Found problems: 405
Three squares $BCDE,CAFG$ and $ABHI$ are constructed outside the triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. Prove that $APQ$ is an isosceles right triangle.
Show that a triangle whose angles $A$, $B$, $C$ satisfy the equality
\[ \frac{\sin^2 A + \sin^2 B + \sin^2 C}{\cos^2 A + \cos^2 B + \cos^2 C} = 2 \]
is a rectangular triangle.
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]
On the sides of a triangle $ABC$ are constructed similar triangles $ABD,BCE,CAF$ with $k=AD/DB=BE/EC=CF/FA$ and $\alpha=\angle ADB=\angle BEC=\angle CFA$. Prove that the midpoints of the segments $AC,BC,CD$ and $EF$ form a parallelogram with an angle $\alpha$ and two sides whose ratio is $k$.
Let $ABC$ be a triangle, and let $P$ be a point inside it such that $\angle PAC = \angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$, respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM.$
In a triangle $ABC$ with $AC\ne BC$, $M$ is the midpoint of $AB$ and $\angle A=\alpha$, $\angle B=\beta$, $\angle ACM=\varphi$ and $\angle BSM=\Psi$. Prove that
$$\frac{\sin\alpha\sin\beta}{\sin(\alpha-\beta)}=\frac{\sin\varphi\sin\Psi}{\sin(\varphi-\Psi)}.$$
Show that it is possible to cut any triangle into several pieces, so that a rectangle is formed when they are joined together.
We say that two triangles are oriented similarly if they are similar and have the same orientation. Prove that if $ALT, ARM, ORT, $ and $ULM$ are four triangles which are oriented similarly, then $A$ is the midpoint of the line segment $OU.$
Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths.
Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent.
[i]Bogdan Enescu, Romania[/i]
Let $ ABCD$ be a regular tetrahedron and $ M,N$ distinct points in the planes $ ABC$ and $ ADC$ respectively. Show that the segments $ MN,BN,MD$ are the sides of a triangle.
Let $A_r,B_r, C_r$ be points on the circumference of a given circle $S$. From the triangle $A_rB_rC_r$, called $\Delta_r$, the triangle $\Delta_{r+1}$ is obtained by constructing the points $A_{r+1},B_{r+1}, C_{r+1} $on $S$ such that $A_{r+1}A_r$ is parallel to $B_rC_r$, $B_{r+1}B_r$ is parallel to $C_rA_r$, and $C_{r+1}C_r$ is parallel to $A_rB_r$. Each angle of $\Delta_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\Delta_1,\Delta_2, \ldots ,\Delta_{15}$ are congruent.
Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram.
Prove that $GR=GS$.
[i]Proposed by Hossein Karke Abadi, Iran[/i]
Each side of a triangle $ABC$ has been divided into $n+1$ equal parts. Find the number of triangles with the vertices at the division points having no side parallel to or lying at a side of $\triangle ABC$.
Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively.
[i]Proposed by United Kingdom.[/i]
In the triangle $ABC$, let $A'$ be the intersection of the perpendicular bisector of $AB$ and the angle bisector of $\angle BAC$ and define $B', C'$ analogously. Prove that
a) The triangle $ABC$ is equilateral if and only if $A' =B'.$
b) If $A', B'$ and $C'$ are distinct, we have $\angle B' A' C' = 90^{\circ} - \frac{1}{2} \angle BAC.$
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.
(Australia)
In $\triangle ABC$ with $AB=AC$, point $D$ lies strictly between $A$ and $C$ on side $\overline{AC}$, and point $E$ lies strictly between $A$ and $B$ on side $\overline{AB}$ such that $AE=ED=DB=BC$. The degree measure of $\angle ABC$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Construct a triangle given the radii of the excircles.
Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly.
(a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$.
(b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.
$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$
Let $ABC$ be a triangle with incenter $I$ and let $D$ be an arbitrary point on the side $BC$. Let the line through $D$ perpendicular to $BI$ intersect $CI$ at $E$. Let the line through $D$ perpendicular to $CI$ intersect $BI$ at $F$. Prove that the reflection of $A$ across the line $EF$ lies on the line $BC$.
For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ($CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$. We denote by $\Sigma$ the set of all triangles $T$ for which
$\widehat{XYZ} = 45$. Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.
Let $\alpha, \beta$ and $\gamma$ denote the angles of the triangle $ABC$. The perpendicular bisector of $AB$ intersects $BC$ at the point $X$, the perpendicular bisector of $AC$ intersects it at $Y$. Prove that $\tan(\beta) \cdot \tan(\gamma) = 3$ implies $BC= XY$ (or in other words: Prove that a sufficient condition for $BC = XY$ is $\tan(\beta) \cdot \tan(\gamma) = 3$). Show that this condition is not necessary, and give a necessary and sufficient condition for $BC = XY$.
A given triangle is divided into $n$ triangles in such a way that any line segment which is a side of a tiling triangle is either a side of another tiling triangle or a side of the given triangle. Let $s$ be the total number of sides and $v$ be the total number of vertices of the tiling triangles (counted without multiplicity).
(a) Show that if $n$ is odd then such divisions are possible, but each of them has the same number $v$ of vertices and the same number $s$ of sides. Express $v$ and $s$ as functions of $n$.
(b) Show that, for $n$ even, no such tiling is possible
In the triangle $ABC$, angle bisector $CD$ intersects the circumcircle of $ABC$ at the point $K$.
(a) Prove the equalities:
$$\frac1{ID}-\frac1{IK}=\frac1{CI},\enspace\frac{CI}{ID}-\frac{ID}{DK}=1$$where $I$ is the center of the inscribed circle of triangle $ABC$.
(b) On the segment $CK$ some point $P$ is chosen whose projections on $AC,BC,AB$ respectively are $P_1,P_2,P_3$. The lines $PP_3$ and $P_1P_2$ intersect at a point $M$. Find the locus of $M$ when $P$ moves around segment $CK$.