Let $ABC$ be a triangle satisfying $BC < CA$. Let $P$ be an arbitrary point on the side $AB$ (different from $A$ and $B$), and let the line $CP$ meet the circumcircle of triangle $ABC$ at a point $S$ (apart from the point $C$). Let the circumcircle of triangle $ASP$ meet the line $CA$ at a point $R$ (apart from $A$), and let the circumcircle of triangle $BPS$ meet the line $CB$ at a point $Q$ (apart from $B$). Prove that the excircle of triangle $APR$ at the side $AP$ is identical with the excircle of triangle $PQB$ at the side $PQ$ if and only if the point $S$ is the midpoint of the arc $AB$ on the circumcircle of triangle $ABC$.

Let $D,E,F$ denote the tangent points of the incircle of $ABC$ with sides $BC,AC,AB$ respectively. Let $M$ be the midpoint of the segment $EF$. Let $L$ be the intersection point of the circle passing through $D,M,F$ and the segment $AB$, $K$ be the intersection point of the circle passing through $D,M,E$ and the segment $AC$. Prove that the circle passing through $A,K,L$ touches the line $BC$

A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs. Juniors) How many vertical sides can there be in this set? Seniors) How many ways are there to do that? [asy] size(120); defaultpen(linewidth(0.8)); path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle; for(int i=0;i<=3;i=i+1) { for(int j=0;j<=2;j=j+1) { real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2; draw(shift(shiftx,shifty)*hex); } } [/asy] [i](T. Doslic)[/i]

Prove that there is no finite set which contains more than $ 2N,$ with $ N > 3,$ pairwise non-collinear vectors of the plane, and to which the following two characteristics apply: 1) for $ N$ arbitrary vectors from this set there are always further $ N\minus{}1$ vectors from this set so that the sum of these is $ 2N\minus{}1$ vectors is equal to the zero-vector; 2) for $ N$ arbitrary vectors from this set there are always further $ N$ vectors from this set so that the sum of these is $ 2N$ vectors is equal to the zero-vector.

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$. [i]Proposed by Estonia[/i]

In an acute triangle $ABC$, an arbitrary point $P$ is chosen on the altitude $AH$. The points $E$ and $F$ are the midpoints of $AC$ and $AB$, respectively. The perpendiculars from $E$ on $CP$ and from $F$ on $BP$ intersect at the point $K$. Show that $KB = KC$.

Compute the radius of the largest circle that fits entirely within a unit cube.

Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DY AZ,DZBX,DXCY$ are circumscribable, so is the third.

The diagonals of a convex quadrilateral $ABCD$ intersect in the point $E$. Let $U$ be the circumcenter of the triangle $ABE$ and $H$ be its orthocenter. Similarly, let $V$ be the circumcenter of the triangle $CDE$ and $K$ be its orthocenter. Prove that $E$ lies on the line $UK$ if and only if it lies on the line $VH$.

Each of two regular polyhedrons $P$ and $Q$ was divided by the plane into two parts. One part of $P$ was attached to one part of $Q$ along the dividing plane and formed a regular polyhedron not equal to $P$ and $Q$. How many faces can it have?

(1) Prove that, on the complex plane, the area of the convex hull of all complex roots of $z^{20}+63z+22=0$ is greater than $\pi$. (2) Let $a_1,a_2,\ldots,a_n$ be complex numbers with sum $1$, and $k_1<k_2<\cdots<k_n$ be odd positive integers. Let $\omega$ be a complex number with norm at least $1$. Prove that the equation \[ a_1 z^{k_1}+a_2 z^{k_2}+\cdots+a_n z^{k_n}=w \] has at least one complex root with norm at most $3n|\omega|$.

Let $ABC$ be a triangle with circumcircle $c$ and circumcenter $O$, and let $D$ be a point on the side $BC$ different from the vertices and the midpoint of $BC$. Let $K$ be the point where the circumcircle $c_1$ of the triangle $BOD$ intersects $c$ for the second time and let $Z$ be the point where $c_1$ meets the line $AB$. Let $M$ be the point where the circumcircle $c_2$ of the triangle $COD$ intersects $c$ for the second time and let $E$ be the point where $c_2$ meets the line $AC$. Finally let $N$ be the point where the circumcircle $c_3$ of the triangle $AEZ$ meets $c$ again. Prove that the triangles $ABC$ and $NKM$ are congruent.

Let $H$ be the orthocenter of triangle $ABC$, $X$ be an arbitrary point. A circle with a diameter of $XH$ intersects lines $AH, BH, CH$ at points $A_1, B_1, C_1$ for the second time, and lines $AX BX, CX$ at points $A_2, B_2, C_2$. Prove that lines A$_1A_2, B_1B_2, C_1C_2$ intersect at one point.

Find the locus of points inside a trihedral angle such that the sum of their distances from the faces of the trihedral angle has a fixed positive value $a$.

The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $P$ and $Q$ respectively. $N$ and $M$ are the midpoints of $AC$ and $BC$ respectively. Let $X=AM\cap BP, Y=BN\cap AQ$. Given $C,X,Y$ are collinear, prove that $CX$ is the angle bisector of the angle $ACB$. I. Gorodnin

Let $ABC$ be a non-right isosceles triangle such that $AC = BC$. Let $D$ be such a point on the perpendicular bisector of $AB$, that $AD$ is tangent on the $ABC$ circumcircle. Let $E$ be such a point on $AB$, that $CE$ and $AD$ are perpendicular and let $F$ be the second intersection of line $AC$ and the circle $CDE$. Prove that $DF$ and $AB$ are parallel.

Consider the function $ f:\mathbb{R}\longrightarrow\mathbb{R} ,\quad f(x)=\max (x-3,2) . $ Find the perimeter and the area of the figure delimited by the lines $ x=-3,x=1, $ the $ Ox $ axis, and the graph of $ f. $

[b]p1.[/b] What is $20\times 19 + 20 \div (2 - 7)$? [b]p2.[/b] Will has three spinners. The first has three equally sized sections numbered $1$, $2$, $3$; the second has four equally sized sections numbered $1$, $2$, $3$, $4$; and the third has five equally sized sections numbered $1$, $2$, $3$, $4$, $5$. When Will spins all three spinners, the probability that the same number appears on all three spinners is $p$. Compute $\frac{1}{p}$. [b]p3.[/b] Three girls and five boys are seated randomly in a row of eight desks. Let $p$ be the probability that the students at the ends of the row are both boys. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m + n$. [b]p4.[/b] Jaron either hits a home run or strikes out every time he bats. Last week, his batting average was $.300$. (Jaron's batting average is the number of home runs he has hit divided by the number of times he has batted.) After hitting $10$ home runs and striking out zero times in the last week, Jaron has now raised his batting average to $.310$. How many home runs has Jaron now hit? [b]p5.[/b] Suppose that the sum $$\frac{1}{1 \cdot 4} +\frac{1}{4 \cdot 7}+ ...+\frac{1}{97 \cdot 100}$$ is expressible as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p6.[/b] Let $ABCD$ be a unit square with center $O$, and $\vartriangle OEF$ be an equilateral triangle with center $A$. Suppose that $M$ is the area of the region inside the square but outside the triangle and $N$ is the area of the region inside the triangle but outside the square, and let $x = |M -N|$ be the positive difference between $M$ and $N$. If $$x =\frac1 8(p -\sqrt{q})$$ for positive integers $p$ and $q$, find $p + q$. [b]p7.[/b] Find the number of seven-digit numbers such that the sum of any two consecutive digits is divisible by $3$. For example, the number $1212121$ satisfies this property. [b]p8.[/b] There is a unique positive integer $x$ such that $x^x$ has $703$ positive factors. What is $x$? [b]p9.[/b] Let $x$ be the number of digits in $2^{2019}$ and let $y$ be the number of digits in $5^{2019}$. Compute $x + y$. [b]p10.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 13$ and $BC = 10$. Consider the set of all points $D$ in three-dimensional space such that $BCD$ is an equilateral triangle. This set of points forms a circle $\omega$. Let $E$ and $F$ be points on $\omega$ such that $AE$ and $AF$ are tangent to $\omega$. If $EF^2$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, determine $m + n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Do there exist points $A, B, C, D$ in space, such that $AB = CD = 8, AC = BD = 10$, and $AD = BC = 13$?

Points $M,N,P,Q$ are taken on the sides $AB,BC,CD,DA$ respectively of a square $ABCD$ such that $AM=BN=CP=DQ=\frac1nAB$. Find the ratio of the area of the square determined by the lines $MN,NP,PQ,QM$ to the ratio of $ABCD$.

In an acute triangle $ABC$ with $AB<AC$, the foot of altitudes from $A,B,C$ to the sides $BC,CA,AB$ are $D,E,F$, respectively. $H$ is the orthocenter. $M$ is the midpoint of segment $BC$. Lines $MH$ and $EF$ intersect at $K$. Let the tangents drawn to circumcircle $(ABC)$ from $B$ and $C$ intersect at $T$. Prove that $T;D;K$ are colinear

In the triangle $ ABC $, the lines $ CP $, $ AP $, $ BP $ are drawn through the internal point $ P $ and intersect the sides $ AB $, $ BC $, $ CA $ at points $ K $, $ L $, $ M$, respectively. Prove that if circles can be inscribed in the quadrilaterals $ AKPM $ and $ KBLP $, then a circle can also be inscribed in the quadrilateral $ LCMP $.

Let $CH$ be the altitude of triangle ABC, $O$ be the center of the circle circumscribed around it. Point $T$ is the projection of point $C$ on the line $TO$. Prove that the line $TH$ bisects the side $BC$.

What is the shortest possible side length of a four-dimensional hypercube that contains a regular octahedron with side 1?

Given a parallelogram \(ABCD\). Let \(\ell_1\) be the line through \(D\), parallel to \(AC\), and \(\ell_2\) the external bisector of \(\angle ACD\). The lines \(\ell_1\) and \(\ell_2\) intersect at \(E\). The lines \(\ell_1\) and \(AB\) intersect at \(F\), and the line \(\ell_2\) intersects the internal bisector of \(\angle BAC\) at \(X\). The line \(BX\) intersects the circumcircle of triangle \(EFX\) at a second point \(Y\). The internal bisector of \(\angle ACD\) intersects the circumcircle of triangle \(ACX\) at a second point \(Z\). Prove that the quadrilateral \(DXYZ\) is inscribed in a circle.