Found problems: 25757
2004 Chile National Olympiad, 3
The perimeter, that is, the sum of the lengths of all sides of a convex quadrilateral $ ABCD $, is equal to $2004$ meters; while the length of its diagonal $ AC $ is equal to $1001$ meters. Find out if the length of the other diagonal $ BD $ can:
a) To be equal to only one meter.
b) Be equal to the length of the diagonal $ AC $.
2019 Regional Olympiad of Mexico Southeast, 4
Let $\Gamma$ a circumference. $T$ a point in $\Gamma$, $P$ and $A$ two points outside $\Gamma$ such that $PT$ is tangent to $\Gamma$ and $PA=PT$. Let $C$ a point in $\Gamma (C\neq T)$, $AC$ and $PC$ intersect again $\Gamma$ in $D$ and $B$, respectively. $AB$ intersect $\Gamma$ in $E$. Prove that $DE$ it´s parallel to $AP$
2014 Romania National Olympiad, 3
Let $ P,Q $ be the midpoints of the diagonals $ BD, $ respectively, $ AC, $ of the quadrilateral $ ABCD, $ and points $ M,N,R,S $ on the segments $ BC,CD,PQ, $ respectively $ AC, $ except their extremities, such that
$$ \frac{BM}{MC}=\frac{DN}{NC}=\frac{PR}{RQ}=\frac{AS}{SC} . $$
Show that the center of mass of the triangle $ AMN $ is situated on the segment $ RS. $
1975 AMC 12/AHSME, 30
Let $x=\cos 36^{\circ} - \cos 72^{\circ}$. Then $x$ equals
$ \textbf{(A)}\ \frac{1}{3} \qquad\textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ 3-\sqrt{6} \qquad\textbf{(D)}\ 2\sqrt{3}-3 \qquad\textbf{(E)}\ \text{none of these} $
1969 Bulgaria National Olympiad, Problem 4
Find the sides of a triangle if it is known that the inscribed circle meets one of its medians in two points and these points divide the median into three equal segments and the area of the triangle is equal to $6\sqrt{14}\text{ cm}^2$.
2018 JHMT, 3
An equilateral triangle $ABC$ is in between two parallel lines $x, y$ that pass through points $A$ and $B$ respectively. Given that $C$ is twice as far from $y$ as $x$, the acute angle that $CA$ makes with $x$ is $\theta$. Then $(\tan \theta)^2$ is of the form $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $p + q$.
1976 IMO Longlists, 27
In a plane three points $P,Q,R,$ not on a line, are given. Let $k, l, m$ be positive numbers. Construct a triangle $ABC$ whose sides pass through $P, Q,$ and $R$ such that
$P$ divides the segment $AB$ in the ratio $1 : k$,
$Q$ divides the segment $BC$ in the ratio $1 : l$, and
$R$ divides the segment $CA$ in the ratio $1 : m.$
1995 IberoAmerican, 3
Let $ r$ and $ s$ two orthogonal lines that does not lay on the same plane. Let $ AB$ be their common perpendicular, where $ A\in{}r$ and $ B\in{}s$(*).Consider the sphere of diameter $ AB$. The points $ M\in{r}$ and $ N\in{s}$ varies with the condition that $ MN$ is tangent to the sphere on the point $ T$. Find the locus of $ T$.
Note: The plane that contains $ B$ and $ r$ is perpendicular to $ s$.
1959 AMC 12/AHSME, 41
On the same side of a straight line three circles are drawn as follows: a circle with a radius of $4$ inches is tangent to the line, the other two circles are equal, and each is tangent to the line and to the other two circles. The radius of the equal circles is:
$ \textbf{(A)}\ 24 \qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 12 $
2011 Sharygin Geometry Olympiad, 7
Points $P$ and $Q$ on sides $AB$ and $AC$ of triangle $ABC$ are such that $PB = QC$. Prove that $PQ < BC$.
1998 North Macedonia National Olympiad, 1
Let $ABCDE$ be a convex pentagon with $AB = BC =CA$ and $CD = DE = EC$.
Let $T$ be the centroid of $\vartriangle ABC$, and $N$ be the midpoint of $AE$. Compute $\angle NT D$
2022 Sharygin Geometry Olympiad, 12
Let $K$, $L$, $M$, $N$ be the midpoints of sides $BC$, $CD$, $DA$, $AB$ respectively of a convex quadrilateral $ABCD$. The common points of segments $AK$, $BL$, $CM$, $DN$ divide each of them into three parts. It is known that the ratio of the length of the medial part to the length of the whole segment is the same for all segments. Does this yield that $ABCD$ is a parallelogram?
2021 Science ON grade VII, 2
In triangle $ABC$, we have $\angle ABC=\angle ACB=44^o$. Point $M$ is in its interior such that $\angle MBC=16^o$ and $\angle MCB=30^o$. Prove that $\angle MAC=\angle MBC$.
[i] (Andra Elena Mircea)[/i]
2001 India IMO Training Camp, 3
Points $B = B_1 , B_2, \cdots , B_n , B_{n+1} = C$ are chosen on side $BC$ of a triangle $ABC$ in that order. Let $r_j$ be the inradius of triangle $AB_jB_{j+1}$ for $j = 1, \cdots, n$ , and $r$ be the inradius of $\triangle ABC$. Show that there is a constant $\lambda$ independent of $n$ such that :
\[(\lambda -r_1)(\lambda -r_2)\cdots (\lambda -r_n) =\lambda^{n-1}(\lambda -r)\]
2018 Harvard-MIT Mathematics Tournament, 9
Po picks $100$ points $P_1,P_2,\cdots, P_{100}$ on a circle independently and uniformly at random. He then draws the line segments connecting $P_1P_2,P_2P_3,\ldots,P_{100}P_1.$ Find the expected number of regions that have all sides bounded by straight lines.
2015 Iran Team Selection Test, 4
Ali puts $5$ points on the plane such that no three of them are collinear. Ramtin adds a sixth point that is not collinear with any two of the former points.Ali wants to eventually construct two triangles from the six points such that one can be placed inside another. Can Ali put the 5 points in such a manner so that he would
always be able to construct the desired triangles? (We say that triangle $T_1$ can be
placed inside triangle $T_2$ if $T_1$ is congruent to a triangle that is located completely
inside $T_2$.)
1978 All Soviet Union Mathematical Olympiad, 261
Given a circle with radius $R$ and inscribed $n$-gon with area $S$. We mark one point on every side of the given polygon. Prove that the perimeter of the polygon with the vertices in the marked points is not less than $2S/R$.
2023 AMC 12/AHSME, 25
A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$
2013 Singapore MO Open, 5
Let $ABC$ be a triangle with integral side lengths such that $\angle A=3\angle B$. Find the minimum value of its perimeter.
2009 Greece JBMO TST, 2
Given convex quadrilateral $ABCD$ inscribed in circle $(O,R)$ (with center $O$ and radius $R$). With centers the vertices of the quadrilateral and radii $R$, we consider the circles $C_A(A,R), C_B(B,R), C_C(C,R), C_D(D,R)$. Circles $C_A$ and $C_B$ intersect at point $K$, circles $C_B$ and $C_C$ intersect at point $L$, circles $C_C$ and $C_D$ intersect at point $M$ and circles $C_D$ and $C_A$ intersect at point $N$ (points $K,L,M,N$ are the second common points of the circles given they all pass through point $O$). Prove that quadrilateral $KLMN$ is a parallelogram.
2024 Korea Junior Math Olympiad, 5
$ABC$ is a right triangle with $\angle C$ the right angle. $X$ is some point inside $ABC$ satisfying $CA=AX$. Let $D$ be the feet of altitude from $C$ to $AB$, and $Y(\neq X)$ the point of intersection of $DX$ and the circumcircle of $ABX$. Prove that $AX=AY$.
2016 India Regional Mathematical Olympiad, 5
Given a rectangle $ABCD$, determine two points $K$ and $L$ on the sides $BC$ and $CD$ such that the triangles $ABK, AKL$ and $ADL$ have same area.
Kvant 2024, M2806
Is it possible to draw a closed $20$-link polyline on the plane and number its links with the numbers $1, 2, 3, \ldots, 20$ in the order of traversal so that for each natural $i = 1, 2, 3, \ldots, 10$ the links numbered $i$ and $10+i$ intersect each other and do not intersect the other links?
[i] I. Efremov[/i]
2013 AMC 12/AHSME, 13
Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $?
$ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $
1949-56 Chisinau City MO, 62
On two intersecting lines $\ell_1$ and $\ell_2$, segments $AB$ and $CD$ of a given length are selected, respectively. Prove that the volume of the tetrahedron $ABCD$ does not depend on the position of the segments $AB$ and $CD$ on the lines $\ell_1$ and $\ell_2$.