Let $P(x)=x^2+4x+1$. What is the product of all real solutions to the equation $P(P(x))=0$?

Triangle $ABC$ has right angle $A$, with $AB=AC=6$. Point $P$ is located within triangle $ABC$ such that the areas of triangles $ABP$, $BCP$, and $ACP$ are equal. Express $CP$ in simplest radical form.

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

In the acute-angled triangle $ABC$ is drawn the altitude $CH$. A ray beginning at point $C$ that lies inside the $\angle BCA$ and intersects for second time the circles circumscribed circles of $\vartriangle BCH$ and $\vartriangle ABC$ at points $X$ and $Y$ respectively. It turned out that $2CX = CY$. Prove that the line $HX$ bisects the segment $AC$. (Hilko Danilo)

[b]p1.[/b] Solve the system of equations $$xy = 2x + 3y$$ $$yz = 2y + 3z$$ $$zx =2z+3x$$ [b]p2.[/b] For any integer $k$ greater than $1$ and any positive integer $n$ , prove that $n^k$ is the sum of $n$ consecutive odd integers. [b]p3.[/b] Determine all pairs of real numbers, $x_1$, $x_2$ with $|x_1|\le 1$ and $|x_2|\le 1$ which satisfy the inequality: $|x^2-1|\le |x-x_1||x-x_2|$ for all $x$ such that $|x| \ge 1$. [b]p4.[/b] Find the smallest positive integer having exactly $100$ different positive divisors. (The number $1$ counts as a divisor). [b]p5.[/b] $ABC$ is an equilateral triangle of side $3$ inches. $DB = AE = 1$ in. and $F$ is the point of intersection of segments $\overline{CD}$ and $\overline{BE}$ . Prove that $\overline{AF} \perp \overline{CD}$. [img]https://cdn.artofproblemsolving.com/attachments/f/a/568732d418f2b1aa8a4e8f53366df9fbc74bdb.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

3 countries $A, B, C$ participate in a competition where each country has 9 representatives. The rules are as follows: every round of competition is between 1 competitor each from 2 countries. The winner plays in the next round, while the loser is knocked out. The remaining country will then send a representative to take on the winner of the previous round. The competition begins with $A$ and $B$ sending a competitor each. If all competitors from one country have been knocked out, the competition continues between the remaining 2 countries until another country is knocked out. The remaining team is the champion. [b]I.[/b] At least how many games does the champion team win? [b]II.[/b] If the champion team won 11 matches, at least how many matches were played?

We have finite number of distinct shapes in plane. A "[i]convex Kearting[/i]" of these shapes is covering plane with convex sets, that each set consists exactly one of the shapes, and sets intersect at most in border. [img]http://aycu30.webshots.com/image/4109/2003791140004582959_th.jpg[/img] In which case Convex kearting is possible? 1) Finite distinct points 2) Finite distinct segments 3) Finite distinct circles

Plane $A$ passes through the points $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Plane $B$ is parallel to plane $A$, but passes through the point $(1, 0, 1)$. Find the distance between planes $A$ and $B$.

Consider a cake in the shape of a circle. It's been divided to some inequal parts by its radii. Arash and Bahram want to eat this cake. At the very first, Arash takes one of the parts. In the next steps, they consecutively pick up a piece adjacent to another piece formerly removed. Suppose that the cake has been divided to 5 parts. Prove that Arash can choose his pieces in such a way at least half of the cake is his.

Let $\phi = \frac{1+\sqrt5}{2}$. Prove that a positive integer appears in the list $$\lfloor \phi \rfloor , \lfloor 2 \phi \rfloor, \lfloor 3\phi \rfloor ,... , \lfloor n\phi \rfloor , ... $$ if and only if it appears exactly twice in the list $$\lfloor 1/ \phi \rfloor , \lfloor 2/ \phi \rfloor, \lfloor 3/\phi \rfloor , ... ,\lfloor n/\phi \rfloor , ... $$

Determine all positive integers $n>2$, such that $n = a^3 + b^3$, where $a$ is the smallest positive divisor of $n$ greater than $1$ and $b$ is an arbitrary positive divisor of $n$.

On sides $ AB$ and $ AC$ of triangle $ ABC$ there are given points $ D,E$ such that $ DE$ is tangent of circle inscribed in triangle $ ABC$ and $ DE \parallel BC$. Prove $ AB\plus{}BC\plus{}CA\geq 8DE$

Find all positive integers $(m,n)$ such that $$11^n+2^n+6=m^3$$

Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.

Three rays $h_1,h_2,h_3$ emanating from a point $O$ are given, not all in the same plane. Show that if for any three points $A_1,A_2,A_3$ on $h_1,h_2,h_3$ respectively, distinct from $O$, the triangle $A_1A_2A_3$ is acute-angled, then the rays $h_1,h_2,h_3$ are pairwise orthogonal.

Let $ABCD$ be a quadrilateral inscribed in a circle. Show that the centroids of triangles $ABC,$ $CDA,$ $BCD,$ $DAB$ lie on one circle.

A finite set $V \in Z^2$ of vectors with integer coordinates is chosen on the plane. Each of them is painted one of the $n$ colors. The color is [i]suitable[/i] for the vector if this vector may be presented as' a linear combination (with integer coefficients) of the vectors from $V$ of this color. It is known,that for any vector from $Z^2$ there exist a suitable color. Find all $n$ such that there must exist a color which is suitable for any vector from $Z^2$ . (V. Lebed)

Find the largest possible number of rooks that can be placed on a $3n \times 3n$ chessboard so that each rook is attacked by at most one rook.

The intersection of $2$ cubes of side length $5$ is a cube of side length $3$. Compute the surface area of the entire figure.

Leaving his house at noon, Jim walks at a constant rate of $4$ miles per hour along a $4$ mile square route returning to his house at $1$ PM. At a randomly chosen time between noon and $1$ PM, Sally chooses a random location along Jim's route and begins running at a constant rate of $7$ miles per hour along Jim's route in the same direction that Jim is walking until she completes one $4$ mile circuit of the square route. The probability that Sally runs past Jim while he is walking is given by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $BD$ and $CE$ be the bisectors of the interior angles $\angle B$ and $\angle C$, respectively ($D\in AC$, $E\in AB$). Consider the circumcircle of $ABC$ with center $O$ and the excircle corresponding to the side $BC$ with center $I_a$. These two circles intersect at points $P$ and $Q$. (a) Prove that $PQ$ is parallel to $DE$. (b) Prove that $I_aO$ is perpendicular to $DE$.

Let $n$ be a positive integer. For a positive integer $m$, we partition the set $\{1, 2, 3,...,m\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$, find the largest positive integer $m$ for which such a partition exists.

Show that the number \[p_n=\left(\frac{3+\sqrt5}{2}\right)^n+\left(\frac{3-\sqrt5}{2}\right)^n-2\] is a positive integer for any positive integer $n.$ Furthermore, show that the numbers $p_{2n-1}$ and $p_{2n}/5$ are perfect squares $($for any positive integer $n).$

[b]a)[/b] Prove that for any natural number $ n\ge 1, $ there is a set $ \mathcal{M} $ of $ n $ points from the Cartesian plane such that the barycenter of every subset of $ \mathcal{M} $ has integral coordinates (both coordinates are integer numbers). [b]b)[/b] Show that if a set $ \mathcal{N} $ formed by an infinite number of points from the Cartesian plane is given such that no three of them are collinear, then there exists a finite subset of $ \mathcal{N} , $ the barycenter of which has non-integral coordinates.

Prove that if the roots of the equation $ x^3 + ax^2 + bx + c = 0 $, with real coefficients, are real, then the roots of the equation $ 3x^2 + 2ax + b = 0 $ are also real.