Let $a,b,c,d,e,f$ be real numbers such that $a^2+b^2+c^2=14, d^2+e^2+f^2=77,$ and $ad+be+cf=32.$ Find $(bf-ce)^2+(cd-af)^2+(ae-bd)^2.$

Let $m$ and $n$ be positive integers such that $m>n$ and $m \equiv n \pmod{2}$. If $(m^2-n^2+1) \mid n^2-1$, then prove that $m^2-n^2+1$ is a perfect square. (proposed by G. Batzaya, folklore)

Find all functions $f: \mathbb{N} \rightarrow [0, +\infty)$ such that $f(1000)=10$ and $f(n+1)= \sum_{k=1}^n \frac{1}{f^2(k) + f(k)f(k+1) + f^2(k+1)}$ for all $n \in \mathbb{N}$. (Here, $f^2(i)$ means $(f(i))^2$.)

Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$. [i]Evan O' Dorney.[/i]

Given the polynomial \[ f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n \] with integer coefficients $a_1,a_2,\ldots,a_n$, and given also that there exist four distinct integers $a$, $b$, $c$ and $d$ such that \[ f(a)=f(b)=f(c)=f(d)=5, \] show that there is no integer $k$ such that $f(k)=8$.

Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.

Find all positive integers $a_1, a_2, \ldots, a_n$ such that \[ \frac{99}{100} = \frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots + \frac{a_{n-1}}{a_n}, \] where $a_0 = 1$ and $(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1)$ for $k = 1,2,\ldots,n-1$.

Assume $n$ is a positive integer and integer $a$ is the root of the equation $$x^4+3ax^2+2ax-2\times 3^n=0.$$ Find all $n$ and $ a$ that satisfy the conditions.

Functions $ f$ and $ g$ are quadratic, $ g(x) \equal{} \minus{} f(100 \minus{} x)$, and the graph of $ g$ contains the vertex of the graph of $ f$. The four $ x$-intercepts on the two graphs have $ x$-coordinates $ x_1$, $ x_2$, $ x_3$, and $ x_4$, in increasing order, and $ x_3 \minus{} x_2 \equal{} 150$. The value of $ x_4 \minus{} x_1$ is $ m \plus{} n\sqrt p$, where $ m$, $ n$, and $ p$ are positive integers, and $ p$ is not divisible by the square of any prime. What is $ m \plus{} n \plus{} p$? $ \textbf{(A)}\ 602\qquad \textbf{(B)}\ 652\qquad \textbf{(C)}\ 702\qquad \textbf{(D)}\ 752\qquad \textbf{(E)}\ 802$

[b]p19.[/b] Sequences $a_n$ and $b_n$ of positive integers satisfy the following properties: (1) $a_1 = b_1 = 1$ (2) $a_5 = 6, b_5 \ge 7$ (3) Both sequences are strictly increasing (4) In each sequence, the difference between consecutive terms is either $1$ or $2$ (5) $\sum^5_{n=1}na_n =\sum^5_{n=1}nb_n = S$ Compute $S$. [b]p20.[/b] Let $A$, $B$, and $C$ be points lying on a line in that order such that $AB = 4$ and $BC = 2$. Let $I$ be the circle centered at B passing through $C$, and let $D$ and $E$ be distinct points on $I$ such that $AD$ and $AE$ are tangent to $I$. Let $J$ be the circle centered at $C$ passing through $D$, and let $F$ and $G$ be distinct points on $J$ such that $AF$ and $AG$ are tangent to $J$ and $DG < DF$. Compute the area of quadrilateral $DEFG$. [b]p21.[/b] Twain is walking randomly on a number line. They start at $0$, and flip a fair coin $10$ times. Every time the coin lands heads, they increase their position by 1, and every time the coin lands tails, they decrease their position by $1$. What is the probability that at some point the absolute value of their position is at least $3$? PS. You should use hide for answers.

Let $f(x)$ be a real-valued function defined on the positive reals such that (1) if $x < y$, then $f(x) < f(y)$, (2) $f\left(2xy\over x+y\right) \geq {f(x) + f(y)\over2}$ for all $x$. Show that $f(x) < 0$ for some value of $x$.

Does there exist a second-degree polynomial $p(x, y)$ in two variables such that every non-negative integer $ n $ equals $p(k,m)$ for one and only one ordered pair $(k,m)$ of non-negative integers? [i]Proposed by Finland.[/i]

Determine the smallest positive real number $\alpha$ such that there exists a sequence of positive real numbers $(a_n)$, $n \in \mathbb{N}$, with the property that for every $n \in \mathbb{N}$ it holds that: \[ a_1 + \cdots + a_{n+1} < \alpha \cdot a_n. \] [i]Proposed by Pavle Martinović[/i]

For each positive integer $n\ge2$, find a polynomial $P_n(x)$ with rational coefficients such that $\displaystyle P_n(\sqrt[n]2)=\frac1{1+\sqrt[n]2}$. (Note that $\sqrt[n]2$ denotes the positive $n^\text{th}$ root of $2$.)

Let $n$ be a positive integer such that there exist positive integers $x_1,x_2,\cdots ,x_n$ satisfying $$x_1x_2\cdots x_n(x_1 + x_2 + \cdots + x_n)=100n.$$ Find the greatest possible value of $n$.

Let $a, b, c$ be positive real numbers satisfying the condition $\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}=1$ . Prove the inequality $$\frac{a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}} \le \frac{3\sqrt3}{2}$$ When does the equality hold? (L. Parts)

Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.

Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$ \[ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}. \] Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.

Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? [i](IMO Problem 4)[/i] [i]Proposed by Vietnam.[/i]

Mark writes the expression $\sqrt{\underline{abcd}}$ on the board, where $\underline{abcd}$ is a four-digit number and $a \neq 0.$ Derek, a toddler, decides to move the $a,$ changing Mark's expression to $a\sqrt{\underline{bcd}}.$ Surprisingly, these two expressions are equal. Compute the only possible four-digit number $\underline{abcd}.$

For a finite set $A$ of integers, define $s(A)$ as the number of values obtained by adding any two elements of $A$, not necessarily different. Analogously, define $r (A)$ as the number of values obtained by subtracting any two elements of $A$, not necessarily different. For example, if $A = \{3,1,-1\}$ $\bullet$ The values obtained by adding any two elements of $A$ are $\{6,4,2,0,-2\}$ and so $s (A) = 5$. $\bullet$ The values obtained by subtracting any two elements of $A$ are $\{4,2,0,-2,-4\}$ and as $r (A) = 5$. Prove that for each positive integer $n$ there is a finite set $A$ of integers such that $r (A) \ge n s (A)$.

On a stormy night ten guests came to dinner party and left their shoes outside the room in order to keep the carpet clean. After the dinner there was a blackout, and the gusts leaving one by one, put on at random, any pair of shoes big enough for their feet. (Each pair of shoes stays together). Any guest who could not find a pair big enough spent the night there. What is the largest number of guests who might have had to spend the night there?

Suppose $a,b,c$ are three rela numbers such that the quadratic equation \[ x^2 - (a +b +c )x + (ab +bc +ca) = 0 \] has roots of the form $\alpha + i \beta$ where $\alpha > 0$ and $\beta \not= 0$ are real numbers. Show that (i) The numbers $a,b,c$ are all positive. (ii) The numbers $\sqrt{a}, \sqrt{b} , \sqrt{c}$ form the sides of a triangle.

On the coordinate plane, at some points with integer coordinates, there is a pebble (a finite number of pebbles). It is allowed to make the following move: select a pair of pebbles, take some vector $\vec{a}$ with integer coordinates and then move one of the selected pebbles to vector $\vec{a}$, and the other to the opposite vector $-\vec{a}$; it is forbidden that there should be more than one pebble at one point. Is it always possible to achieve a situation in which all the pebbles lie on the same straight line in a few moves? [i] K. Ivanov [/i]

Given positive real numbers $a, b, c, d$ that satisfy equalities \[a^2 + d^2 - ad = b^2 + c^2 + bc \ \ \text{and} \ \ a^2 + b^2 = c^2 + d^2\] find all possible values of the expression $\frac{ab+cd}{ad+bc}.$