Found problems: 3597
2023 Romanian Master of Mathematics Shortlist, A2
Fix an integer $n \geq 2$ and let $a_1, \ldots, a_n$ be integers, where $a_1 = 1$. Let
$$ f(x) = \sum_{m=1}^n a_mm^x. $$ Suppose that $f(x) = 0$ for some $K$ consecutive positive integer values of $x$. In terms of $n$, determine the maximum possible value of $K$.
2010 Indonesia MO, 7
Given 2 positive reals $a$ and $b$. There exists 2 polynomials $F(x)=x^2+ax+b$ and $G(x)=x^2+bx+a$ such that all roots of polynomials $F(G(x))$ and $G(F(x))$ are real. Show that $a$ and $b$ are more than $6$.
[i]Raja Oktovin, Pekanbaru[/i]
2002 Moldova Team Selection Test, 4
Let $P(x)$ be a polynomial with integer coefficients for which there exists a positive integer n such that the real parts of all roots of $P(x)$ are less than $n- \frac{1}{2}$ , polynomial $x-n+1$ does not divide $P(x)$, and $P(n)$ is a prime number. Prove that the polynomial $P(x)$ is irreducible (over $Z[x]$).
2016 Saint Petersburg Mathematical Olympiad, 7
A polynomial $P$ with real coefficients is called [i]great,[/i] if for some integer $a>1$ and for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all [i]great[/i] polynomials.
[i]Proposed by A. Golovanov[/i]
1974 Miklós Schweitzer, 7
Given a positive integer $ m$ and $ 0 < \delta <\pi$, construct a trigonometric polynomial $ f(x)\equal{}a_0\plus{} \sum_{n\equal{}1}^m (a_n \cos nx\plus{}b_n \sin nx)$ of degree $ m$ such that $ f(0)\equal{}1, \int_{ \delta \leq |x| \leq \pi} |f(x)|dx \leq c/m,$ and $ \max_{\minus{}\pi \leq x \leq \pi}|f'(x)| \leq c/{\delta}$, for some universal constant $ c$.
[i]G. Halasz[/i]
1977 Germany Team Selection Test, 2
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
MathLinks Contest 7th, 4.3
Let $ a,b,c$ be positive real numbers such that $ ab\plus{}bc\plus{}ca\equal{}3$. Prove that
\[ \frac 1{1\plus{}a^2(b\plus{}c)} \plus{} \frac 1{1\plus{}b^2(c\plus{}a)} \plus{} \frac 1 {1\plus{}c^2(a\plus{}b) } \leq \frac 3 {1\plus{}2abc} .\]
1962 Swedish Mathematical Competition, 1
Find all polynomials $f(x)$ such that $f(2x) = f'(x) f''(x)$.
BIMO 2022, 4
Given a polynomial $P\in \mathbb{Z}[X]$ of degree $k$, show that there always exist $2d$ distinct integers $x_1, x_2, \cdots x_{2d}$ such that $$P(x_1)+P(x_2)+\cdots P(x_{d})=P(x_{d+1})+P(x_{d+2})+\cdots + P(x_{2d})$$ for some $d\le k+1$.
[Extra: Is this still true if $d\le k$? (Of course false for linear polynomials, but what about higher degree?)]
2007 Thailand Mathematical Olympiad, 8
Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$. Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$.
1986 Miklós Schweitzer, 8
Let $a_0=0$, $a_1, \ldots, a_k$ and $b_1, \ldots, b_k$ be arbitrary real numbers.
(i) Show that for all sufficiently large $n$ there exist polynomials $p_n$ of degree at most $n$ for which
$$p_n^{(i)} (-1)=a_i,\,\,\,\,\, p_n^{(i)} (1)=b_i,\,\,\,\,\, i=0, 1, \ldots, k$$
and
$$\max_{|x|\leq 1} |p_n (x)|\leq \frac{c}{n^2}\,\,\,\,\,\,\,\,\,\, (*)$$
where the constant $c$ depends only on the numbers $a_i, b_i$.
(ii) Prove that, in general, (*) cannot be replaced by the relation
$$\lim_{n\to\infty} n^2\cdot \max_{|x|\leq 1} |p_n (x)| = 0$$
[J. Szabados]
2014 Putnam, 5
Let $P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.$ Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j\ne k.$
2006 Pan African, 2
Let $a, b, c$ be three non-zero integers. It is known that the sums $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are integers. Find these sums.
2001 Poland - Second Round, 1
Let $k,n>1$ be integers such that the number $p=2k-1$ is prime. Prove that, if the number $\binom{n}{2}-\binom{k}{2}$ is divisible by $p$, then it is divisible by $p^2$.
2021 Saudi Arabia BMO TST, 1
Do there exist two polynomials $P$ and $Q$ with integer coefficient such that
i) both $P$ and $Q$ have a coefficient with absolute value bigger than $2021$,
ii) all coefficients of $P \cdot Q$ by absolute value are at most $1$.
2003 India IMO Training Camp, 6
A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find $z_n$, the maximum number of regions into which $n$ zig-zags can divide the plane. For example, $z_1=2,z_2=12$(see the diagram). Of these $z_n$ regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in $n$ of degree not exceeding $2$.
[asy]
draw((30,0)--(-70,0), Arrow);
draw((30,0)--(-20,-40));
draw((-20,-40)--(80,-40), Arrow);
draw((0,-60)--(-40,20), dashed, Arrow);
draw((0,-60)--(0,15), dashed);
draw((0,15)--(40,-65),dashed, Arrow);
[/asy]
2021 JHMT HS, 10
A polynomial $P(x)$ of some degree $d$ satisfies $P(n) = n^3 + 10n^2 - 12$ and $P'(n) = 3n^2 + 20n - 1$ for $n = -2, -1, 0, 1, 2.$ Also, $P$ has $d$ distinct (not necessarily real) roots $r_1, r_2, \ldots, r_d.$ The value of
\[ \sum_{k=1}^{d}\frac{1}{4 - r_k^2} \]
can be expressed as a common fraction $\tfrac{p}{q}.$ What is the value of $p + q?$
1916 Eotvos Mathematical Competition, 1
If $ a$ and $b$ are positive numbers, prove that the equation
$$\frac{1}{x}+\frac{1}{x - a}+\frac{1}{x+ b}= 0$$
has two rea] roots, one between $ a/3$ and $2a/3$, and one between $-2b/3$ and $-b/3$.
1990 IMO Longlists, 95
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.
1985 Traian Lălescu, 1.2
Find the first degree polynomial function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equation
$$ f(x-1)=-3x-5-f(2), $$
for all real numbers $ x. $
1998 All-Russian Olympiad, 1
The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$. Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc.
2004 District Olympiad, 4
Let $A=(a_{ij})\in \mathcal{M}_p(\mathbb{C})$ such that $a_{12}=a_{23}=\ldots=a_{p-1,p}=1$ and $a_{ij}=0$ for any other entry.
a)Prove that $A^{p-1}\neq O_p$ and $A^p=O_p$.
b)If $X\in \mathcal{M}_{p}(\mathbb{C})$ and $AX=XA$, prove that there exist $a_1,a_2,\ldots,a_p\in \mathbb{C}$ such that:
\[X=\left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_p \\ 0 & a_1 & a_2 & \ldots & a_{p-1} \\ 0 & 0 & a_1 & \ldots & a_{p-2} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & a_1 \end{array} \right)\]
c)If there exist $B,C\in \mathcal{M}_p(\mathbb{C})$ such that $(I_p+A)^n=B^n+C^n,\ (\forall)n\in \mathbb{N}^*$, prove that $B=O_p$ or $C=O_p$.
1994 Moldova Team Selection Test, 1
Let $P(X)=X^n+a_1X^{n-1}+\ldots+a_n$ be a plynomial with real roots $x_1. x_2,\ldots,x_n$. Denote $E_k=x_1^k+x_2^k+\ldots+x_n^k, \forall k\in\mathbb{N}$. There exists an $m\in\mathbb{N}$ such that $E_m=E_{m+1}=E_{m+2}=1$. Find $\max\{P(-2),P(2)\}$.
2023 Kurschak Competition, 1
Let $f(x)$ be a non-constant polynomial with non-negative integer coefficients. Prove that there are infinitely many positive integers $n$, for which $f(n)$ is not divisible by any of $f(2)$, $f(3)$, ..., $f(n-1)$.
STEMS 2021 Math Cat C, Q4
Let $n$ be a fixed positive integer.
- Show that there exist real polynomials $p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]$ such that
\[(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)\]
- Find the least natural number $k$, depending on $n$, such that the above polynomials $p_1, p_2, \cdots, p_k$ exist.