The polynomial $R(x)$ is the remainder upon dividing $x^{2007}$ by $x^2-5x+6$. $R(0)$ can be expressed as $ab(a^c-b^c)$. Find $a+c-b$.

What is the largest number $N$ for which there exist $N$ consecutive positive integers such that the sum of the digits in the $k$-th integer is divisible by $k$ for $1 \le k \le N$ ? (S Tokarev)

Find all real numbers $x,y$ greater than $1$, satisfying the condition that the numbers $\sqrt{x-1}+\sqrt{y-1}$ and $\sqrt{x+1}+\sqrt{y+1}$ are nonconsecutive integers.

A sphere with ray $R$ is cut by two parallel planes. such that the center of the sphere is outside the region determined by these planes. Let $S_{1}$ and $S_{2}$ be the areas of the intersections, and $d$ the distance between these planes. Find the area of the intersection of the sphere with the plane parallel with these two planes, with equal distance from them.

Consider the expanded form of $\left(x+\frac{1}{2\sqrt[4]{x}}\right)^n$, put all items in number (from high power to low power). If the coefficients of the first three items are arithmetic sequence, then the number of items with an integral power is________.

Let $k>1$ be a real number. Calculate: (a) $L=\lim_{n\to \infty} \int_0^1\left( \frac{k}{\sqrt[n]{x}+k-1}\right)^n\text{d} x.$ (b) $\lim_{n\to \infty} n\left\lbrack L- \int_0^1\left( \frac{k}{\sqrt[n]{x}+k-1}\right)^n\text{d} x\right\rbrack.$

[b]a)[/b] Show that the number $ \sqrt{9-\sqrt{77}}\cdot\sqrt {2}\cdot\left(\sqrt{11}-\sqrt{7}\right)\cdot\left( 9+\sqrt{77}\right) $ is natural. [b]b)[/b] Consider two real numbers $ x,y $ such that $ xy=6 $ and $ x,y>2. $ Show that $ x+y<5. $

Let $\Gamma$ be a circle and $P$ a point outside of $\Gamma$ . A tangent from $P$ to the circle intersects it in $A$. Another line through $P$ intersects $\Gamma$ at the points $B$ and $C$. The bisector of $\angle APB$ intersects $AB$ at $D$ and $AC$ at $E$. Prove that the triangle $ADE$ is isosceles.

Let $n$ be a natural number. Prove that the number of all non-isosceles triangles with lengths of their sides equal to natural numbers and a perimeter $2n$ is $[\frac{n^2-6n+12}{12}]$.

Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$

Let $M$ be the set of natural numbers $k$ for which there exists a natural number $n$ such that $$3^n \equiv k\pmod n.$$ Prove that $M$ has infinitely many elements.

How many valence electrons are in the pyrophosphate ion, $\ce{P2O7}^{4-}?$ $ \textbf{(A) } 48\qquad\textbf{(B) } 52\qquad\textbf{(C) } 54\qquad\textbf{(D) } 56\qquad $

A sequence $ (x_n)$ is given by $ x_1\equal{}2$ and $ nx_n\equal{}2(2n\minus{}1)x_{n\minus{}1}$ for $ n>1$. Prove that $ x_n$ is an integer for every $ n \in \mathbb{N}$.

Let $x,y,z$ be real numbers such that, $x \geq y \geq z >0$. Prove that $$\frac{x^2-y^2}{z}+\frac{z^2-y^2}{x}+\frac{x^2-z^2}{y} \geq 3x-4y+z$$

Julia bakes a cake in the shape of a unit square. Each minute, Julia makes two cuts through the cake as follows: [list] [*] she picks a [b]square[/b] piece $\mathcal{S}$ of the cake with no cuts through its interior; then [*] she slices the entire cake along the two lines parallel to the sides of the cake passing through the center of $\mathcal{S}$. [/list] She does not move any pieces of cake during this process. After eight minutes, she has a grid of $9^2 = 81$ pieces of cake. (The pieces can be various sizes.) Compute the number of distinct grids that she could have ended up with. Two grids are the same if they have the same set of cuts; in particular, two grids that differ by a rotation or reflection are distinct. [i]Proposed by Sean Li[/i]

Find all integers $x,y$ such that $x^2(y-1)+y^2(x-1) = 1$.

Let \( P_1(x) \) and \( P_2(x) \) be monic quadratic trinomials, and let \( A_1 \) and \( A_2 \) be the vertices of the parabolas \( y = P_1(x) \) and \( y = P_2(x) \), respectively. Let \( m(g(x)) \) denote the minimum value of the function \( g(x) \). It is known that the differences \( m(P_1(P_2(x))) - m(P_1(x)) \) and \( m(P_2(P_1(x))) - m(P_2(x)) \) are equal positive numbers. Find the angle between the line \( A_1A_2 \) and the $x$-axis. \\

Find the maximum value of $n$, such that there exist $n$ pairwise distinct positive numbers $x_1,x_2,\cdots,x_n$, satisfy $$x_1^2+x_2^2+\cdots+x_n^2=2017$$

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

Given regular $45$-gon. Can you mark its corners with the digits $\{0,1,...,9\}$ in such a way, that for every pair of digits there would be a side with both ends marked with those digits?

Given a natural number $n\ge 3$, determine all strictly increasing sequences $a_1<a_2<\cdots<a_n$ such that $\text{gcd}(a_1,a_2)=1$ and for any pair of natural numbers $(k,m)$ satisfy $n\ge m\ge 3$, $m\ge k$, $$\frac{a_1+a_2+\cdots +a_m}{a_k}$$ is a positive integer.

Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable. [i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]

The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams? $\textbf{(A) }60\qquad\textbf{(B) }88\qquad\textbf{(C) }96\qquad\textbf{(D) }144\qquad \textbf{(E) }160$

The numbers $1$ through $9$ are written on a $3 \times 3$ board, without repetitions. A valid operation is to choose a row or a column of the board, and replace its three numbers $a, b, c$ (in order, i.e., the first number of the row/column is $a$, the second number of the row/column is $b$, the third number of the row/column is $c$) with either the three non-negative numbers $a-x, b-x, c+x$ (in order) or with the three non-negative numbers $a+x, b-x, c-x$ (in order), where $x$ is a real positive number which may vary in each operation . a) Determine if there is a way of getting all $9$ numbers on the board to be the same, starting with the following board: $\begin{array}{|c|c|c|c|c|c|c|c|} \hline 1 & 2 & 3 \\ \hline 4 & 5 & 6 \\ \hline 7 & 8 & 9 \\ \hline \end{array}$ b) For all posible configurations such that it is possible to get all $9$ numbers to be equal to a number $m$ using the valid operations, determine the maximum value of $m$.

In a group of $m$ girls and $n$ boys, any two persons either know each other or do not know each other. For any two boys and any two girls, there are at least one boy and one girl among them,who do not know each other. Prove that the number of unordered pairs of (boy, girl) who know each other does not exceed $m+\frac{n(n-1)}{2}$.