Found problems: 25757
2017 Germany, Landesrunde - Grade 11/12, 5
In a right-angled triangle let $r$ be the inradius and $s_a,s_b$ be the lengths of the medians of the legs $a,b$.
Prove the inequality \[ \frac{r^2}{s_a^2+s_b^2} \leq \frac{3-2 \sqrt2}{5}. \]
2007 AMC 8, 12
A unit hexagon is composed of a regular haxagon of side length 1 and its equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?
[asy]
defaultpen(linewidth(0.7));
draw(polygon(3));
pair D=origin+1*dir(270), E=origin+1*dir(150), F=1*dir(30);
draw(D--E--F--cycle);[/asy]
$\textbf{(A)}\: 1:1\qquad \textbf{(B)}\: 6:5\qquad \textbf{(C)}\: 3:2\qquad \textbf{(D)}\: 2:1\qquad \textbf{(E)}\: 3:1\qquad $
1997 Akdeniz University MO, 5
A $ABC$ triangle divide by a $d$ line such that, new two pieces' areas are equal. $d$ line intersects with $[AB]$ at $D$, $[AC]$ at $E$. Prove that
$$\frac{AD+AE}{BD+DE+EC+CB} > \frac{1}{4}$$
2004 Purple Comet Problems, 11
How far is it from the point $(9, 17)$ to its reflection across the line \[3x + 4y = 15?\]
2003 Baltic Way, 14
Equilateral triangles $AMB,BNC,CKA$ are constructed on the exterior of a triangle $ABC$. The perpendiculars from the midpoints of $MN, NK, KM$ to the respective lines $CA, AB, BC$ are constructed. Prove that these three perpendiculars pass through a single point.
1993 Dutch Mathematical Olympiad, 4
Let $ C$ be a circle with center $ M$ in a plane $ V$, and $ P$ be a point not on the circle $ C$.
$ (a)$ If $ P$ is fixed, prove that $ AP^2\plus{}BP^2$ is a constant for every diameter $ AB$ of the circle $ C$.
$ (b)$ Let $ AB$ be a fixed diameter of $ C$ and $ P$ a point on a fixed sphere $ S$ not intersecting $ V$. Determine the points $ P$ on $ S$ that minimize $ AP^2\plus{}BP^2$.
1992 IMO Shortlist, 20
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
2015 BMT Spring, 13
There exist right triangles with integer side lengths such that the legs differ by $ 1$. For example, $3-4-5$ and $20-21-29$ are two such right triangles. What is the perimeter of the next smallest Pythagorean right triangle with legs differing by $ 1$?
2006 Austrian-Polish Competition, 6
Let $D$ be an interior point of the triangle $ABC$.
$CD$ and $AB$ intersect at $D_{c}$,
$BD$ and $AC$ intersect at $D_{b}$,
$AD$ and $BC$ intersect at $D_{a}$.
Prove that there exists a triangle $KLM$ with orthocenter $H$ and the feet of altitudes $H_{k}\in LM, H_{l}\in KM, H_{m}\in KL$, so that
$(AD_{c}D) = (KH_{m}H)$
$(BD_{c}D) = (LH_{m}H)$
$(BD_{a}D) = (LH_{k}H)$
$(CD_{a}D) = (MH_{k}H)$
$(CD_{b}D) = (MH_{l}H)$
$(AD_{b}D) = (KH_{l}H)$
where $(PQR)$ denotes the area of the triangle $PQR$
1980 USAMO, 4
The inscribed sphere of a given tetrahedron touches all four faces of the tetrahedron at their respective centroids. Prove that the tetrahedron is regular.
2024 Thailand TST, 1
Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.
Prove that lines $AD, PM$, and $BC$ are concurrent.
2012 Online Math Open Problems, 16
Let $A_1B_1C_1D_1A_2B_2C_2D_2$ be a unit cube, with $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ opposite square faces, and let $M$ be the center of face $A_2 B_2 C_2 D_2$. Rectangular pyramid $MA_1B_1C_1D_1$ is cut out of the cube. If the surface area of the remaining solid can be expressed in the form $a + \sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime, find $a+b$.
[i]Author: Alex Zhu[/i]
2014 Junior Balkan Team Selection Tests - Romania, 5
Let $D$ and $E$ be the midpoints of sides $[AB]$ and $[AC]$ of the triangle $ABC$. The circle of diameter $[AB]$ intersects the line $DE$ on the opposite side of $AB$ than $C$, in $X$. The circle of diameter $[AC]$ intersects $DE$ on the opposite side of $AC$ than $B$ in $Y$ . Let $T$ be the intersection of $BX$ and $CY$.
Prove that the orthocenter of triangle $XY T$ lies on $BC$.
2022 Novosibirsk Oral Olympiad in Geometry, 7
The diagonals of the convex quadrilateral $ABCD$ intersect at the point $O$. The points $X$ and $Y$ are symmetrical to the point $O$ with respect to the midpoints of the sides $BC$ and $AD$, respectively. It is known that $AB = BC = CD$. Prove that the point of intersection of the perpendicular bisectors of the diagonals of the quadrilateral lies on the line $XY$.
2024 AIME, 15
Let $\mathcal{B}$ be the set of rectangular boxes that have volume $23$ and surface area $54$. Suppose $r$ is the least possible radius of a sphere that can fit any element of $\mathcal{B}$ inside it. Then $r^{2}$ can be expressed as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2008 Junior Balkan Team Selection Tests - Moldova, 3
Rhombuses $ABCD$ and $A_1B_1C_1D_1$ are equal. Side $BC$ intersects sides $B_1C_1$ and $C_1D_1$ at points $M$ and $N$ respectively. Side $AD$ intersects sides $A_1B_1$ and $A_1D_1$ at points $Q$ and $P$ respectively. Let $O$ be the intersection point of lines $MP$ and $QN$. Find $\angle A_1B_1C_1$ , if $\angle QOP = \frac12 \angle B_1C_1D_1$.
1990 IMO Longlists, 68
In coordinate plane, a variable point $M$, starting from the origin $O(0, 0)$, moves on the line $l$ with slope $k$, where $k$ is an irrational number.
[b](i)[/b] Prove that point $O(0, 0)$ is the only rational point (namely, the coordinates of which are both rationals) on the line $l.$
[b](ii)[/b] Prove that for any number $\varepsilon > 0$, there exist integers $m, n$ such that the distance between $l$ and the point $(m, n)$ is less than $\varepsilon.$
2025 Bulgarian Winter Tournament, 10.2
Let $D$ be an arbitrary point on the side $BC$ of the non-isosceles acute triangle $ABC$. The circle with center $D$ and radius $DA$ intersects the rays $AB^\to$ (after $B$) and $AC^\to$ (after $C$) at $M$ and $N$. Prove that the orthocenter of triangle $AMN$ lies on a fixed line, independent of the choice of $D$.
2007 Oral Moscow Geometry Olympiad, 3
Construct a parallelogram $ABCD$, if three points are marked on the plane: the midpoints of its altitudes $BH$ and $BP$ and the midpoint of the side $AD$.
2021 CMIMC, 1.6
Let circles $\omega$ and $\Gamma$, centered at $O_1$ and $O_2$ and having radii $42$ and $54$ respectively, intersect at points $X,Y$, such that $\angle O_1XO_2 = 105^{\circ}$. Points $A$, $B$ lie on $\omega$ and $\Gamma$ respectively such that $\angle O_1XA = \angle AXB = \angle BXO_2$ and $Y$ lies on both minor arcs $XA$ and $XB$. Define $P$ to be the intersection of $AO_2$ and $BO_1$. Suppose $XP$ intersects $AB$ at $C$. What is the value of $\frac{AC}{BC}$?
[i]Proposed by Puhua Cheng[/i]
2016 Baltic Way, 16
In triangle $ABC,$ the points $D$ and $E$ are the intersections of the angular bisectors from $C$ and $B$ with the sides $AB$ and $AC,$ respectively. Points $F$ and $G$ on the extensions of $AB$ and $AC$ beyond $B$ and $C,$ respectively, satisfy $BF = CG = BC.$ Prove that $F G \parallel DE.$
2020 Princeton University Math Competition, 11
Three (not necessarily distinct) points in the plane which have integer coordinates between $ 1$ and $2020$, inclusive, are chosen uniformly at random. The probability that the area of the triangle with these three vertices is an integer is $a/b$ in lowest terms. If the three points are collinear, the area of the degenerate triangle is $0$. Find $a + b$.
2001 Bundeswettbewerb Mathematik, 3
Let $ ABC$ an acute triangle with circumcircle center $ O.$ The line $ (BO)$ intersects the circumcircle again in $ D,$ and the extension of the altitude from $ A$ intersects the circle in $ E.$ Prove that the quadrilateral $ BECD$ and the triangle $ ABC$ have the same area.
2021 Brazil National Olympiad, 5
Let $ABC$ be an acute-angled triangle. Let $A_1$ be the midpoint of the arc $BC$ which contain the point $A$. Let $A_2$ and $A_3$ be the foot(s) of the perpendicular(s) of the point $A_1$ to the lines $AB$ and $AC$, respectively. Define $B_2,B_3,C_2,C_3$ analogously.
a) Prove that the line $A_2A_3$ cuts $BC$ in the midpoint.
b) Prove that the lines $A_2A_3,B_2B_3$ and $C_2C_3$ are concurrents.
2023 All-Russian Olympiad, 4
Given is a triangle $ABC$ and a point $X$ inside its circumcircle. If $I_B, I_C$ denote the $B, C$ excenters, then prove that $XB \cdot XC <XI_B \cdot XI_C$.