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 $A_c \Delta B_c$.

Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles. [b]a.)[/b] What is the volume of this polyhedron ? [b]b.)[/b] Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?

Is $\sum_{n=1}^{+\infty}\frac{\cos n}{n}(1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}})$ convergent? why?

Let $\mathcal F = \{ f: [0,1] \to [0,\infty) \mid f$ continuous $\}$ and $n$ an integer, $n\geq 2$. Find the smallest real constant $c$ such that for any $f\in \mathcal F$ the following inequality takes place \[ \int^1_0 f \left( \sqrt [n] x \right) dx \leq c \int^1_0 f(x) dx. \]

In a handbook of plants each plant is characterized by $100$ attributes (each attribute may either be present in a plant or not). Two plants are called [i]dissimilar [/i] if they differ by no less than $51$ attributes. (a) Prove that the handbook cannot describe more than $50$ pair-wise dissimilar plants. (b) Can it describe $50$ pairwise dissimilar plants? (Dima Tereshin)

Patty has $ 20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $ 70$ cents more. How much are her coins worth? $ \textbf{(A)}\ \$1.15\qquad \textbf{(B)}\ \$1.20\qquad \textbf{(C)}\ \$1.25\qquad \textbf{(D)}\ \$1.30\qquad \textbf{(E)}\ \$1.35$

Let $ABCD$ be a quadrilateral with $\angle DAB=60^o$, $\angle ABC=60^o$ and $\angle BCD=120^o$. Diagonals $AC$, $BD$ intersect at point $M$ and $BM=a, MD=2a$. Let $O$ be the midpoint of side $AC$ and draw $OH \perp BD, H \in BD$ and $MN\perp OB, N \in OB$. Prove that i) $HM=MN=\frac{a}{2}$ ii) $AD=DC$ iii) $S_{ABCD}=\frac{9a^2}{2}$

Let $ G$ be a transitive subgroup of the symmetric group $ S_{25}$ different from $ S_{25}$ and $ A_{25}$. Prove that the order of $ G$ is not divisible by $ 23$. [i]J. Pelikan[/i]

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

Solve the equation for the positive integers: $$(x+2y)^2+2x+5y+9=(y+z)^2$$

Determine all complex numbers $\lambda$ for which there exists a positive integer $n$ and a real $n\times n$ matrix $A$ such that $A^2=A^T$ and $\lambda$ is an eigenvalue of $A$.

Let $a, b, c$ be non-negative real numbers. Prove that $$a\sqrt{3a^2+6b^2}+b\sqrt{3b^2+6c^2}+c\sqrt{3c^2+6a^2}=>(a+b+c)^2$$

In the plane are given a circle $k$ with radii $R$ and the points $A_1,A_2,\ldots,A_n$, lying on $k$ or outside $k$. Prove that there exist infinitely many points $X$ from the given circumference for which $$\sum_{i=1}^n A_iX^2\ge2nR^2.$$ Does there exist a pair of points on different sides of some diameter, $X$ and $Y$ from $k$, such that $$\sum_{i=1}^n A_iX^2\ge2nR^2\text{ and }\sum_{i=1}^n A_iY^2\ge2nR^2?$$ [i]H. Lesov[/i]

Determine whether there are infinitely many triples $(u,v,w)$ of positive integers such that $u,v,w$ form an arithmetic progression and the numbers $uv+1, vw+1$ and $wu+1$ are all perfect squares.

A number is [i]special[/i] if its tens digit is $9$. For example, $499$ and $1092$ are special, but $509$ is not. Diego has several cards. On each of them, he wrote a special number (he may write the same number on more than one card). When he adds up the numbers on the cards, the total is $2024$. What is the smallest number of cards Diego can have?

What is the smallest positive integer value of $x$ for which $x \equiv 4$ (mod $9$) and $x \equiv 7$ (mod $8$)?

Find all positive integers $n$ that can be represented in the form $$n=\mathrm{lcm}(a,b)+\mathrm{lcm}(b,c)+\mathrm{lcm}(c,a)$$ where $a,b,c$ are positive integers.

Solve the system of equations: $$\begin{cases} xy+zu=14 \\ xz+yu=11 \\ xu+yz=10 \\ x+y+z+u=10 \end{cases}$$

The Wonder Island is inhabited by Hedgehogs. Each Hedgehog consists of three segments of unit length having a common endpoint, with all three angles between them $120^{\circ}$. Given that all Hedgehogs are lying flat on the island and no two of them touch each other, prove that there is a finite number of Hedgehogs on Wonder Island.

Evaluate the sum $\frac{3!+4!}{2(1!+2!)}+\frac{4!+5!}{3(2!+3!)}+\cdots+\frac{12!+13!}{11(10!+11!)}$

A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$? [asy] import graph3; import solids; real h=2+2*sqrt(7); currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); currentlight=light(4,-4,4); draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0)); draw(shift((1,3,1))*unitsphere,gray(0.85)); draw(shift((3,3,1))*unitsphere,gray(0.85)); draw(shift((3,1,1))*unitsphere,gray(0.85)); draw(shift((1,1,1))*unitsphere,gray(0.85)); draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85)); draw(shift((1,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,1,h-1))*unitsphere,gray(0.85)); draw(shift((1,1,h-1))*unitsphere,gray(0.85)); draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); [/asy] $\textbf{(A) }2+2\sqrt 7\qquad \textbf{(B) }3+2\sqrt 5\qquad \textbf{(C) }4+2\sqrt 7\qquad \textbf{(D) }4\sqrt 5\qquad \textbf{(E) }4\sqrt 7\qquad$

A rectangular wooden block has a square top and bottom, its volume is $576$, and the surface area of its vertical sides is $384$. Find the sum of the lengths of all twelve of the edges of the block.

$ n\geq 2$ cars are participating in a rally. The cars leave the start line at different times and arrive at the finish line at different times. During the entire rally each car takes over any other car at most once , the number of cars taken over by each car is different and each car is taken over by the same number of cars. Find all possible values of $ n$

The lengths of two line segments are $ a$ units and $ b$ units respectively. Then the correct relation between them is: $ \textbf{(A)}\ \frac{a\plus{}b}{2} > \sqrt{ab} \qquad\textbf{(B)}\ \frac{a\plus{}b}{2} < \sqrt{ab} \qquad\textbf{(C)}\ \frac{a\plus{}b}{2} \equal{} \sqrt{ab}\\ \textbf{(D)}\ \frac{a\plus{}b}{2} \leq \sqrt{ab} \qquad\textbf{(E)}\ \frac{a\plus{}b}{2} \geq \sqrt{ab}$