Let $ABC$ be an acute triangle. Let $H_A,H_B,H_C$ be points on $BC,AC,AB$ respectively such that $AH_A\perp BC, BH_B\perp AC, CH_C\perp AB$. Let the circumcircles $AH_BH_C,BH_AH_C,CH_AH_B$ be $\omega_A,\omega_B,\omega_C$ with circumcenters $O_A,O_B,O_C$ respectively and define $O_AB\cap \omega_B=P_{AB}\neq B$. Define $P_{AC},P_{BA},P_{BC},P_{CA},P_{CB}$ similarly. Define circles $\omega_{AB},\omega_{AC}$ to be $O_AP_{AB}H_C,O_AP_{AC}H_B$ respectively. Define circles $\omega_{BA},\omega_{BC},\omega_{CA},\omega_{CB}$ similarly. Prove that there are $6$ pairs of tangent circles in the $6$ circles of the form $\omega_{xy}$.

Given an isosceles triangle $ABC$ with $AB = AC$. Let $O$ be the circumcenter of $ABC$, $D$ the midpoint of $AB$ and $E$ the centroid of $ACD$. Prove that $CD \perp EO$.

$ABC$ is a triangle with points $D$, $E$ on $BC$ with $D$ nearer $B$; $F$, $G$ on $AC$, with $F$ nearer $C$; $H$, $K$ on $AB$, with $H$ nearer $A$. Suppose that $AH=AG=1$, $BK=BD=2$, $CE=CF=4$, $\angle B=60^\circ$ and that $D$, $E$, $F$, $G$, $H$ and $K$ all lie on a circle. Find the radius of the incircle of triangle $ABC$.

Which polygon inscribed in a given circle has the property that the sum of the squares of the lengths of its sides is maximum?

On sides $AB$ and $BC$ of a square $ABCD$ the respective points $E$ and $F$ have been chosen so that $BE = BF$. Let $BN$ be the altitude in triangle $BCE$. Prove that $\angle DNF = 90$.

$ABCD$ is a square. $E$ is a point on the side $BC$ such that $BE =1/3 BC$, and $F$ is a point on the ray $DC$ such that $CF =1/2 DC$. Prove that the lines $AE$ and $BF$ intersect on the circumcircle of the square. [img]https://cdn.artofproblemsolving.com/attachments/e/d/09a8235d0748ce4479e21a3bb09b0359de54b5.png[/img]

Let $ABC$ be a triangle with $AB < AC,$ $I$ its incenter, and $M$ the midpoint of the side $BC$. If $IA=IM,$ determine the smallest possible value of the angle $AIM$.

The triangle $ ABC$ is inscribed in a circle. The interior bisectors of the angles $ A,B$ and $ C$ meet the circle again at $ A', B'$ and $ C'$ respectively. Prove that the area of triangle $ A'B'C'$ is greater than or equal to the area of triangle $ ABC.$

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $17$. What is the greatest possible perimeter of the triangle?

In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$ $\textbf{(A) } \frac{\sqrt5}{12} \qquad \textbf{(B) } \frac{\sqrt5}{10} \qquad \textbf{(C) } \frac{\sqrt5}{9} \qquad \textbf{(D) } \frac{\sqrt5}{8} \qquad \textbf{(E) } \frac{2\sqrt5}{15}$

Let $PQRS$ be a square piece of paper. $P$ is folded onto $R$ and then $Q$ is folded onto $S$. The area of the resulting figure is 9 square inches. Find the perimeter of square $PQRS$. [asy] draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); label("$P$",(0,2),SE); label("$Q$",(2,2),SW); label("$R$",(2,0),NW); label("$S$",(0,0),NE);[/asy] $ \text{(A)}\ 9\qquad\text{(B)}\ 16\qquad\text{(C)}\ 18\qquad\text{(D)}\ 24\qquad\text{(E)}\ 36 $

Let $ G$ be a finite non-commutative group of order $ t \equal{} 2^nm$, where $ n, m$ are positive and $ m$ is odd. Prove, that if the group contains an element of order $ 2^n$, then (i) $ G$ is not simple; (ii) $ G$ contains a normal subgroup of order $ m$.

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? $ \textbf{(A) } \frac {1}{2187} \qquad \textbf{(B) } \frac {1}{729} \qquad \textbf{(C) } \frac {2}{243} \qquad \textbf{(D) } \frac {1}{81} \qquad \textbf{(E) } \frac {5}{243}$

Parts of a pentagon have areas $x,y,z$ as shown in the picture. Given the area $x$, find the areas $y$ and $z$ and the area of the entire pentagon. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS9mLzM5NjNjNDcwY2ZmMzgzY2QwYWM0YzI1NmYzOWU2MWY1NTczZmYxLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0wOCBhdCA0LjMwLjU1IFBNLnBuZw[/img]

Acute-angled triangle ${ABC}$ with ${AB<AC<BC}$ and let be ${c(O,R)}$ it’s circumcircle. Diameters ${BD}$ and ${CE}$ are drawn. Circle ${c_1(A,AE)}$ interescts ${AC}$ at ${K}$. Circle ${{c}_{2}(A,AD)}$ intersects ${BA}$ at ${L}$ .(${A}$ lies between ${B}$ and ${L}$). Prove that lines ${EK}$ and ${DL}$ intersect at circle $c$ . by Evangelos Psychas (Greece)

Let $ABC$ be a triangle with altitudes $AD, BE, CF$ and orthocenter $H$. The perpendicular bisector of $HD$ meets $EF$ at $P$ and $N$ is the center of the nine-point circle. Let $L$ be a point on the circumcircle of $ABC$ such that $\angle PLN=90^{\circ}$ and $A, L$ are in distinct sides of the line $PN$. Show that $ANDL$ is cyclic.

Define a figure which is constructed by unit squares "cross star" if it satisfies the following conditions: $(1)$Square bar $AB$ is bisected by square bar $CD$ $(2)$At least one square of $AB$ lay on both sides of $CD$ $(3)$At least one square of $CD$ lay on both sides of $AB$ There is a rectangular grid sheet composed of $38\times 53=2014$ squares,find the number of such cross star in this rectangle sheet

Let $\triangle ABC$ be a triangle with side‐lengths $a,b,c$, incenter $I$, and circumradius $R$. Denote by $P$ the area of $\triangle ABC$, and let $P_1,\;P_2,\;P_3$ be the areas of triangles $\triangle ABI$, $\triangle BCI$, and $\triangle CAI$, respectively. Prove that \[ \frac{abc}{12R} \;\le\; \frac{P_1^2 + P_2^2 + P_3^2}{P} \;\le\; \frac{3R^3}{4\sqrt[3]{abc}}. \]

A solid box is $ 15$ cm by $ 10$ cm by $ 8$ cm. A new solid is formed by removing a cube $ 3$ cm on a side from each corner of this box. What percent of the original volume is removed? $ \textbf{(A)}\ 4.5 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$

Let $\Gamma$ be the excircle of triangle $ABC$ opposite to the vertex $A$ (namely, the circle tangent to $BC$ and to the prolongations of the sides $AB$ and $AC$ from the part $B$ and $C$). Let $D$ be the center of $\Gamma$ and $E$, $F$, respectively, the points in which $\Gamma$ touches the prolongations of $AB$ and $AC$. Let $J$ be the intersection between the segments $BD$ and $EF$. Prove that $\angle CJB$ is a right angle.

In an acute-angled triangle \(ABC\) \((AC > BC)\) with altitude \(AD\), the following points are marked: \(H\) - the orthocenter, \(O\) - the circumcenter, \(K\) - the midpoint of side \(AB\). Inside the triangle \(\triangle ADC\), there is a point \(P\) such that the following equality holds: \[ \angle KPD + \angle ACB = 2 \angle OPH = 180^{\circ} \] Prove that \[ BH = 2PD \] [i]Proposed by Vadym Solomka[/i]

[b]7.1.[/b] Prove that from the sides of an arbitrary quadrilateral you can fold a trapezoid. [b]7.2 / 6.2[/b] The numbers $A$ and $B$ are relatively prime. What common divisors can have the numbers $A+B$ and $A-B$? [b]7.3. / 6.4[/b] $15$ magazines lie on the table, completely covering it. Prove that it is possible to remove eight of them so that the remaining magz cover at least $7/15$ of the table area. [b]7.4[/b] In a six-digit number that is divisible by $7$, the last digit has been moved to the beginning. Prove that the resulting number is also divisible at $7$. [url=https://artofproblemsolving.com/community/c6h3391057p32066818]7.5*[/url] (asterisk problems in separate posts) [b]7.6 [/b] On sides $AB$ and $ BC$ of triangle $ABC$ , are constructed squares $ABDE$ and $BCKL$ with centers $O_1$ and $O_2$. $M_1$ and $M_2$ are midpoints of segments $DL$ and $AC$. Prove that $O_1M_1O_2M_2$ is a square. [img]https://cdn.artofproblemsolving.com/attachments/8/1/8aa816a84c5ac9de78b396096cf718063de390.png[/img] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].

Find the sum of the fiftieth powers of all sides and diagonals of a regular $100$-gon inscribed in a circle of radius $R.$

Point $A$, $B$, $C$, and $D$ form a rectangle in that order. Point $X$ lies on $CD$, and segments $\overline{BX}$ and $\overline{AC}$ intersect at $P$. If the area of triangle $BCP$ is 3 and the area of triangle $PXC$ is 2, what is the area of the entire rectangle?

[b]6.1[/b] The student bought a briefcase, a fountain pen and a book. If the briefcase cost 5 times cheaper, the fountain pen was 2 times cheaper, and the book was 2 1/2 times cheaper cheaper, then the entire purchase would cost 2 rubles. If the briefcase was worth 2 times cheaper, a fountain pen is 4 times cheaper, and a book is 3 times cheaper, then the whole the purchase would cost 3 rubles. How much does it really cost? ´ [b]6.2.[/b] Which number is greater: $$\underbrace{888...88}_{19 \, digits} \cdot \underbrace{333...33}_{68 \, digits} \,\,\, or \,\,\, \underbrace{444...44}_{19 \, digits} \cdot \underbrace{666...67}_{68 \, digits} \, ?$$ [b]6.3[/b] Distance between Luga and Volkhov 194 km, between Volkhov and Lodeynoye Pole 116 km, between Lodeynoye Pole and Pskov 451 km, between Pskov and Luga 141 km. What is the distance between Pskov and Volkhov? [b]6.4 [/b] There are $4$ objects in pairs of different weights. How to use a pan scale without weights Using five weighings, arrange all these objects in order of increasing weights? [b]6.5 [/b]. Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams. [b]6.6 [/b] In task 6.1, determine what is more expensive: a briefcase or a fountain pen. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].