Found problems: 25757
Kyiv City MO Juniors Round2 2010+ geometry, 2021.7.4
The sides of the triangle $ABC$ are extended in both directions and on these extensions $6$ equal segments $AA_1 , AA_2, BB_1,BB_2, CC_1, CC_2$ are drawn (fig.). It turned out that all $6$ points $A_1,A_2,B_1,B_2,C_1, C_2$ lie on the same circle, is $\vartriangle ABC$ necessarily equilateral?
(Bogdan Rublev)
[img]https://cdn.artofproblemsolving.com/attachments/0/3/a499f6e6d978ce63d2ab40460dc73b62882863.png[/img]
2023 Czech-Polish-Slovak Junior Match, 4
In triangle $ABC$, the points $M$ and $N$ are the midpoints of the sides $AB$ and $AC$, respectively. The bisectors of interior angles $\angle ABC$ and $\angle BCA$ intersect the line $MN$ at points $P$ and $Q$, respectively. Let $p$ be the tangent to the circumscribed circle of the triangle $AMP$ passing through point $P$, and $q$ be the tangent to the circumscribed circle of the triangle $ANQ$ passing through point $Q$. Prove that the lines $p$ and $q$ intersect on line $BC$.
Durer Math Competition CD Finals - geometry, 2021.C3
In the isosceles triangle $ABC$ we have $AC = BC$. Let $X$ be an arbitrary point of the segment $AB$. The line parallel to $BC$ and passing through $X$ intersects the segment $AC$ in $N$, and the line parallel to $AC$ and passing through $BC$ intersects the segment $BC$ in $M$. Let $k_1$ be the circle with center $N$ and radius $NA$. Similarly, let $k_2$ be the circle with center $M$ and radius $MB$. Let $T$ be the intersection of the circles $k_1$ and $k_2$ different from $X$. Show that the angles $\angle NCM$ and $\angle NTM$ are equal.
2010 National Olympiad First Round, 1
Let $D$ be a point inside of equilateral $\triangle ABC$, and $E$ be a point outside of equilateral $\triangle ABC$ such that $m(\widehat{BAD})=m(\widehat{ABD})=m(\widehat{CAE})=m(\widehat{ACE})=5^\circ$. What is $m(\widehat{EDC})$ ?
$ \textbf{(A)}\ 45^\circ
\qquad\textbf{(B)}\ 40^\circ
\qquad\textbf{(C)}\ 35^\circ
\qquad\textbf{(D)}\ 30^\circ
\qquad\textbf{(E)}\ 25^\circ
$
2000 Balkan MO, 3
How many $1 \times 10\sqrt 2$ rectangles can be cut from a $50\times 90$ rectangle using cuts parallel to its edges?
1985 IMO Longlists, 35
We call a coloring $f$ of the elements in the set $M = \{(x, y) | x = 0, 1, \dots , kn - 1; y = 0, 1, \dots , ln - 1\}$ with $n$ colors allowable if every color appears exactly $k$ and $ l$ times in each row and column and there are no rectangles with sides parallel to the coordinate axes such that all the vertices in $M$ have the same color. Prove that every allowable coloring $f$ satisfies $kl \leq n(n + 1).$
2006 Switzerland - Final Round, 2
Let $ABC$ be an equilateral triangle and let $D$ be an inner point of the side $BC$. A circle is tangent to $BC$ at $D$ and intersects the sides $AB$ and $AC$ in the inner points $M, N$ and $P, Q$ respectively. Prove that $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.
2005 AMC 8, 4
A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are 6.1 cm, 8.2 cm and 9.7 cm. What is the area of the square in square centimeters?
$ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 64 $
1990 All Soviet Union Mathematical Olympiad, 515
The point $P$ lies inside the triangle $ABC$. A line is drawn through $P$ parallel to each side of the triangle. The lines divide $AB$ into three parts length $c, c', c"$ (in that order), and $BC$ into three parts length $a, a', a"$ (in that order), and $CA$ into three parts length $b, b', b"$ (in that order). Show that $abc = a'b'c' = a"b"c"$.
2017-IMOC, C5
We say a finite set $S$ of points with $|S|\ge3$ is [i]good[/i] if for any three distinct elements of $S$, they are non-collinear and the orthocenter of them is also in $S$. Find all good sets.
2022 Indonesia TST, G
Given that $ABC$ is a triangle, points $A_i, B_i, C_i \hspace{0.15cm} (i \in \{1,2,3\})$ and $O_A, O_B, O_C$ satisfy the following criteria:
a) $ABB_1A_2, BCC_1B_2, CAA_1C_2$ are rectangles not containing any interior points of the triangle $ABC$,
b) $\displaystyle \frac{AB}{BB_1} = \frac{BC}{CC_1} = \frac{CA}{AA_1}$,
c) $AA_1A_3A_2, BB_1B_3B_2, CC_1C_3C_2$ are parallelograms, and
d) $O_A$ is the centroid of rectangle $BCC_1B_2$, $O_B$ is the centroid of rectangle $CAA_1C_2$, and $O_C$ is the centroid of rectangle $ABB_1A_2$.
Prove that $A_3O_A, B_3O_B,$ and $C_3O_C$ concur at a point.
[i]Proposed by Farras Mohammad Hibban Faddila[/i]
1986 Bulgaria National Olympiad, Problem 5
Let $A$ be a fixed point on a circle $k$. Let $B$ be any point on $k$ and $M$ be a point such that $AM:AB=m$ and $\angle BAM=\alpha$, where $m$ and $\alpha$ are given. Find the locus of point $M$ when $B$ describes the circle $k$.
2010 USA Team Selection Test, 7
In triangle ABC, let $P$ and $Q$ be two interior points such that $\angle ABP = \angle QBC$ and $\angle ACP = \angle QCB$. Point $D$ lies on segment $BC$. Prove that $\angle APB + \angle DPC = 180^\circ$ if and only if $\angle AQC + \angle DQB = 180^\circ$.
1987 China National Olympiad, 5
Let $A_1A_2A_3A_4$ be a tetrahedron. We construct four mutually tangent spheres $S_1,S_2,S_3,S_4$ with centers $A_1,A_2,A_3,A_4$ respectively. Suppose that there exists a point $Q$ such that we can construct two spheres centered at $Q$ satisfying the following conditions:
i) One sphere with radius $r$ is tangent to $S_1,S_2,S_3,S_4$;
ii) One sphere with radius $R$ is tangent to every edges of tetrahedron $A_1A_2A_3A_4$.
Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.
1983 IMO Longlists, 35
Let $P_1, P_2, \dots , P_n$ be distinct points of the plane, $n \geq 2$. Prove that
\[ \max_{1\leq i<j\leq n} P_iP_j > \frac{\sqrt 3}{2}(\sqrt n -1) \min_{1\leq i<j\leq n} P_iP_j \]
1986 India National Olympiad, 3
Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles. Prove that
\[ \frac{1}{\sqrt{c}}\equal{}\frac{1}{\sqrt{a}}\plus{}\frac{1}{\sqrt{b}}\]
1969 IMO Longlists, 5
$(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$
$(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas.
$(b)$ Find the locus of the centers of these hyperbolas.
2016 Romania National Olympiad, 4
Consider the isosceles right triangle $ABC$, with $\angle A = 90^o$ and the point $M \in (BC)$ such that $\angle AMB = 75^o$. On the inner bisector of the angle $MAC$ take a point $F$ such that $BF = AB$. Prove that:
a) the lines $AM$ and $BF$ are perpendicular;
b) the triangle $CFM$ is isosceles.
2024 Kurschak Competition, 1
The quadrilateral $ABCD$ is divided into cyclic quadrilaterals with pairwise disjoint interiors. None of the vertices of the cyclic quadrilaterals in the decomposition is an interior point of a side of any cyclic quadrilateral in the decomposition or of a side of the quadrilateral $ABCD$. Prove that $ABCD$ is also a cyclic quadrilateral.
2010 Sharygin Geometry Olympiad, 8
Given is a regular polygon. Volodya wants to mark $k$ points on its perimeter so that any another regular polygon (maybe having a different number of sides) doesn’t contain all marked points on its perimeter. Find the minimal $k$ sufficient for any given polygon.
2015 AoPS Mathematical Olympiad, 7
Let $ABC$ be a right triangle with $\angle C = 90^\circ$. Let $P_A$, $P_B$, and $P_C$ be regular pentagons with side lengths $BC$, $CA$, and $AB$, respectively. Prove that $[P_A]+[P_B]=[P_C]$.
[i]Proposed by CaptainFlint[/i]
Denmark (Mohr) - geometry, 2015.3
Triangle $ABC$ is equilateral. The point $D$ lies on the extension of $AB$ beyond $B$, the point $E$ lies on the extension of $CB$ beyond $B$, and $|CD| = |DE|$. Prove that $|AD| = |BE|$.
[img]https://1.bp.blogspot.com/-QnAXFw3ijn0/XzR0YjqBQ3I/AAAAAAAAMU0/0TvhMQtBNjolYHtgXsQo2OPGJzEYSfCwACLcBGAsYHQ/s0/2015%2BMohr%2Bp3.png[/img]
1976 Bundeswettbewerb Mathematik, 4
Each vertex of the 3-dimensional Euclidean space either is coloured red or blue. Prove that within those squares being possible in this space with edge length 1 there is at least one square either with three red vertices or four blue vertices !
2022 MOAA, 3
The area of the figure enclosed by the $x$-axis, $y$-axis, and line $7x + 8y = 15$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1987 Dutch Mathematical Olympiad, 4
On each side of a regular tetrahedron with edges of length $1$ one constructs exactly such a tetrahedron. This creates a dodecahedron with $8$ vertices and $18$ edges. We imagine that the dodecahedron is hollow. Calculate the length of the largest line segment that fits entirely within this dodecahedron.