Let $P_c(x)=x^4+ax^3+bx^2+cx+1$ and $Q_c(x)=x^4+cx^3+bx^2+ax+1$ with $a,b$ real numbers, $c \in \{1,2, \dots, 2017\}$ an integer and $a \ne c$. Define $A_c=\{\alpha | P_c(\alpha)=0\}$ and $B_c=\{\beta | P(\beta)=0\}$. (a) Find the number of unordered pairs of polynomials $P_c(x), Q_c(x)$ with exactly two common roots. (b) For any $1 \le c \le 2017$, find the sum of the elements of $A_c \Delta B_c$.

Let $ABC$ be a triangle. Suppose that $D$, $E$, and $F$ are points on segments $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively such that triangles $AEF$, $BFD$, and $CDE$ have equal inradii. Prove that the sum of the inradii of $\triangle AEF$ and $\triangle DEF$ is equal to the inradius of $\triangle ABC$. [i]Aprameya Tripathy[/i]

In an acute-angled triangle $ABC$, $CF$ is an altitude, with $F$ on $AB$, and $BM$ is a median, with $M$ on $CA$. Given that $BM=CF$ and $\angle MBC=\angle FCA$, prove that triangle $ABC$ is equilateral.

A certain island has some ports along the coast and some towns inland. All roads on this island are one-way, and they do not meet except at a port or a town. Moreover, once you leave a certain port or town by road, there is no way you can return there by road. For any two ports $i$ and $j$, let $f_{ij}$ denote the number of different routes along the roads between $i$ and $j$. (a) Suppose there are four ports on the island: $1, 2, 3$ and $4$, in clockwise order. Show that $$f_{14}f_{23} \ge f_{13}f_{24}$$ (b) Suppose there were six ports on the island: $1, 2, 3, 4, 5$ and $6$, in clockwise order. Show that $$f_{16}f_{25}f_{34} + f_{15}f_{24}f_{36} + f_{14}f_{26}f_{35}\ge f_{16}f_{24}f_{35}+ f_{15}f_{26}f_{34} + f_{14}f_{25}f_{36}$$ (S Fomin}

Prove that for all natural $n$ there exists a polynomial $P(x)$ divisible by $(x-1)^n$ such that its degree is not greater than $2^n$ and each of its coefficients is equal to $1$, $0$ or $-1$. (D. Fomin, Leningrad)

[b]p1.[/b] One hundred hobbits sit in a circle. The hobbits realize that whenever a hobbit and his two neighbors add up their total rubles, the sum is always $2007$. Prove that each hobbit has $669$ rubles. [b]p2.[/b] There was a young lady named Chris, Who, when asked her age, answered this: "Two thirds of its square Is a cube, I declare." Now what was the age of the miss? (a) Find the smallest possible age for Chris. You must justify your answer. (Note: ages are positive integers; "cube" means the cube of a positive integer.) (b) Find the second smallest possible age for Chris. You must justify your answer. (Ignore the word "young.") [b]p3.[/b] Show that $$\sum_{n=1}^{2007}\frac{1}{n^3+3n^2+2n}<\frac14$$ [b]p4.[/b] (a) Show that a triangle $ABC$ is isosceles if and only if there are two distinct points $P_1$ and $P_2$ on side $BC$ such that the sum of the distances from $P_1$ to the sides $AB$ and $AC$ equals the sum of the distances from $P_2$ to the sides $AB$ and $AC$. (b) A convex quadrilateral is such that the sum of the distances of any interior point to its four sides is constant. Prove that the quadrilateral is a parallelogram. (Note: "distance to a side" means the shortest distance to the line obtained by extending the side.) [b]p5.[/b] Each point in the plane is colored either red or green. Let $ABC$ be a fixed triangle. Prove that there is a triangle $DEF$ in the plane such that $DEF$ is similar to $ABC$ and the vertices of $DEF$ all have the same color. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The numbers $0, 1, 2, . . . , 9$ are written (in some order) on the circumference. Prove that a) there are three consecutive numbers with the sum being at least $15$, b) it is not necessarily the case that there exist three consecutive numbers with the sum more than $15$.

$A$ and $B$ are two opposite vertices of an $n \times n$ board. Within each small square of the board, the diagonal parallel to $AB$ is drawn, so that the board is divided in $2n^{2}$ equal triangles. A coin moves from $A$ to $B$ along the grid, and for every segment of the grid that it visits, a seed is put in each triangle that contains the segment as a side. The path followed by the coin is such that no segment is visited more than once, and after the coins arrives at $B$, there are exactly two seeds in each of the $2n^{2}$ triangles of the board. Determine all the values of $n$ for which such scenario is possible.

Let $ABC$ be an acute-angled triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_a$ passes in the points $A$ and $H$ and is tangent to the circumcircle of $ABC$. Define $X_b, X_c$ analogously, let $O_a, O_b, O_c$ the symmetric of $O$ to the sides $BC, AC$ and $AB$, respectively. Prove that the lines $O_aX_a, O_bX_b, O_cX_c$ are concurrents.

Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be a monic polynomial of degree $4$. It is known that all the roots of $P$ are real, distinct and belong to the interval $[-1, 1]$. (a) Prove that $P(x) > -4$ for all real $x$. (b) Find the highest value of the real constant $k$ such that $P(x) > k$ for every real $x$ and for every polynomial $P(x)$ satisfying the given conditions.

Lev scores $91, 89, 88, 94, 87, 85$ on his first $6$ tests. After having his final exam, he (correctly) states that the average of all $7$ of his test scores is equal to his final exam score. What was Lev’s final exam score?

Consider the string of length $ 6$ composed of three characters $ a$, $ b$, $ c$. For each string, if two $ a$s are next to each other, or two $ b$s are next to each other, then replace $ aa$ by $ b$, and replace $ bb$ by $ a$. Also, if $ a$ and $ b$ are next to each other, or two $ c$s are next to each other, remove all two of them (i.e. delete $ ab$, $ ba$, $ cc$). Determine the number of strings that can be reduced to $ c$, the string of length 1, by the reducing processes mentioned above.

Show that there are no triples of positive integers $(x,y,z)$ satisfying the equation \[x^2= 5^y+3^z\]

In a $2\times n$ array we have positive reals s.t. the sum of the numbers in each of the $n$ columns is $1$. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most $\frac{n+1}4$.

Let $n\ge 1$ be a fixed integer. Calculate the distance $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$ , where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x)= \sum_{k=n}^{\infty} c_k x^k$ defined on the closed interval $[0,1]$ , where $c_k \ge 0$ and $\sum_{k=n}^{\infty} c_k=1$.

Let $a,b,c$ denote the sides of a triangle and $[ABC]$ the area of the triangle as usual. $(a)$ If $6[ABC] = 2a^2+bc$, determine $A,B,C$. $(b)$ For all triangles, prove that $3a^2+3b^2 - c^2 \ge 4 \sqrt{3} [ABC]$.

Eight problems were given to each of $30$ students. After the test was given, point values of the problems were determined as follows: a problem is worth $n$ points if it is not solved by exactly $n$ contestants (no partial credit is given, only zero marks or full marks). (a) Is it possible that the contestant having got more points that any other contestant had also solved less problems than any other contestant? (b) Is it possible that the contestant having got less points than any other contestant has solved more problems than any other contestant?

Let $p,q$ be positive reals with sum 1. Show that for any $n$-tuple of reals $(y_1,y_2,...,y_n)$, there exists an $n$-tuple of reals $(x_1,x_2,...,x_n)$ satisfying $$p\cdot \max\{x_i,x_{i+1}\} + q\cdot \min\{x_i,x_{i+1}\} = y_i$$ for all $i=1,2,...,2017$, where $x_{2018}=x_1$.

Let $BC$ be a chord not passing through the center of a circle $\omega$. Point $A$ varies on the major arc $BC$. Let $E$ and $F$ be the projection of $B$ onto $AC$, and of $C$ onto $AB$ respectively. The tangents to the circumcircle of $\vartriangle AEF$ at $E, F$ intersect at $P$. (a) Prove that $P$ is independent of the choice of $A$. (b) Let $H$ be the orthocenter of $\vartriangle ABC$, and let $T$ be the intersection of $EF$ and $BC$. Prove that $TH \perp AP$.

Positive integers $ a,b,$ and $ c$ are randomly and independently selected with replacement from the set $ \{ 1,2,3,\dots,2010 \}.$ What is the probability that $ abc \plus{} ab \plus{} a$ is divisible by $ 3$? $ \textbf{(A)}\ \dfrac{1}{3} \qquad\textbf{(B)}\ \dfrac{29}{81} \qquad\textbf{(C)}\ \dfrac{31}{81} \qquad\textbf{(D)}\ \dfrac{11}{27} \qquad\textbf{(E)}\ \dfrac{13}{27}$

$11$ girls and $n$ boys went for mushrooms. They have found $n^2+9n -2$ in total, and each child has found the same quantity. Which is greater: the number of girls or the number of boys? (A. Tolpygo, Kiev)

Prove that any real number $0<x<1$ can be written as a difference of two positive and less than $1$ irrational numbers.

Prove that diagonals of a convex quadrilateral are perpendicular if and only if inside of the quadrilateral there is a point, whose orthogonal projections on sides of the quadrilateral are vertices of a rectangle.

The numbers $a, b, c$ satisfy the condition $| a - b | = 2 | b - c | = 3 | c - a |$. Prove that $a = b = c$.

Find all non-constant polynomials $P(x)$ with integer coefficients and positive leading coefficient, such that $P^{2mn}(m^2)+n^2$ is a perfect square for all positive integers $m, n$.