It is given that $r=\left(3\left(\sqrt6-1\right)-4\left(\sqrt3+1\right)+5\sqrt2\right)R$ where $r$ and $R$ are the radii of the inscribed and circumscribed spheres in a regular $n$-angled pyramid. If it is known that the centers of the spheres given coincide, (a) find $n$; (b) if $n=3$ and the lengths of all edges are equal to a find the volumes of the parts from the pyramid after drawing a plane $\mu$, which intersects two of the edges passing through point $A$ respectively in the points $E$ and $F$ in such a way that $|AE|=p$ and $|AF|=q$ $(p<a,q<a)$, intersects the extension of the third edge behind opposite of the vertex $A$ wall in the point $G$ in such a way that $|AG|=t$ $(t>a)$.

Each of the two opposite sides of a convex quadrilateral is divided into seven equal parts, and corresponding division points are connected by a segment, thus dividing the quadrilateral into seven smaller quadrilaterals. Prove that the area of at least one of the small quadrilaterals equals $1\slash 7$ slash of the area of the large quadrilateral.

Let $ABC$ be a scalene triangle with incenter $I$ and circumcircle $\omega$. The internal and external bisectors of angle $\angle BAC$ intersect $BC$ at $D$ and $E$, respectively. Let $M$ be the point on segment $AC$ such that $MC = MB$. The tangent to $\omega$ at $B$ meets $MD$ at $S$. The circumcircles of triangles $ADE$ and $BIC$ intersect each other at $P$ and $Q$. If $AS$ meets $\omega$ at a point $K$ other than $A$, prove that $K$ lies on $PQ$. [i]Proposed by Alexandru Lopotenco (Moldova)[/i]

A point $P$ is given on the curve $x^4+y^4=1$. Find the maximum distance from the point $P$ to the origin.

In the triangle $ABC$ with circumcircle $\Gamma$, the incircle $\omega$ touches sides $BC, CA$, and $AB$ at $D, E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $K\neq D$. Line $AK$ meets $\Gamma$ at $L\neq A$. Rays $KI$ and $IL$ meet the circumcircle of triangle $BIC$ at $Q\neq I$ and $P\neq I$, respectively. The circumcircles of triangles $KFB$ and $KEC$ meet $EF$ at $R\neq F$ and $S\neq E$, respectively. Prove that $PQRS$ is cyclic. [i]India, Anant Mugdal[/i]

In an acute-angled triangle $ABC$, the circle $S$ with the altitude $BK$ as the diameter intersects $AB$ at $E$ and $BC$ at $F$. Prove that the tangents to $S$ at $E$ and $F$ meet on the median from $B$.

[b]p1.[/b] It takes $12$ months for Santa Claus to pack gifts. It would take $20$ months for his apprentice to do the job. If they work together, how long will it take for them to pack the gifts? [b]p2.[/b] All passengers on a bus sit in pairs. Exactly $2/5$ of all men sit with women, exactly $2/3$ of all women sit with men. What part of passengers are men? [b]p3.[/b] There are $100$ colored balls in a box. Every $10$-tuple of balls contains at least two balls of the same color. Show that there are at least $12$ balls of the same color in the box. [b]p4.[/b] There are $81$ wheels in storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that one can determine which wheel is incorrectly marked with four measurements. [b]p5.[/b] Remove from the figure below the specified number of matches so that there are exactly $5$ squares of equal size left: (a) $8$ matches (b) $4$ matches [img]https://cdn.artofproblemsolving.com/attachments/4/b/0c5a65f2d9b72fbea50df12e328c024a0c7884.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $O$ be the circumcenter and $G$ be the centroid of a triangle $ABC$. If $R$ and $r$ are the circumcenter and incenter of the triangle, respectively, prove that \[ OG \leq \sqrt{ R ( R - 2r ) } . \] [i]Greece[/i]

Consider an acute triangle $ABC$ in which $A_1, B_1,$ and $C_1$ are the feet of the altitudes from $A, B,$ and $C,$ respectively, and $H$ is the orthocenter. The perpendiculars from $H$ onto $A_1C_1$ and $A_1B_1$ intersect lines $AB$ and $AC$ at $P$ and $Q,$ respectively. Prove that the line perpendicular to $B_1C_1$ that passes through $A$ also contains the midpoint of the line segment $PQ$.

Prove that for positive integers $ x,y,z$ the number $ x^2 \plus{} y^2 \plus{} z^2$ is not divisible by $ 3(xy \plus{} yz \plus{} zx)$.

Let $ABC$ be a triangle with circumcircle $\omega$ . Point $D$ lies on the arc $BC$ of $\omega$ and is different than $B,C$ and the midpoint of arc $BC$. Tangent of $\Gamma$ at $D$ intersects lines $BC$, $CA$, $AB$ at $A',B',C'$, respectively. Lines $BB'$ and $CC'$ intersect at $E$. Line $AA'$ intersects the circle $\omega$ again at $F$. Prove that points $D,E,F$ are collinear. (Saudi Arabia)

Arbitrary triangle $ABC$ is given (with $AB<BC$). Let $M$ - midpoint of $AC$, $N$- midpoint of arc $AC$ of circumcircle $ABC$, which is contains point $B$. Let $I$ - in-center of $ABC$. Proved, that $ \angle IMA = \angle INB$

There are 8 members in a a bridge committee (committee for making bridges). Of these 8 members, 3 are chosen to be in special "approval" committee with 1 of 3 members being the "boss." In how many ways can this happen?

Let $P$ be a point in the interior of the acute-angled triangle $ABC$. Prove that if the reflections of $P$ with respect to the sides of the triangle lie on the circumcircle of the triangle, then $P$ is the orthocenter of $ABC$.

In an acute-angled triangle $ ABC,$ let $ AD,BE$ be altitudes and $ AP,BQ$ internal bisectors. Denote by $ I$ and $ O$ the incenter and the circumcentre of the triangle, respectively. Prove that the points $ D, E,$ and $ I$ are collinear if and only if the points $ P, Q,$ and $ O$ are collinear.

Determine the maximal area triangle such that all of its vertices satisfy $\frac{x^2}{9} + \frac{y^2}{16} = 1$.

Let $n \in \mathbb{N}^*$ and consider a circle of length $6n$ along with $3n$ points on the circle which divide it into $3n$ arcs: $n$ of them have length $1,$ some other $n$ have length $2$ and the remaining $n$ have length $3.$ Prove that among these points there must be two such that the line that connects them passes through the center of the circle.

Let $ABCD$ be a unit square. A semicircle with diameter $AB$ is drawn so that it lies outside of the square. If $E$ is the midpoint of arc $AB$ of the semicircle, what is the area of triangle $CDE$

The inscribed circle $\omega$ of the triangle $ABC$ touches its sides $BC, CA$, and $AB$ at the points $D, E$, and $F$, respectively. Let the points $X, Y$, and $Z$ of the circle $\omega$ be diametrically opposite to the points $D, E$, and $F$, respectively. Line $AX, BY$ and $CZ$ intersect the sides $BC, CA$ and $AB$ at the points $D', E'$ and $F'$, respectively. On the segments $AD', BE'$ and $CF'$ noted the points $X', Y'$ and $Z'$, respectively, so that $D'X'= AX$, $E'Y' = BY$, $F'Z' = CZ$. Prove that the points $X', Y'$ and $Z'$ coincide.

Give the answer about the following propositions $(p),\ (q)$ whether they are true or not. If the answer is true, then give the proof and if the answer is false, then give the proof by giving the counter example. $(p)$ If we can form a triangle such that one of inner angles of the triangle is $60^\circ$ by choosing 3 points from the vertices of a regular $n$-polygon, then $n$ is a multiple of 3. $(q)$ In $\triangle{ABC},\ \triangle{A'B'C'}$, if $AB=A'B',\ BC=B'C',\ \angle{A}=\angle{A'}$, then these triangles are congruent. 30 points

On the plane circles $k$ and $\ell$ are intersected at points $C$ and $D$, where circle $k$ passes through the center $L$ of circle $\ell$. The straight line passing through point $D$ intersects circles $k$ and $\ell$ for the second time at points $A$ and $B$ respectively in such a way that $D$ is the interior point of segment $AB$. Show that $AB = AC$.

A point $ (x,y)$ in the plane is called a lattice point if both $ x$ and $ y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to $ \textbf{(A)}\ 4.0\qquad \textbf{(B)}\ 4.2\qquad \textbf{(C)}\ 4.5\qquad \textbf{(D)}\ 5.0\qquad \textbf{(E)}\ 5.6$

(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A_c$, $ B_c$; $ C_1$ is the common point of $ AA_c$ and $ BB_c$. Points $ A_1$, $ B_1$ are defined similarly. Prove that circle $ A_1B_1C_1$ passes through the circumcenter of triangle $ ABC$.

A point $D$ lies inside triangle $ABC$. Let $A_1, B_1, C_1$ be the second intersection points of the lines $AD$, $BD$, and $CD$ with the circumcircles of $BDC$, $CDA$, and $ADB$, respectively. Prove that $$\frac{AD}{AA_1} + \frac{BD}{BA_1} + \frac{CD}{CC_1} = 1.$$

Construct a triangle $ABC$ given angle $A$ and the medians drawn from vertices $B$ and $C$.