Found problems: 239
Let $ABC$ be an acute-angled triangle, and let $F$ be the foot of its altitude from the vertex $C$. Let $M$ be the midpoint of the segment $CA$. Assume that $CF=BM$. Then the angle $MBC$ is equal to angle $FCA$ if and only if the triangle $ABC$ is equilateral.
Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.
For a convex hexagon $ ABCDEF$ with an area $ S$, prove that:
\[ AC\cdot(BD\plus{}BF\minus{}DF)\plus{}CE\cdot(BD\plus{}DF\minus{}BF)\plus{}AE\cdot(BF\plus{}DF\minus{}BD)\geq 2\sqrt{3}S
\]
Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.
See all the problems from 5-th Kyiv math festival
[url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.
Circles $ C_1,C_2$ intersect at $ P$. A line $ \Delta$ is drawn arbitrarily from $ P$ and intersects with $ C_1,C_2$ at $ B,C$. What is locus of $ A$ such that the median of $ AM$ of triangle $ ABC$ has fixed length $ k$.
Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersection of $AD,BE$, $H$ is the orthocentre of $\triangle ABC$. Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$.
(Edited)
In $\triangle ABC$, the length of altitude $AD$ is $12$, and the bisector $AE$ of $\angle A$ is $13$. Denote by $m$ the length of median $AF$. Find the range of $m$ when $\angle A$ is acute, orthogonal and obtuse respectively.
Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$.
Prove that $\angle {APB}=\angle {APD}$
Equiangular hexagon $ ABCDEF$ has side lengths $ AB \equal{} CD \equal{} EF \equal{} 1$ and $ BC \equal{} DE \equal{} FA \equal{} r$. The area of $ \triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $ r$?
$ \textbf{(A)}\ \frac {4\sqrt {3}}{3} \qquad
\textbf{(B)}\ \frac {10}{3} \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ \frac {17}{4} \qquad
\textbf{(E)}\ 6$
Let $\Delta ABC$ be a triangle. The internal bisectors of angles $\angle CAB$ and $\angle ABC$ intersect segments $BC$, respectively $AC$ in $D$, respectively $E$. Prove that \[DE\leq (3-2\sqrt{2})(AB+BC+CA).\]
Let $ ABC$ be a triangle with $ \angle A \equal{} 45^\circ$. Let $ P$ be a point on side $ BC$ with $ PB \equal{} 3$ and $ PC \equal{} 5$. Let $ O$ be the circumcenter of $ ABC$. Determine the length $ OP$.
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?
$\textbf{(A) }\dfrac{\sqrt3}4\qquad
\textbf{(B) }\dfrac{\sqrt3}3\qquad
\textbf{(C) }\dfrac23\qquad
\textbf{(D) }\dfrac{\sqrt2}2\qquad
\textbf{(E) }\dfrac{\sqrt3}2$
Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.
Let $x,y,z$ be real numbers such that $x^2+y^2+z^2-2xyz=1$. Prove that
\[ (1+x)(1+y)(1+z)\le 4+4xyz. \]
Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $q$.
Triangle $ ABC$ has $ \angle C \equal{} 60^{\circ}$ and $ BC \equal{} 4$. Point $ D$ is the midpoint of $ BC$. What is the largest possible value of $ \tan{\angle BAD}$?
$ \textbf{(A)} \ \frac {\sqrt {3}}{6} \qquad \textbf{(B)} \ \frac {\sqrt {3}}{3} \qquad \textbf{(C)} \ \frac {\sqrt {3}}{2\sqrt {2}} \qquad \textbf{(D)} \ \frac {\sqrt {3}}{4\sqrt {2} \minus{} 3} \qquad \textbf{(E)}\ 1$
In this figure $\angle RFS = \angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\dfrac{1}{2}$ inches. The length of $RS$, in inches, is:
[asy]
import olympiad;
pair F,R,S,D;
F=origin;
R=5*dir(aCos(9/16));
S=(7.5,0);
D=4*dir(aCos(9/16)+aCos(1/8));
label("$F$",F,SW);label("$R$",R,N); label("$S$",S,SE); label("$D$",D,W);
label("$7\frac{1}{2}$",(F+S)/2.5,SE);
label("$4$",midpoint(F--D),SW);
label("$5$",midpoint(F--R),W);
label("$6$",midpoint(D--R),N);
draw(F--D--R--F--S--R);
markscalefactor=0.1;
draw(anglemark(S,F,R)); draw(anglemark(F,D,R));
//Credit to throwaway1489 for the diagram[/asy]
$\textbf{(A)}\ \text{undetermined} \qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\dfrac{1}{2} \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 6\dfrac{1}{4}$
Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter of quadrilateral $ ABCD$.
Compute $\tan\left(\frac{\pi}{7}\right)\tan\left(\frac{2\pi}{7}\right)\tan\left(\frac{3\pi}{7}\right)$.
In triangle $ABC$, $\angle A = 90^\circ$ and $\angle C = 70^\circ$. $F$ is point on $AB$ such that $\angle ACF = 30^\circ$, and $E$ is a point on $CA$ such that $\angle CF E = 20^\circ$. Prove that $BE$ bisects $\angle B$.
Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?
$\textbf{(A) }6\sqrt 3+3\qquad
\textbf{(B) }\dfrac{27}2\qquad
\textbf{(C) }8\sqrt 3\qquad
\textbf{(D) }14\qquad
\textbf{(E) }3\sqrt 3+9\qquad$
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths $3$, $4$, and $5$. What is the area of the triangle?
$\textbf{(A)}\ 6 \qquad
\textbf{(B)}\ \frac{18}{\pi^2} \qquad
\textbf{(C)}\ \frac{9}{\pi^2}\left(\sqrt{3}-1\right) \qquad
\textbf{(D)}\ \frac{9}{\pi^2}\left(\sqrt{3}+1\right) \qquad
\textbf{(E)}\ \frac{9}{\pi^2}\left(\sqrt{3}+3\right)$
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?