Found problems: 25757
2012 JBMO ShortLists, 4
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ , and let $O$ , $H$ be the triangle's circumcenter and orthocenter respectively . Let also $A^{'}$ be the point where the angle bisector of the angle $BAC$ meets $\omega$ . If $A^{'}H=AH$ , then find the measure of the angle $BAC$.
1998 May Olympiad, 2
There are $1998$ rectangular pieces $2$ cm wide and $3$ cm long and with them squares are assembled (without overlapping or gaps). What is the greatest number of different squares that can be had at the same time?
2000 Junior Balkan Team Selection Tests - Romania, 4
On the hypotenuse $ BC $ of an isosceles right triangle $ ABC $ let $ M,N $ such that $ BM^2-MN^2+NC^2=0. $
Show that $ \angle MAN= 45^{\circ } . $
[i]Cristinel Mortici[/i]
1998 Croatia National Olympiad, Problem 3
Points $E$ and $F$ are chosen on the sides $AB$ and $BC$ respectively of a square $ABCD$ such that $BE=BF$. Let $BN$ be an altitude of the triangle $BCE$. Prove that the triangle $DNF$ is right-angled.
2020 Junior Balkan Team Selection Tests - Moldova, 11
Let $\triangle ABC$ be an acute triangle. The bisector of $\angle ACB$ intersects side $AB$ in $D$. The circumcircle of triangle $ADC$ intersects side $BC$ in $C$ and $E$ with $C \neq E$. The line parallel to $AE$ which passes through $B$ intersects line $CD$ in $F$. Prove that the triangle $\triangle AFB$ is isosceles.
2013 Today's Calculation Of Integral, 870
Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$
(1) Find all points of intersection of $E$ and $H$.
(2) Find the area of the region expressed by the system of inequality
\[\left\{
\begin{array}{ll}
3x^2+y^2\leq 3 &\quad \\
xy\geq \frac 34 , &\quad
\end{array}
\right.\]
2017 Adygea Teachers' Geometry Olympiad, 1
Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively.
1993 AMC 8, 25
A checkerboard consists of one-inch squares. A square card, $1.5$ inches on a side, is placed on the board so that it covers part or all of the area of each of $n$ squares. The maximum possible value of $n$ is
$\text{(A)}\ 4\text{ or }5 \qquad \text{(B)}\ 6\text{ or }7\qquad \text{(C)}\ 8\text{ or }9 \qquad \text{(D)}\ 10\text{ or }11 \qquad \text{(E)}\ 12\text{ or more}$
Kharkiv City MO Seniors - geometry, 2012.11.4
The incircle $\omega$ of triangle $ABC$ touches its sides $BC, CA$ and $AB$ at points $D, E$ and $E$, respectively. Point $G$ lies on circle $\omega$ in such a way that $FG$ is a diameter. Lines $EG$ and $FD$ intersect at point $H$. Prove that $AB \parallel CH$.
2019 HMNT, 9
Let $ABCD$ be an isosceles trapezoid with $AD = BC = 255$ and $AB = 128$. Let $M$ be the midpoint of $CD$ and let $N$ be the foot of the perpendicular from $A$ to $CD$. If $\angle MBC = 90^o$, compute $\tan\angle NBM$.
2022 Adygea Teachers' Geometry Olympiad, 4
In a regular hexagonal pyramid $SABCDEF$, a plane is drawn through vertex $A$ and the midpoints of edges $SC$ and $CE$. Find the ratio in which this plane divides the volume of the pyramid.
2019 Jozsef Wildt International Math Competition, W. 41
For $n \in \mathbb{N}$, consider in $\mathbb{R}^3$ the regular tetrahedron with vertices $O(0, 0, 0)$, $A(n, 9n, 4n)$, $B(9n, 4n, n)$ and $C(4n, n, 9n)$. Show that the number $N$ of points $(x, y, z)$, $[x, y, z \in \mathbb{Z}]$ inside or on the boundary of the tetrahedron $OABC$ is given by$$N=\frac{343n^3}{3}+\frac{35n^2}{2}+\frac{7n}{6}+1$$
2018 PUMaC Geometry B, 4
Let $\triangle ABC$ satisfy $AB = 17, AC = \frac{70}{3}$ and $BC = 19$. Let $I$ be the incenter of $\triangle ABC$ and $E$ be the excenter of $\triangle ABC$ opposite $A$. (Note: this means that the circle tangent to ray $AB$ beyond $B$, ray $AC$ beyond $C$, and side $BC$ is centered at $E$.) Suppose the circle with diameter $IE$ intersects $AB$ beyond $B$ at $D$. If $BD = \frac{a}{b}$ where $a, b$ are coprime positive integers, find $a + b$.
2020 Switzerland - Final Round, 3
We are given $n$ distinct rectangles in the plane. Prove that between the $4n$ interior angles formed by these rectangles at least $4\sqrt n$ are distinct.
1993 India National Olympiad, 9
Show that there exists a convex hexagon in the plane such that
(i) all its interior angles are equal;
(ii) its sides are $1,2,3,4,5,6$ in some order.
2004 Chile National Olympiad, 6
The $ AB, BC $ and $ CD $ segments of the polygon $ ABCD $ have the same length and are tangent to a circle $ S $, centered on the point $ O $. Let $ P $ be the point of tangency of $ BC $ with $ S $, and let $ Q $ be the intersection point of lines $ AC $ and $ BD $. Show that the point $ Q $ is collinear with the points $ P $ and $ O $.
2013 National Olympiad First Round, 13
Let $D$ and $E$ be points on side $[BC]$ of a triangle $ABC$ with circumcenter $O$ such that $D$ is between $B$ and $E$, $|AD|=|DB|=6$, and $|AE|=|EC|=8$. If $I$ is the incenter of triangle $ADE$ and $|AI|=5$, then what is $|IO|$?
$
\textbf{(A)}\ \dfrac {29}{5}
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ \dfrac {23}{5}
\qquad\textbf{(D)}\ \dfrac {21}{5}
\qquad\textbf{(E)}\ \text{None of above}
$
2014 CHMMC (Fall), 2
Consider two overlapping regular tetrahedrons of side length $2$ in space. They are centered at the same point, and the second one is oriented so that the lines from its center to its vertices are perpendicular to the faces of the first tetrahedron. What is the volume encompassed by the combined solid?
2020 Thailand TST, 2
Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$.
Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$.
Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$.
(Hungary)
1999 Junior Balkan Team Selection Tests - Romania, 4
Let be three discs $ D_1,D_2,D_3. $ For each $ i,j\in\{1,2,3\} , $ denote $ a_{ij} $ as being the area of $ D_i\cap D_j. $
If $ x_1,x_2,x_3\in\mathbb{R} $ such that $ x_1x_2x_3\neq 0, $ then
$$ a_{11} x_1^2+a_{22} x_2^2+a_{33} x_3^2+2a_{12} x_1x_2+2a_{23 }x_2x_3+2a_{31} x_3x_1>0. $$
[i]Vasile Pop[/i]
2012 Online Math Open Problems, 23
Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF.$ The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4} {5}$ the area of $ABC.$ The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[i]Author: Ray Li[/i]
2024 Oral Moscow Geometry Olympiad, 2
Petya drew a pentagon $ABCDE$ on the plane. After that, Vasya marked all the points $S$ in a given half-space relative to the plane of the pentagon so that in the pyramid $SABCD$ exactly two side faces are perpendicular to the plane of the base $ABCD$, and the height is $1$. How many points could have Vasya?
1989 India National Olympiad, 6
Triangle $ ABC$ has incentre $ I$ and the incircle touches $ BC, CA$ at $ D, E$ respectively. Let $ BI$ meet $ DE$ at $ G$. Show that $ AG$ is perpendicular to $ BG$.
2013 Stars Of Mathematics, 2
Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. Prove that at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle.
[i](Dan Schwarz)[/i]
2019 PUMaC Geometry A, 4
Let $BC=6$, $BX=3$, $CX=5$, and let $F$ be the midpoint of $\overline{BC}$. Let $\overline{AX}\perp\overline{BC}$ and $AF=\sqrt{247}$. If $AC$ is of the form $\sqrt{b}$ and $AB$ is of the form $\sqrt{c}$ where $b$ and $c$ are nonnegative integers, find $2c+3b$.