Found problems: 25757
1995 Tournament Of Towns, (459) 4
Some points with integer coordinates in the plane are marked. It is known that no four of them lie on a circle. Show that there exists a circle of radius 1995 without any marked points inside.
(AV Shapovelov)
1999 Mongolian Mathematical Olympiad, Problem 6
Two circles in the plane intersect at $C$ and $D$. A chord $AB$ of the first circle and a chord $EF$ of the second circle pass through a point on the common chord $CD$. Show that the points $A,B,E,F$ lie on a circle.
2019 Jozsef Wildt International Math Competition, W. 59
In the any $[ABCD]$ tetrahedron we denote with $\alpha$, $\beta$, $\gamma$ the measures, in radians, of the angles of the three pairs of opposite edges and with $r$, $R$ the lengths of the rays of the sphere inscribed and respectively circumscribed the tetrahedron. Demonstrate inequality$$\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}$$(A refinement of inequality $R \geq 3r$).
2024 JHMT HS, 9
Compute the smallest positive integer $k$ such that the area of the region bounded by
\[ k\min(x,y)+x^2+y^2=0 \]
exceeds $100$.
1986 Traian Lălescu, 2.4
Prove that $ ABCD $ is a rectangle if and only if $ MA^2+MC^2=MB^2+MD^2, $ for all spatial points $ M. $
MathLinks Contest 2nd, 3.2
Let $ABC$ be a triangle with altitudes $AD, BE, CF$. Choose the points $A_1, B_1, C_1$ on the lines $AD, BE, CF$ respectively, such that $$\frac{AA_1}{AD}= \frac{BB_1}{BE}= \frac{CC_1}{CF} = k.$$
Find all values of $k$ such that the triangle $A_1B_1C_1$ is similar to the triangle $ABC$ for all triangles $ABC$.
2021 Moldova EGMO TST, 9
Let $ABCD$ be a square and $E$ a on point diagonal $(AC)$, different from its midpoint. $H$ and $K$ are the orthoceneters of triangles $ABE$ and $ADE$. Prove that $AH$ and $CK$ are parallel.
2015 Peru Cono Sur TST, P7
In the plan $6$ points were located such that the distance between two damages of them is greater than or equal to $1$. Prove that it is possible to choose two of those points such that their distance is greater than or equal to $2 \cos{18}$
Observation: It might help you to know that $\cos{18} = 0.95105\ldots$ and $\cos{24} = 0.91354\ldots$
2013 Greece Junior Math Olympiad, 2
Let $ABC$ be an acute angled triangle with $AB<AC$. Let $M$ be the midpoint of side $BC$. On side $AB$, consider a point $D$ such that, if segment $CD$ intersects median $AM$ at point $E$, then $AD=DE$. Prove that $AB=CE$.
2002 Moldova National Olympiad, 4
The circumradius of a tetrahedron $ ABCD$ is $ R$, and the lenghts of the segments connecting the vertices $ A,B,C,D$ with the centroids of the opposite faces are equal to $ m_a,m_b,m_c$ and $ m_d$, respectively. Prove that:
$ m_a\plus{}m_b\plus{}m_c\plus{}m_d\leq \dfrac{16}{3}R$
1984 Brazil National Olympiad, 3
Given a regular dodecahedron of side $a$. Take two pairs of opposite faces: $E, E' $ and $F, F'$. For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.
2023 Polish Junior Math Olympiad First Round, 3.
Let $ABCD$ be a rectangle. Point $E$ lies on side $AB$, and point $F$ lies on segment $CE$. Prove that if triangles $ADE$ and $CDF$ have equal areas, then triangles $BCE$ and $DEF$ also have equal areas.
2015 Saudi Arabia IMO TST, 1
Let $ABC$ be an acute-angled triangle inscribed in the circle $(O)$, $H$ the foot of the altitude of $ABC$ at $A$ and $P$ a point inside $ABC$ lying on the bisector of $\angle BAC$. The circle of diameter $AP$ cuts $(O)$ again at $G$. Let $L$ be the projection of $P$ on $AH$. Prove that if $GL$ bisects $HP$ then $P$ is the incenter of the triangle $ABC$.
Lê Phúc Lữ
2023 UMD Math Competition Part I, #9
The Amazing Prime company ships its products in boxes whose length, width, and height (in inches) are prime numbers. If the volume of one of their boxes is $105$ cubic inches, what is its surface area (that is, the sum of the areas of the 6 sides of the box) in square inches?
$$
\mathrm a. ~ 21\qquad \mathrm b.~71\qquad \mathrm c. ~77 \qquad \mathrm d. ~05 \qquad \mathrm e. ~142
$$
1999 Singapore Senior Math Olympiad, 2
In $\vartriangle ABC$ with edges $a, b$ and $c$, suppose $b + c = 6$ and the area $S$ is $a^2 - (b -c)^2$. Find the value of $\cos A$ and the largest possible value of $S$.
2019 CMIMC, 3
Let $ABC$ be an equilateral triangle with side length $2$, and let $M$ be the midpoint of $\overline{BC}$. Points $X$ and $Y$ are placed on $AB$ and $AC$ respectively such that $\triangle XMY$ is an isosceles right triangle with a right angle at $M$. What is the length of $\overline{XY}$?
1995 AMC 12/AHSME, 8
In $\triangle ABC$, $\angle C = 90^\circ, AC = 6$ and $BC = 8$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED = 90^\circ$. If $DE = 4$, then $BD =$
[asy]
size(100); pathpen = linewidth(0.7); pointpen = black+linewidth(3);
pair A = (0,0), C = (6,0), B = (6,8), D = (2*A+B)/3, E = (2*C+B)/3; D(D("A",A,SW)--D("B",B,NW)--D("C",C,SE)--cycle); D(D("D",D,NW)--D("E",E,plain.E)); D(rightanglemark(D,E,B,16)); D(rightanglemark(A,C,B,16));[/asy]
$\mathbf{(A)}\;5\qquad
\mathbf{(B)}\;\frac{16}{3}\qquad
\mathbf{(C)}\; \frac{20}{3}\qquad
\mathbf{(D)}\; \frac{15}{2}\qquad
\mathbf{(E)}\; 8$
2025 Kosovo National Mathematical Olympiad`, P4
Let $D$ be a point on the side $AC$ of triangle $\triangle ABC$ such that $AB=AD=DC$ and let $E$ be a point on the side $BC$ such that $BE=2CE$. Prove that $\angle BDE = 90 ^{\circ}$.
2009 Postal Coaching, 3
Let $ABC$ be a triangle with circumcentre $O$ and incentre $I$ such that $O$ is different from $I$. Let $AK, BL, CM$ be the altitudes of $ABC$, let $U, V , W$ be the mid-points of $AK, BL, CM$ respectively. Let $D, E, F$ be the points at which the in-circle of $ABC$ respectively touches the sides $BC, CA, AB$. Prove that the lines $UD, VE, WF$ and $OI$ are concurrent.
2017 Saudi Arabia BMO TST, 3
Let $ABC$ be an acute triangle and $(O)$ be its circumcircle. Denote by $H$ its orthocenter and $I$ the midpoint of $BC$. The lines $BH, CH$ intersect $AC,AB$ at $E, F$ respectively. The circles $(IBF$) and $(ICE)$ meet again at $D$.
a) Prove that $D, I,A$ are collinear and $HD, EF, BC$ are concurrent.
b) Let $L$ be the foot of the angle bisector of $\angle BAC$ on the side $BC$. The circle $(ADL)$ intersects $(O)$ again at $K$ and intersects the line $BC$ at $S$ out of the side $BC$. Suppose that $AK,AS$ intersects the circles $(AEF)$ again at $G, T$ respectively. Prove that $TG = TD$.
2013 CentroAmerican, 3
Let $ABCD$ be a convex quadrilateral and let $M$ be the midpoint of side $AB$. The circle passing through $D$ and tangent to $AB$ at $A$ intersects the segment $DM$ at $E$. The circle passing through $C$ and tangent to $AB$ at $B$ intersects the segment $CM$ at $F$. Suppose that the lines $AF$ and $BE$ intersect at a point which belongs to the perpendicular bisector of side $AB$. Prove that $A$, $E$, and $C$ are collinear if and only if $B$, $F$, and $D$ are collinear.
2012 AMC 10, 23
A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
$ \textbf{(A)}\ \dfrac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \dfrac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \dfrac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} $
2007 Harvard-MIT Mathematics Tournament, 6
Triangle $ABC$ has $\angle A=90^\circ$, side $BC=25$, $AB>AC$, and area $150$. Circle $\omega$ is inscribed in $ABC$, with $M$ its point of tangency on $AC$. Line $BM$ meets $\omega$ a second time at point $L$. Find the length of segment $BL$.
2018 Sharygin Geometry Olympiad, 5
The vertex $C$ of equilateral triangles $ABC$ and $CDE$ lies on the segment $AE$, and the vertices $B$ and $D$ lie on the same side with respect to this segment. The circumcircles of these triangles centered at $O_1$ and $O_2$ meet for the second time at point $F$. The lines $O_1O_2$ and $AD$ meet at point $K$. Prove that $AK = BF$.
2020 Taiwan TST Round 2, 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)