a triangle with blue vertices. $O$ is a point inside both triangles such that the distance from $O$ to any red vertex is less than the distance from $O$ to any blue vertex. Can the three red vertices and the three blue vertices all lie on the same circle?
1992 Bundeswettbewerb Mathematik, 3
Given is a triangle $ABC$ with side lengths $a, b,c$. Three spheres touch each other in pairs and also touch the plane of the triangle at points $A,B$ and $C$, respectively. Determine the radii of these spheres.