An acute-angled triangle $ABC$ is inscribed in a circle with center $O$. The bisector of $\angle A$ meets $BC$ at $D$, and the perpendicular to $AO$ through $D$ meets the segment $AC$ in a point $P$. Show that $AB = AP$.

In a tetrahedron we know that sum of angles of all vertices is $180^\circ.$ (e.g. for vertex $A$, we have $\angle BAC + \angle CAD + \angle DAB=180^\circ.$) Prove that faces of this tetrahedron are four congruent triangles.

In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$? $\textbf{(A) }6\sqrt{2}\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }9\sqrt{2}\qquad\textbf{(E) }32$ [asy] draw((-4,6*sqrt(2))--(4,6*sqrt(2))); draw((-4,-6*sqrt(2))--(4,-6*sqrt(2))); draw((-8,0)--(-4,6*sqrt(2))); draw((-8,0)--(-4,-6*sqrt(2))); draw((4,6*sqrt(2))--(8,0)); draw((8,0)--(4,-6*sqrt(2))); draw((-4,6*sqrt(2))--(4,6*sqrt(2))--(4,8+6*sqrt(2))--(-4,8+6*sqrt(2))--cycle); draw((-8,0)--(-4,-6*sqrt(2))--(-4-6*sqrt(2),-4-6*sqrt(2))--(-8-6*sqrt(2),-4)--cycle); label("$I$",(-4,8+6*sqrt(2)),dir(100)); label("$J$",(4,8+6*sqrt(2)),dir(80)); label("$A$",(-4,6*sqrt(2)),dir(280)); label("$B$",(4,6*sqrt(2)),dir(250)); label("$C$",(8,0),W); label("$D$",(4,-6*sqrt(2)),NW); label("$E$",(-4,-6*sqrt(2)),NE); label("$F$",(-8,0),E); draw((4,8+6*sqrt(2))--(4,6*sqrt(2))--(4+4*sqrt(3),4+6*sqrt(2))--cycle); label("$K$",(4+4*sqrt(3),4+6*sqrt(2)),E); draw((4+4*sqrt(3),4+6*sqrt(2))--(8,0),dashed); label("$H$",(-4-6*sqrt(2),-4-6*sqrt(2)),S); label("$G$",(-8-6*sqrt(2),-4),W); label("$32$",(-10,-8),N); label("$18$",(0,6*sqrt(2)+2),N); [/asy]

Let $P$ be an arbitrary point on the incircle $k$ of triangle $ABC$ with center $I$, different from the points of tangency with its sides. The tangent to $k$ at $P$ intersects the lines $BC$, $AC$, $AB$ at points $A_0$, $B_0$, $C_0$, respectively. The lines through $A_0$, $B_0$, $C_0$, parallel to the bisectors of the angles $\angle BAC$, $\angle ABC$, $\angle ACB$, form a triangle $\Delta$. Prove that the line $PI$ is tangent to the circumcircle of $\Delta$.

The base of the pyramid is a convex quadrangle. Is there necessarily a section of this pyramid that does not intersect the base and is an inscribed quadrangle? (M. Volchkevich)

Points $A_1, B_1, C_1$ inside an acute-angled triangle $ABC$ are selected on the altitudes from $A, B, C$ respectively so that the sum of the areas of triangles $ABC_1, BCA_1$, and $CAB_1$ is equal to the area of triangle $ABC$. Prove that the circumcircle of triangle $A_1B_1C_1$ passes through the orthocenter $H$ of triangle $ABC$.

Compute the area of an octagon inscribed in a circle, whose four sides have length $1$ and the other four sides have length $2$.

The two heights in the triangle are not less than the respective sides. Find the angles.

$ABC$ is a triangle with $AB = 15$, $BC = 14$, and $CA = 13$. The altitude from $A$ to $BC$ is extended to meet the circumcircle of $ABC$ at $D$. Find $AD$.

Let $E$ and $F$ on $AC$ and $AB$ respectively in $\triangle ABC$ such that $DE || BC$ then draw line $l$ through $A$ such that $l || BC$ let $D'$ and $E'$ reflection of $D$ and $E$ to $l$ respectively prove that $D'B, E'C$ and $l$ are congruence.

Point $M$ on side $AB$ of quadrilateral $ABCD$ is such that quadrilaterals $AMCD$ and $BMDC$ are circumscribed around circles centered at $O_1$ and $O_2$ respectively. Line $O_1O_2$ cuts an isosceles triangle with vertex $M$ from angle $CMD$. prove that $ABCD$ is a cyclc quadrilateral.

A cyclic $n$-gon is divided by non-intersecting (inside the $n$-gon) diagonals to $n-2$ triangles. Each of these triangles is similar to at least one of the remaining ones. For what $n$ this is possible?

Three circles are constructed on the medians of a triangle as on diameters. It is known that they intersect in pairs. Let $C_1$ be the intersection point of the circles built on the medians $AM_1$ and $BM_2$, which is more distant from the vertex $C$. Points $A_1$ and $B_1$ are defined similarly. Prove that the lines $AA_1, BB_1$ and $CC_1$ intersect at one point. (D. Tereshin)

Let $ABC$ be an acute triangle such that $\angle B > \angle C$. Let $D$ and $E$ be the points on the segments $BC$ and $CA$, respectively, such that $AD$ bisects $\angle A$ and $BE\perp AC$. Finally, let $M$ be the midpoint of the side $BC$. Suppose that the circumcircle of $\triangle CDE$ intersects $AD$ again at a point $X$ different from $D$. Prove that $\angle XME = 90^{\circ} - \angle BAC$.

[b]7.1[/b] Prove that a natural number with an odd number of divisors is a perfect square. [b]7.2[/b] In a triangle $ABC$ with area $S$, medians $AK$ and $BE$ are drawn, intersecting at the point $O$. Find the area of the quadrilateral $CKOE$. [img]https://cdn.artofproblemsolving.com/attachments/0/f/9cd32bef4f4459dc2f8f736f7cc9ca07e57d05.png[/img] [b]7.3 .[/b] The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers. When you need to swap tires so that the car can travel the longest possible distance with the same tires? [b]7.4 [/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it? [b]7.5 / 8.4[/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ . [b]7.6[/b] Black paint was sprayed onto a white surface. Prove that there are two points of the same color, the distance between which is $1965$ meters. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].

The graph of $r=2+\cos2\theta$ and its reflection over the line $y=x$ bound five regions in the plane. Find the area of the region containing the origin.

Show that any triangle has two sides whose lengths $a$ and $b$ satisfy $\frac{\sqrt{5}-1}{2}<\frac{a}{b}<\frac{\sqrt{5}+1}{2}$.

Anita plays the following single-player game: She is given a circle in the plane. The center of this circle and some point on the circle are designated “known points”. Now she makes a series of moves, each of which takes one of the following forms: (i) She draws a line (infinite in both directions) between two “known points”; or (ii) She draws a circle whose center is a “known point” and which intersects another “known point”. Once she makes a move, all intersections between her new line/circle and existing lines/circles become “known points”, unless the new/line circle is identical to an existing one. In other words, Anita is making a ruler-and-compass construction, starting from a circle. What is the smallest number of moves that Anita can use to construct a drawing containing an equilateral triangle inscribed in the original circle?

Let $ABCD$ be a tetrahedron and $O$ its incenter, and let the line $OD$ be perpendicular to $AD$. Find the angle between the planes $DOB$ and $DOC.$

Let $C_1$ and $C_2$ be two circles with centers $O_1$ and $O_2$, respectively, intersecting at $A$ and $B$. Let $l_1$ be the line tangent to $C_1$ passing trough $A$, and $l_2$ the line tangent to $C_2$ passing through $B$. Suppose that $l_1$ and $l_2$ intersect at $P$ and $l_1$ intersects $C_2$ again at $Q$. Show that $PO_1B$ and $PO_2Q$ are similar triangles. [i]Proposed by Pablo Valeriano[/i]

If in $\bigtriangleup ABC$, $AD$ is the altitude and $AE$ is the diameter of the circumcircle through $A$, then prove that $AB\cdot AC = AD \cdot AE$. Use this result to show that if $ABCD$ is a cyclic quadrilateral then show that $AC \cdot (AB \cdot BC + CD \cdot DA) = BD\cdot (DA\cdot AB + BC \cdot CD)$.

Side $ AC$ of right triangle $ ABC$ is divide into $ 8$ equal parts. Seven line segments parallel to $ BC$ are drawn to $ AB$ from the points of division. If $ BC \equal{} 10$, then the sum of the lengths of the seven line segments: $ \textbf{(A)}\ \text{cannot be found from the given information} \qquad \textbf{(B)}\ \text{is }{33}\qquad \textbf{(C)}\ \text{is }{34}\qquad \textbf{(D)}\ \text{is }{35}\qquad \textbf{(E)}\ \text{is }{45}$

Let $ABC$ be an acute triangle and let $O$ be its circumcentre. Now, let the diameter $PQ$ of circle $ABC$ intersects sides $AB$ and $AC$ in their interior at$ D$ and $E$, respectively. Now, let $F$ and $G$ be the midpoints of $CD$ and $BE$. Prove that $\angle FOG=\angle BAC$

In parallelogram $ABCD$, $\overline{DE}$ is the altitude to the base $\overline{AB}$ and $\overline{DF}$ is the altitude to the base $\overline{BC}$. ['''Note:''' ''Both pictures represent the same parallelogram.''] If $DC=12$, $EB=4$, and $DE=6$, then $DF=$ [asy] unitsize(12); pair A,B,C,D,P,Q,W,X,Y,Z; A = (0,0); B = (12,0); C = (20,6); D = (8,6); W = (18,0); X = (30,0); Y = (38,6); Z = (26,6); draw(A--B--C--D--cycle); draw(W--X--Y--Z--cycle); P = (8,0); Q = (758/25,6/25); dot(A); dot(B); dot(C); dot(D); dot(W); dot(X); dot(Y); dot(Z); dot(P); dot(Q); draw(A--B--C--D--cycle); draw(W--X--Y--Z--cycle); draw(D--P); draw(Z--Q); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$E$",P,S); label("$A$",W,SW); label("$B$",X,S); label("$C$",Y,NE); label("$D$",Z,NW); label("$F$",Q,E); [/asy] $\text{(A)}\ 6.4 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 7.2 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10$

There is a $11\times 11$ square grid. We divided this in $5$ rectangles along unit squares. How many ways that one of the rectangles doesn’t have a edge on basic circumference. Note that we count as different ways that one way coincides with another way by rotating or reversing.