Found problems: 77
2020 Jozsef Wildt International Math Competition, W57
In all triangles $ABC$ does it hold that:
$$\sum\sin^2\frac A2\cos^2A\ge\frac{3\left(s^2-(2R+r)^2\right)}{8R^2}$$
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
1987 Bulgaria National Olympiad, Problem 6
Let $\Delta$ be the set of all triangles inscribed in a given circle, with angles whose measures are integer numbers of degrees different than $45^\circ,90^\circ$ and $135^\circ$. For each triangle $T\in\Delta$, $f(T)$ denotes the triangle with vertices at the second intersection points of the altitudes of $T$ with the circle.
(a) Prove that there exists a natural number $n$ such that for every triangle $T\in\Delta$, among the triangles $T,f(T),\ldots,f^n(T)$ (where $f^0(T)=T$ and $f^k(T)=f(f^{k-1}(T))$) at least two are equal.
(b) Find the smallest $n$ with the property from (a).
1977 Bulgaria National Olympiad, Problem 6
A Pythagorean triangle is any right-angled triangle for which the lengths of two legs and the length of the hypotenuse are integers. We are observing all Pythagorean triangles in which may be inscribed a quadrangle with sidelength integer number, two of which sides lie on the cathets and one of the vertices of which lies on the hypotenuse of the triangle given. Find the side lengths of the triangle with minimal surface from the observed triangles.
[i]St. Doduneko[/i]
2016 AMC 10, 11
What is the area of the shaded region of the given $8 \times 5$ rectangle?
[asy]
size(6cm);
defaultpen(fontsize(9pt));
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));
label("$1$",(1/2,5),dir(90));
label("$7$",(9/2,5),dir(90));
label("$1$",(8,1/2),dir(0));
label("$4$",(8,3),dir(0));
label("$1$",(15/2,0),dir(270));
label("$7$",(7/2,0),dir(270));
label("$1$",(0,9/2),dir(180));
label("$4$",(0,2),dir(180));
[/asy]
$\textbf{(A)}\ 4\dfrac{3}{5} \qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \qquad \textbf{(D)}\ 6\dfrac{1}{2} \qquad \textbf{(E)}\ 8$
2017 OMMock - Mexico National Olympiad Mock Exam, 2
Alice and Bob play on an infinite board formed by equilateral triangles. In each turn, Alice first places a white token on an unoccupied cell, and then Bob places a black token on an unoccupied cell. Alice's goal is to eventually have $k$ white tokens on a line. Determine the maximum value of $k$ for which Alice can achieve this no matter how Bob plays.
[i]Proposed by Oriol Solé[/i]
1958 February Putnam, B1
i) Given line segments $A,B,C,D$ with $A$ the longest, construct a quadrilateral with these sides and with $A$ and $B$ parallel, when possible.
ii) Given any acute-angled triangle $ABC$ and one altitude $AH$, select any point $D$ on $AH$, then draw $BD$ and extend until it intersects $AC$ in $E$, and draw $CD$ and extend until it intersects $AB$ in $F$. Prove that $\angle AHE = \angle AHF$.
2009 Postal Coaching, 4
Let $ABC$ be a triangle, and let $DEF$ be another triangle inscribed in the incircle of $ABC$. If $s$ and $s_1$ denote the semiperimeters of $ABC$ and $DEF$ respectively, prove that $2s_1 \le s$. When does equality hold?
2002 Kurschak Competition, 3
Prove that the edges of a complete graph with $3^n$ vertices can be partitioned into disjoint cycles of length $3$.
2021 EGMO, 5
A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane such that
[list]
[*] no three points in $P$ lie on a line and
[*] no two points in $P$ lie on a line through the origin.
[/list]
A triangle with vertices in $P$ is [i]fat[/i] if $O$ is strictly inside the triangle. Find the maximum number of fat triangles.
1971 Czech and Slovak Olympiad III A, 5
Let $ABC$ be a given triangle. Find the locus $\mathbf M$ of all vertices $Z$ such that triangle $XYZ$ is equilateral where $X$ is any point of segment $AB$ and $Y\neq X$ lies on ray $AC.$
1979 Bulgaria National Olympiad, Problem 5
A convex pentagon $ABCDE$ satisfies $AB=BC=CA$ and $CD=DE=EC$. Let $S$ be the center of the equilateral triangle $ABC$ and $M$ and $N$ be the midpoints of $BD$ and $AE$, respectively. Prove that the triangles $SME$ and $SND$ are similar.
1990 IMO Longlists, 5
Given the condition that there exist exactly $1990$ triangles $ABC$ with integral side-lengths satisfying the following conditions:
(i) $\angle ABC =\frac 12 \angle BAC;$
(ii) $AC = b.$
Find the minimal value of $b.$
1998 Croatia National Olympiad, Problem 3
Let $AA_1,BB_1,CC_1$ be the altitudes of a triangle $ABC$. If $\overrightarrow{AA_1}+\overrightarrow{BB_1}+\overrightarrow{CC_1}=0$ prove that the triangle $ABC$ is equilateral.
1998 Croatia National Olympiad, Problem 1
Let $a,b,c$ be the sides and $\alpha,\beta,\gamma$ be the corresponding angles of a triangle. Prove the equality
$$\left(\frac bc+\frac cb\right)\cos\alpha+\left(\frac ca+\frac ac\right)\cos\beta+\left(\frac ab+\frac ba\right)\cos\gamma=3.$$
2002 Croatia National Olympiad, Problem 3
If two triangles with side lengths $a,b,c$ and $a',b',c'$ and the corresponding angle $\alpha,\beta,\gamma$ and $\alpha',\beta',\gamma'$ satisfy $\alpha+\alpha'=\pi$ and $\beta=\beta'$, prove that $aa'=bb'+cc'$.
2015 ISI Entrance Examination, 7
Let $\gamma_1, \gamma_2,\gamma_3 $ be three circles of unit radius which touch each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be $\triangle XYZ$ . Find the length of each side of $\triangle XYZ$
2015 China Team Selection Test, 4
Prove that : For each integer $n \ge 3$, there exists the positive integers $a_1<a_2< \cdots <a_n$ , such that for $ i=1,2,\cdots,n-2 $ , With $a_{i},a_{i+1},a_{i+2}$ may be formed as a triangle side length , and the area of the triangle is a positive integer.
1968 Yugoslav Team Selection Test, Problem 3
Each side of a triangle $ABC$ is divided into three equal parts, and the middle segment in each of the sides is painted green. In the exterior of $\triangle ABC$ three equilateral triangles are constructed, in such a way that the three green segments are sides of these triangles. Denote by $A',B',C'$ the vertices of these new equilateral triangles that don’t belong to the edges of $\triangle ABC$, respectively. Let $A'',B'',C''$ be the points symmetric to $A',B',C'$ with respect to $BC,CA,AB$.
(a) Prove that $\triangle A'B'C'$ and $\triangle A''B''C''$ are equilateral.
(b) Prove that $ABC,A'B'C'$, and $A''B''C''$ have a common centroid.
Kyiv City MO Juniors 2003+ geometry, 2008.8.4
There are two triangles $ABC$ and $BKL$ on the plane so that the segment $AK$ is divided into three equal parts by the point of intersection of the medians $\vartriangle ABC$ and the point of intersection of the bisectors $ \vartriangle BKL $ ($AK $ - median $ \vartriangle ABC$, $KA$ - bisector $\vartriangle BKL $) and quadrilateral $KALC $ is trapezoid. Find the angles of the triangle $BKL$.
(Bogdan Rublev)
2010 VTRMC, Problem 4
Let $\triangle ABC$ be a triangle with sides $a,b,c$ and corresponding angles $A,B,C$ (so $a=BC$ and $A=\angle BAC$ etc.). Suppose that $4A+3C=540^\circ$. Prove that $(a-b)^2(a+b)=bc^2$.
2025 6th Memorial "Aleksandar Blazhevski-Cane", P6
There are $n \ge 7$ points in the plane, no $3$ of which are collinear. At least $7$ pairs of points are joined by line segments. For every aforementioned line segment $s$, let $t(s)$ be the number of triangles for which the segment $s$ is a side. Prove that there exist different line segments $s_1, s_2, s_3,$ and $s_4$ such that
\[t(s_1) = t(s_2) = t(s_3) = t(s_4)\]
holds.
Proposed by [i]Viktor Simjanoski[/i]
1992 Bulgaria National Olympiad, Problem 5
Points $D,E,F$ are midpoints of the sides $AB,BC,CA$ of triangle $ABC$. Angle bisectors of the angles $BDC$ and $ADC$ intersect the lines $BC$ and $AC$ respectively at the points $M$ and $N$, and the line $MN$ intersects the line $CD$ at the point $O$. Let the lines $EO$ and $FO$ intersect respectively the lines $AC$ and $BC$ at the points $P$ and $Q$. Prove that $CD=PQ$. [i](Plamen Koshlukov)[/i]
1962 Bulgaria National Olympiad, Problem 4
There are given a triangle and some internal point $P$. $x,y,z$ are distances from $P$ to the vertices $A,B$ and $C$. $p,q,r$ are distances from $P$ to the sides $BC,CA,AB$ respectively. Prove that:
$$xyz\ge(q+r)(r+p)(p+q).$$
2019 Istmo Centroamericano MO, 5
Gabriel plays to draw triangles using the vertices of a regular polygon with $2019$ sides, following these rules:
(i) The vertices used by each triangle must not have been previously used.
(ii) The sides of the triangle to be drawn must not intersect with the sides of the triangles previously drawn.
If Gabriel continues to draw triangles until it is no longer possible, determine the minimum number of triangles that he drew.
1991 Bulgaria National Olympiad, Problem 1
Let $M$ be a point on the altitude $CD$ of an acute-angled triangle $ABC$, and $K$ and $L$ the orthogonal projections of $M$ on $AC$ and $BC$. Suppose that the incenter and circumcenter of the triangle lie on the segment $KL$.
(a) Prove that $CD=R+r$, where $R$ and $r$ are the circumradius and inradius, respectively.
(b) Find the minimum value of the ratio $CM:CD$.