A polynomial $f$ with integer coefficients is written on the blackboard. The teacher is a mathematician who has $3$ kids: Andrew, Beth and Charles. Andrew, who is $7$, is the youngest, and Charles is the oldest. When evaluating the polynomial on his kids' ages he obtains: [list]$f(7) = 77$ $f(b) = 85$, where $b$ is Beth's age, $f(c) = 0$, where $c$ is Charles' age.[/list] How old is each child?

Let $a_1$ , $\ldots$ , $a_n$ be positive integers and $a$ a positive integer that is greater than $1$ and is divisible by the product $a_1a_2\ldots a_n$. Prove that $a^{n+1}+a-1$ is not divisible by the product $(a+a_1-1)(a+a_2-1)\ldots(a+a_n-1)$.

Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ? [i]Proposed by Hungary.[/i] [hide="Remark"]I'm not sure I'm posting this in a right Forum.[/hide]

Suppose $a_1, a_2, ..., a_r$ are integers with $a_i \geq 2$ for all $i$ such that $a_1 + a_2 + ... + a_r = 2010$. Prove that the set $\{1,2,3,...,2010\}$ can be partitioned in $r$ subsets $A_1, A_2, ..., A_r$ each with $a_1, a_2, ..., a_r$ elements respectively, such that the sum of the numbers on each subset is divisible by $2011$. Decide whether this property still holds if we replace $2010$ by $2011$ and $2011$ by $2012$ (that is, if the set to be partitioned is $\{1,2,3,...,2011\}$).

The sequence of integers an is given by $a_0 = 0, a_n = p(a_n-1)$, where $p(x)$ is a polynomial whose coefficients are all positive integers. Show that for any two positive integers $m, k$ with greatest common divisor $d$, the greatest common divisor of $a_m$ and $a_k$ is $a_d$.

Suppose that $F,G,H$ are polynomials of degree at most $2n+1$ with real coefficients such that: i) For all real $x$ we have $F(x)\le G(x)\le H(x)$. ii) There exist distinct real numbers $x_1,x_2,\ldots ,x_n$ such that $F(x_i)=H(x_i)\quad\text{for}\ i=1,2,3,\ldots ,n$. iii) There exists a real number $x_0$ different from $x_1,x_2,\ldots ,x_n$ such that $F(x_0)+H(x_0)=2G(x_0)$. Prove that $F(x)+H(x)=2G(x)$ for all real numbers $x$.

Let $f(x)=\sum_{i=0}^{n}a_ix^i$ and $g(x)=\sum_{i=0}^{n}b_ix^i$, where $a_n$,$b_n$ can be zero. Called $f(x)\ge g(x)$ if exist $r$ such that $\forall i>r,a_i=b_i,a_r>b_r$ or $f(x)=g(x)$. Prove that: if the leading coefficients of $f$ and $g$ are positive, then $f(f(x))+g(g(x))\ge f(g(x))+g(f(x))$

Show that all complex roots of the polynomial $P(z)=a_0z^n+a_1z^{n-1}+\ldots+a_{n-1}z+a_n$, where $0<a_0<\ldots<a_n$, satisfy $|z|>1$.

Suppose the polynomial $P(x) \equiv x^3 + ax^2 + bx +c$ has only real zeroes and let $Q(x) \equiv 5x^2 - 16x + 2004$. Assume that $P(Q(x)) = 0$ has no real roots. Prove that $P(2004) > 2004$

Denote $f_n(X) \in \Bbb Z [X]$ the polynomial $\Pi_{j=1}^n ( X + j -1)$. Show that if the numbers $\alpha$ and $\beta$ satisfy $f'_{1997} (\alpha) = f'_{1999} (\beta) = 0$ , then $f_{1997} (\alpha ) \neq f_{1999} (\beta)$ .

The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$. A polynomial $P$ of degree $2007$ has the property that $P\left(r_j+\dfrac{1}{r_j}\right)=0$ for $j=1,\ldots,2007$. Determine the value of $P(1)/P(-1)$.

Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.

Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

Let $P(x)$ be a nonconstant polynomial of degree $n$ with rational coefficients which can not be presented as a product of two nonconstant polynomials with rational coefficients. Prove that the number of polynomials $Q(x)$ of degree less than $n$ with rational coefficients such that $P(x)$ divides $P(Q(x))$ a) is finite b) does not exceed $n$.

Determine all pairs of real numbers $ a, b $ for which the polynomials $ x^4 + 2ax^2 + 4bx + a^2 $ and $ x^3 + ax - b $ have two different common real roots.

Let $P(x)$ be a polynomial with the coefficients being $0$ or $1$ and degree $2023$. If $P(0)=1$, then prove that every real root of this polynomial is less than $\frac{1-\sqrt{5}}{2}$.

Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which \[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\] is a rational number.

The set of polynomials $f_1, f_2, \ldots, f_n$ with real coefficients is called [i]special [/i], if for any different $i,j,k \in \{ 1,2, \ldots, n\}$ polynomial $\dfrac{2}{3}f_i + f_j + f_k$ has no real roots, but for any different $p,q,r,s \in \{ 1,2, \ldots, n\}$ of a polynomial $f_p + f_q + f_r + f_s$ there is a real root. a) Give an example of a [i]special [/i] set of four polynomials whose sum is not a zero polynomial. b) Is there a [i]special [/i] set of five polynomials?

Consider the sequence of polynomials $P_0(x) = 2$, $P_1(x) = x$ and $P_n(x) = xP_{n-1}(x) - P_{n-2}(x)$ for $n \geq 2$. Let $x_n$ be the greatest zero of $P_n$ in the the interval $|x| \leq 2$. Show that $$\lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac{\pi^3}{12}$$

Consider a sequence of polynomials $P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots$, where $P_0(x) = 2, P_1(x) = x$ and for every $n \geq 1$ the following equality holds: \[P_{n+1}(x) + P_{n-1}(x) = xP_n(x).\] Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1,$ \[(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.\]

Two polynomials $ f(x)=a_{100}x^{100}+a_{99}x^{99}+\dots+a_{1}x+a_{0}$ and $ g(x)=b_{100}x^{100}+b_{99}x^{99}+\dots+b_{1}x+b_{0}$ of degree $ 100$ differ from each other by a permutation of coefficients. It is known that $ a_{i}\ne b_{i}$ for $ i=0, 1, 2, \dots, 100$. Is it possible that $ f(x)\geq g(x)$ for all real $ x$?

Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \dfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $ a\ge 3 $ and a polynom $ P. $ Show that: $$ \max_{1\le k\le \text{grad} P} \left| a^{k-1}-P(k-1) \right| \ge 1 $$

Let $p$ be a prime number, and define a sequence by: $x_i=i$ for $i=,0,1,2...,p-1$ and $x_n=x_{n-1}+x_{n-p}$ for $n \geq p$ Find the remainder when $x_{p^3}$ is divided by $p$.