Found problems: 25757
2018 India National Olympiad, 1
Let $ABC$ be a non-equilateral triangle with integer sides. Let $D$ and $E$ be respectively the mid-points of $BC$ and $CA$ ; let $G$ be the centroid of $\Delta{ABC}$. Suppose, $D$, $C$, $E$, $G$ are concyclic. Find the least possible perimeter of $\Delta{ABC}$.
2005 Indonesia MO, 4
Let $ M$ be a point in triangle $ ABC$ such that $ \angle AMC\equal{}90^{\circ}$, $ \angle AMB\equal{}150^{\circ}$, $ \angle BMC\equal{}120^{\circ}$. The centers of circumcircles of triangles $ AMC,AMB,BMC$ are $ P,Q,R$, respectively. Prove that the area of $ \triangle PQR$ is greater than the area of $ \triangle ABC$.
2021 Science ON all problems, 2
Is it possible for an isosceles triangle with all its sides of positive integer lengths to have an angle of $36^o$?
[i] (Adapted from Archimedes 2011, Traian Preda)[/i]
2013 Tournament of Towns, 5
In a quadrilateral $ABCD$, angle $B$ is equal to $150^o$, angle $C$ is right, and sides $AB$ and $CD$ are equal. Determine the angle between $BC$ and the line connecting the midpoints of sides $BC$ and $AD$.
2017 Kazakhstan National Olympiad, 4
The acute triangle $ABC$ $(AC> BC)$ is inscribed in a circle with the center at the point $O$, and $CD$ is the diameter of this circle. The point $K$ is on the continuation of the ray $DA$ beyond the point $A$. And the point $L$ is on the segment $BD$ $(DL> LB)$ so that $\angle OKD = \angle BAC$, $\angle OLD = \angle ABC$. Prove that the line $KL$ passes through the midpoint of the segment $AB$.
2005 JBMO Shortlist, 6
Let $C_1,C_2$ be two circles intersecting at points $A,P$ with centers $O,K$ respectively. Let $B,C$ be the symmetric of $A$ wrt $O,K$ in circles $C_1,C_2 $ respectively. A random line passing through $A$ intersects circles $C_1,C_2$ at $D,E$ respectively. Prove that the center of circumcircle of triangle $DEP$ lies on the circumcircle of triangle $OKP$.
2011 Oral Moscow Geometry Olympiad, 1
The bisector of angle $B$ and the bisector of external angle $D$ of rectangle $ABCD$ intersect side $AD$ and line $AB$ at points $M$ and $K$, respectively. Prove that the segment $MK$ is equal and perpendicular to the diagonal of the rectangle.
2012 India National Olympiad, 1
Let $ABCD$ be a quadrilateral inscribed in a circle. Suppose $AB=\sqrt{2+\sqrt{2}}$ and $AB$ subtends $135$ degrees at center of circle . Find the maximum possible area of $ABCD$.
2003 Junior Macedonian Mathematical Olympiad, Problem 3
Let $ABC$ be a given triangle. The circumcircle of the triangle has radius $R$, the incircle has radius $r$, the longest side of the triangle is $a$, while the shortest altitude is $h$. Show that: $\frac{R}{r} > \frac{a}{h}$.
2021 XVII International Zhautykov Olympiad, #4
Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$.
Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$
1994 AMC 12/AHSME, 17
An $8$ by $2\sqrt{2}$ rectangle has the same center as a circle of radius $2$. The area of the region common to both the rectangle and the circle is
$ \textbf{(A)}\ 2\pi \qquad\textbf{(B)}\ 2\pi+2 \qquad\textbf{(C)}\ 4\pi-4 \qquad\textbf{(D)}\ 2\pi+4 \qquad\textbf{(E)}\ 4\pi-2 $
1999 Moldova Team Selection Test, 11
Let $ABC$ be a triangle. Show that there exists a lin $l$ in the plane of $ABC$ such that the overlapping area of $ABC$ and $A^{'}B^{'}C^{'}$, the symmetric of $ABC$ with respect to $l$, is greater than $\frac{2}{3}$ of area of $ABC$.
2010 Canadian Mathematical Olympiad Qualification Repechage, 8
Consider three parallelograms $P_1,~P_2,~ P_3$. Parallelogram $P_3$ is inside parallelogram $P_2$, and the vertices of $P_3$ are on the edges of $P_2$. Parallelogram $P_2$ is inside parallelogram $P_1$, and the vertices of $P_2$ are on the edges of $P_1$. The sides of $P_3$ are parallel to the sides of $P_1$. Prove that one side of $P_3$ has length at least half the length of the parallel side of $P_1$.
2022 Moscow Mathematical Olympiad, 2
In a Cartesian coordinate system (with the same scale on the x and y axes)there is a graph of the exponential function $y=3^x$. Then the y-axis and all marks on the x-axis erased. Only the graph of the function and the x-axis remained without a scale and a mark of $0$.
How can you restore the y-axis using a compass and ruler?
1997 Bulgaria National Olympiad, 2
Let $M$ be the centroid of $\Delta ABC$
Prove the inequality
$\sin \angle CAM + \sin\angle CBM \le \frac{2}{\sqrt 3}$
(a) if the circumscribed circle of $\Delta AMC$ is tangent to the line $AB$
(b) for any $\Delta ABC$
1995 Tournament Of Towns, (456) 1
Does there exist a sphere passing through only one rational point? (A rational point is a point whose Cartesian coordinates are all rational numbers.)
(A Rubin)
1996 IMC, 6
Upper content of a subset $E$ of the plane $\mathbb{R}^{2}$ is defined as
$$\mathcal{C}(E)=\inf\{\sum_{i=1}^{n} \text{diam}(E_{i})\}$$
where $\inf$ is taken over all finite families of sets $E_{1},\dots,E_{n}$ $n\in \mathbb{N}$, in $\mathbb{R}^{2}$
such that $E\subset \bigcup_{i=1}^{n}E_{i}$.
Lower content of $E$ is defined as
$$\mathcal{K}(E)=\sup\{\text{length}(L) |\, L \text{ is a closed line segment onto which $E$ can be contracted}\}$$.
Prove that
i) $\mathcal{C}(L)=\text{length}(L)$ if $L$ is a closed line segment;
ii) $\mathcal{C}(E) \geq \mathcal{K}(E)$;
iii) the equality in ii) is not always true even if $E$ is compact.
2008 Junior Balkan Team Selection Tests - Romania, 4
Let $ ABC$ be a triangle, and $ D$ the midpoint of the side $ BC$. On the sides $ AB$ and $ AC$ we consider the points $ M$ and $ N$, respectively, both different from the midpoints of the sides, such that \[ AM^2\plus{}AN^2 \equal{}BM^2 \plus{} CN^2 \textrm{ and } \angle MDN \equal{} \angle BAC.\] Prove that $ \angle BAC \equal{} 90^\circ$.
2019 Romanian Masters In Mathematics, 2
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.
[i]Jakob Jurij Snoj, Slovenia[/i]
2003 All-Russian Olympiad, 1
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
Geometry Mathley 2011-12, 11.4
Let $ABC$ be a triangle and $P$ be a point in the plane of the triangle. The lines $AP,BP, CP$ meets $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let $A_2,B_2,C_2$ be the Miquel point of the complete quadrilaterals $AB_1PC_1BC$, $BC_1PA_1CA$, $CA_1PB_1AB$. Prove that the circumcircles of the triangles $APA_2$,$BPB_2$, $CPC_2$, $BA_2C$, $AB_2C$, $AC_2B$ have a point of concurrency.
Nguyễn Văn Linh
1949-56 Chisinau City MO, 58
On the plane $n$ points are chosen so that exactly $m$ of them lie on one straight line and no three points not included in these $m$ points lie on one straight line. What is the number of all lines, each of which contains at least two of these points?
2006 Junior Balkan MO, 2
The triangle $ABC$ is isosceles with $AB=AC$, and $\angle{BAC}<60^{\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\angle{ABD}\equiv\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\angle{BDC}$ and $\angle{ACB}$. Compute $\angle{COD}$.
Geometry Mathley 2011-12, 4.3
Let $ABC$ be a triangle not being isosceles at $A$. Let $(O)$ and $(I)$ denote the circumcircle and incircle of the triangle. $(I)$ touches $AC$ and $AB$ at $E, F$ respectively. Points $M$ and $N$ are on the circle $(I)$ such that $EM \parallel FN \parallel BC$. Let $P,Q$ be the intersections of $BM,CN$ and $(I)$. Prove that
i) $BC,EP, FQ$ are concurrent, and denote by $K$ the point of concurrency.
ii) the circumcircles of triangle $BPK, CQK$ are all tangent to $(I)$ and all pass through a common point on the circle $(O)$.
Nguyễn Minh Hà
2015 AMC 8, 8
What is the smallest whole number larger than the perimeter of any triangle with a side of length $ 5$ and a side of length $19$?
$\textbf{(A) }24\qquad\textbf{(B) }29\qquad\textbf{(C) }43\qquad\textbf{(D) }48\qquad \textbf{(E) }57$