Olympiad problems

2020 GQMO, 6

For every integer $n$ not equal to $1$ or $-1$, define $S(n)$ as the smallest integer greater than $1$ that divides $n$. In particular, $S(0)=2$. We also define $S(1) = S(-1) = 1$. Let $f$ be a non-constant polynomial with integer coefficients such that $S(f(n)) \leq S(n)$ for every positive integer $n$. Prove that $f(0)=0$. [b]Note:[/b] A non-constant polynomial with integer coefficients is a function of the form $f(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_k x^k$, where $k$ is a positive integer and $a_0,a_1,\ldots,a_k$ are integers such that $a_k \neq 0$. [i]Pitchayut Saengrungkongka, Thailand[/i]

1996 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 7

Tags: algebra , polynomial

If 1,2, and 3 are solutions to the equation $ x^4 \plus{} ax^2 \plus{} bx \plus{} c \equal{} 0,$ then $ a\plus{}c$ equals A. -12 B. 24 C. 35 D. -61 E. -63

2021 AMC 12/AHSME Fall, 25

Tags: algebra , polynomial

Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $\big(a_1, a_2, a_3, a_4\big)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial $$q(x) = c_3x^3+c_2x^2+c_1x+c_0$$such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c_1?$ $(\textbf{A})\: {-}6\qquad(\textbf{B}) \: {-}1\qquad(\textbf{C}) \: 4\qquad(\textbf{D}) \: 6\qquad(\textbf{E}) \: 11$

2007 IMC, 3

Tags: quadratic , algebra , polynomial , analytic geometry , linear algebra , matrix

Call a polynomial $ P(x_{1}, \ldots, x_{k})$ [i]good[/i] if there exist $ 2\times 2$ real matrices $ A_{1}, \ldots, A_{k}$ such that $ P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).$ Find all values of $ k$ for which all homogeneous polynomials with $ k$ variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)

1989 Iran MO (2nd round), 2

Tags: polynomial , algebra

Let $n$ be a positive integer. Prove that the polynomial \[P(x)= \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+...+x+1 \] Does not have any rational root.

2024 BAMO, 4

Tags: algebra , polynomial

Find all polynomials $f$ that satisfy the equation \[\frac{f(3x)}{f(x)} = \frac{729 (x-3)}{x-243}\] for infinitely many real values of $x$.

2014 Taiwan TST Round 1, 1

Tags: algebra , polynomial , inequalities , absolute value

Let $f(x) = x^n + a_{n-2} x^{n-2} + a_{n-3}x^{n-3} + \dots + a_1x + a_0$ be a polynomial with real coefficients $(n \ge 2)$. Suppose all roots of $f$ are real. Prove that the absolute value of each root is at most $\sqrt{\frac{2(1-n)}n a_{n-2}}$.

2009 Postal Coaching, 5

Tags: polynomial , sequence , algebra

Define a sequence $<x_n>$ by $x_1 = 1, x_2 = x, x_{n+2} = xx_{n+1} + nx_n, n \ge 1$. Consider the polynomial $P_n(x) = x_{n-1}x_{n+1} - x_n^2$, for each $n \ge 2$. Prove or disprove that the coefficients of $P_n(x)$ are all non-negative, except for the constant term when $n$ is odd.

2019 PUMaC Algebra B, 5

Tags: algebra , quadratic , polynomial

Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i\geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.

1991 Spain Mathematical Olympiad, 3

Tags: algebra , polynomial , triangle inequality

What condition must be satisfied by the coefficients $u,v,w$ if the roots of the polynomial $x^3 -ux^2+vx-w$ are the sides of a triangle

2019 Romanian Master of Mathematics Shortlist, original P6

Tags: polynomial , multivariate polynomial , algebra

Let $P(x)$ be a nonconstant complex coefficient polynomial and let $Q(x,y)=P(x)-P(y).$ Suppose that polynomial $Q(x,y)$ has exactly $k$ linear factors unproportional two by tow (without counting repetitons). Let $R(x,y)$ be factor of $Q(x,y)$ of degree strictly smaller than $k$. Prove that $R(x,y)$ is a product of linear polynomials. [b]Note: [/b] The [i]degree[/i] of nontrivial polynomial $\sum_{m}\sum_{n}c_{m,n}x^{m}y^{n}$ is the maximum of $m+n$ along all nonzero coefficients $c_{m,n}.$ Two polynomials are [i]proportional[/i] if one of them is the other times a complex constant. [i]Proposed by Navid Safaie[/i]

2014 NIMO Problems, 1

Tags: algebra , polynomial , induction

Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$. [i]Proposed by Evan Chen[/i]

2017 Korea USCM, 4

Tags: algebra , polynomial , linear algebra , matrix

For a real coefficient cubic polynomial $f(x)=ax^3+bx^2+cx+d$, denote three roots of the equation $f(x)=0$ by $\alpha,\beta,\gamma$. Prove that the three roots $\alpha,\beta,\gamma$ are distinct real numbers iff the real symmetric matrix $$\begin{pmatrix} 3 & p_1 & p_2 \\ p_1 & p_2 & p_3 \\ p_2 & p_3 & p_4 \end{pmatrix},\quad p_i = \alpha^i + \beta^i + \gamma^i$$ is positive definite.

2013 QEDMO 13th or 12th, 10

Tags: polynomial , algebra , integer polynomial

Let $p$ be a prime number gretater then $3$. What is the number of pairs $(m, n)$ of integers with $0 <m <n <p$, for which the polynomial $x^p + px^n + px^m +1$ is not a product of two non-constant polynomials with integer coefficients can be written?

2000 Moldova National Olympiad, Problem 4

Tags: functional equation , fe , polynomial , algebra

Find all polynomials $P(x)$ with real coefficients that satisfy the relation $$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$

2003 Iran MO (3rd Round), 5

Tags: polynomial , number theory , algebra

Let $p$ be an odd prime number. Let $S$ be the sum of all primitive roots modulo $p$. Show that if $p-1$ isn't squarefree (i. e., if there exist integers $k$ and $m$ with $k>1$ and $p-1=k^2m$), then $S \equiv 0 \mod p$. If not, then what is $S$ congruent to $\mod p$ ?

2002 USAMO, 3

Tags: polynomial , lagrange s interpolation

Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.

2007 Harvard-MIT Mathematics Tournament, 10

Tags: algebra , polynomial

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 $P(1)/P(-1)$.

2007 Balkan MO Shortlist, N3

Tags: polynomial , trigonometry , algebra

i thought that this problem was in mathlinks but when i searched i didn't find it.so here it is: Find all positive integers m for which for all $\alpha,\beta \in \mathbb{Z}-\{0\}$ \[ \frac{2^m \alpha^m-(\alpha+\beta)^m-(\alpha-\beta)^m}{3 \alpha^2+\beta^2} \in \mathbb{Z} \]

1986 All Soviet Union Mathematical Olympiad, 439

Tags: polynomial , algebra , integer polynomial

Let us call a polynomial [i]admissible[/i] if all it's coefficients are $0, 1, 2$ or $3$. For given $n$ find the number of all the [i]admissible [/i] polynomials $P$ such, that $P(2) = n$.

2016 Chile TST IMO, 4

Tags: algebra , polynomial

Let $ f $ and $ g $ be two nonzero polynomials with integer coefficients such that $ \deg(f) > \deg(g) $. Suppose that for infinitely many prime numbers $ p $, the polynomial $ pf + g $ has a rational root. Prove that $ f $ has a rational root. Clarification: A rational root of a polynomial $ f $ is a number $ q \in \mathbb{Q} $ such that $ f(q) = 0 $.

2003 Federal Competition For Advanced Students, Part 1, 3

Tags: quadratic , polynomial , algebra

Given a positive real number $t$, find the number of real solutions $a, b, c, d$ of the system \[a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t.\]

2020 Romanian Master of Mathematics Shortlist, A1

Tags: algebra , polynomial

Prove that for all sufficiently large positive integers $d{}$, at least $99\%$ of the polynomials of the form \[\sum_{i\leqslant d}\sum_{j\leqslant d}\pm x^iy^j\]are irreducible over the integers.

2007 Moldova Team Selection Test, 2

Tags: polynomial , calculus , derivative , induction , algebra

If $b_{1}, b_{2}, \ldots, b_{n}$ are non-negative reals not all zero, then prove that the polynomial \[x^{n}-b_{1}x^{n-1}-b_{2}x^{n-2}-\ldots-b_{n}=0\] has only one positive root $p$, which is simple. Moreover prove that any root of the polynomial does not exceed $p$ in absolute value.

2021 Grand Duchy of Lithuania, 1

Tags: algebra , polynomial

Prove that for any polynomial $f(x)$ (with real coefficients) there exist polynomials $g(x)$ and $h(x)$ (with real coefficients) such that $f(x) = g(h(x)) - h(g(x))$.