Found problems: 721
Points $C_1, B_1$ on sides $AB, AC$ respectively of triangle $ABC$ are such that $BB_1 \perp CC_1$. Point $X$ lying inside the triangle is such that $\angle XBC = \angle B_1BA, \angle XCB = \angle C_1CA$. Prove that $\angle B_1XC_1 =90^o- \angle A$.
(A. Antropov, A. Yakubov)
Let $ABC$ be an isosceles triangle with $AC = BC$, and let $M$ be the midpoint of $AB$. Let $P$ be a point inside the triangle such that $\angle PAB =\angle PBC$. Prove that $\angle APM + \angle BPC = 180^o$.
Suppose $L$ is the circle inscribed in the square $T_1$, and $T_2$ is the square inscribed in $L$, so that vertices of $T_1$ lie on the straight lines containing the sides of $T_2$. Find the angles of the convex octagon whose vertices are at the tangency points of $L$ with the sides of $T_1$ and at the vertices of $T_2$.
(S Markelov)
Let $ABC$ be an isosceles triangle with $\angle A=\angle B=80^\circ$. A straight line passes through $B$
and through the circumcenter of the triangle and intersects the side $AC$ at $D$. Prove that $AB=CD$.
Rhombuses $ABCD$ and $A_1B_1C_1D_1$ are equal. Side $BC$ intersects sides $B_1C_1$ and $C_1D_1$ at points $M$ and $N$ respectively. Side $AD$ intersects sides $A_1B_1$ and $A_1D_1$ at points $Q$ and $P$ respectively. Let $O$ be the intersection point of lines $MP$ and $QN$. Find $\angle A_1B_1C_1$ , if $\angle QOP = \frac12 \angle B_1C_1D_1$.
Let $n>1$ be an integer. Suppose we are given $2n$ points in the plane such that no three of them are collinear. The points are to be labelled $A_1, A_2, \dots , A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2$. We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.
A square is given. Lines divide it into $n$ polygons.
What is he the largest possible sum of the internal angles of all polygons?
Inside the convex quadrilateral $ABCD$ there is a point $M$ such that $\angle AMB = \angle ADM + \angle BCM$ and $\angle AMD = \angle ABM + \angle DCM$. Prove that $AM \cdot CM + BM \cdot DM \ge \sqrt{AB \cdot BC\cdot CD \cdot DA}$
In acute triangle $ABC$, $\angle A = 20^o$. Prove that the triangle is isosceles if and only if $$\sqrt[3]{a^3 + b^3 + c^3 -3abc} = \min\{b, c\}$$, where $a,b, c$ are the side lengths of triangle $ABC$.
On the diagonal $AC$ of a convex quadrilateral $ABCD$ is chosen such a point $K$ such that $KD = DC$, $\angle BAC = \frac12 \angle KDC$, $\angle DAC = \frac12 \angle KBC$. Prove that $\angle KDA = \angle BCA$ or $\angle KDA = \angle KBA$.
A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.
In triangle $ABC$, angle $B$ is obtuse and equal to $a$. The bisectors of angles $A$ and $C$ intersect opposite sides at points $P$ and $M$, respectively. On the side $AC$, points $K$ and $L$ are taken so that $\angle ABK = \angle CBL = 2a - 180^o$. What is the angle between straight lines $KP$ and $LM$?
In the quadrilateral $ABCD$, the angles $B$ and $D$ are right . The diagonal $AC$ forms with the side $AB$ the angle of $40^o$, as well with side $AD$ an angle of $30^o$. Find the acute angle between the diagonals $AC$ and $BD$.
In the diagram, $\angle AOB = \angle BOC$ and$\angle COD = \angle DOE = \angle EOF$. Given that $\angle AOD = 82^o$ and $\angle BOE = 68^o$. Find $\angle AOF$.
[img]https://cdn.artofproblemsolving.com/attachments/b/2/deba6cd740adbf033ad884fff8e13cd21d9c5a.png[/img]
In triangle $ABC,\angle B > \angle C, T$ is the midpoint of arc $BAC$ of the circumcicle of $ABC$, and $I$ is the incentre of $ABC$. Let $E$ be point such that $\angle AEI = 90^0$ and $AE$ is parallel to $BC$. If $TE$ intersects the circumcircle of $ABC$ at $P(\ne T)$ and $\angle B = \angle IPB$, determine $\angle A$.
A parallelogram $ABCD$ with $|AD| =|BD|$ has been given. A point $E$ lies on line segment $|BD|$ in such a way that $|AE| = |DE|$. The (extended) line $AE$ intersects line segment $BC$ in $F$. Line $DF$ is the angle bisector of angle $CDE$. Determine the size of angle $ABD$.
[asy]
unitsize (3 cm);
pair A, B, C, D, E, F;
D = (0,0);
A = dir(250);
B = dir(290);
C = B + D - A;
E = extension((A + D)/2, (A + D)/2 + rotate(90)*(A - D), B, D);
F = extension(A, E, B, C);
draw(A--B--C--D--cycle);
draw(A--F--D--B);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, NE);
dot("$D$", D, NW);
dot("$E$", E, S);
dot("$F$", F, SE);
[/asy]
Let $\alpha,\beta,\gamma$ be angles of a triangle. Determine all real triplets $x,y,z$ satisfying the system
\begin{align*}
x\cos\beta+\frac1z\cos\alpha &=1, \\
y\cos\gamma+\frac1x\cos\beta &=1, \\
z\cos\alpha+\frac1y\cos\gamma &=1.
\end{align*}
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)
Let $D$ be a point inside an acute triangle $ABC$ such that $\angle ADC = \angle A +\angle B$, $\angle BDA = \angle B + \angle C$ and $\angle CDB = \angle C + \angle A$. Prove that $\frac{AB \cdot CD}{AD} = \frac{AC \cdot CB} {AB}$.
$ABCD$ is a quadrilateral with $\angle A = 90o$, $AD = a$, $BC = b$, $AB = h$, and area $\frac{(a+b)h}{2}$. What can we say about $\angle B$?
Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.
The pentagon $ABCDE$ is inscribed in the circle. Line segments $AC$ and $BD$ intersect at point $K$. Line segment $CE$ touches the circumcircle of triangle $ABK$ at point $N$. Find the angle $CNK$ if $\angle ECD = 40^o.$
Into an angle $A$ of size $a$, a circle is inscribed tangent to its sides at points $B$ and $C$. A line tangent to this circle at a point M meets the segments $AB$ and $AC$ at points $P$ and $Q$ respectively. What is the minimum $a$ such that the inequality $S_{PAQ}<S_{BMC}$ is possible?
Let $O$ be a point inside a convex polygon $A_1A_2... A_n$ such that $$\angle OA_1A_n \le \angle OA_1A_2, \angle OA_2A_1 \le \angle OA_2A_3, ..., \angle OA_{n-1}A_{n-2} \le \angle OA_{n-1}A_n, \angle OA_nA_{n-1} \le \angle OA_nA_1$$ and all of these angles are acute. Prove that $O$ is the centre of the circle inscribed in the polygon.
(V Proizvolov)
In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$. If angle $D$ is $60^o$ and angle $E$ is $40^o$, how large is then angle $u$?
[img]https://1.bp.blogspot.com/-6Fq2WUcP-IA/Xzb9G7-H8jI/AAAAAAAAMWY/hfMEAQIsfTYVTdpd1Hfx15QPxHmfDLEkgCLcBGAsYHQ/s0/2009%2BMohr%2Bp1.png[/img]