Olympiad problems

1985 Traian Lălescu, 1.1

Prove that for all $ n\ge 2 $ natural numbers there exist $ a_n\in\mathbb{Q} $ such that $$ X^{2n}+a_nX^n+1\Huge\vdots X^2+\frac{1}{2}X+1, $$ and that there isn´t any $ a_n\in\mathbb{R}\setminus\mathbb{Q} $ with this property.

1998 Baltic Way, 4

Tags: modular arithmetic , polynomial , algebra , number theory

Let $P$ be a polynomial with integer coefficients. Suppose that for $n=1,2,3,\ldots ,1998$ the number $P(n)$ is a three-digit positive integer. Prove that the polynomial $P$ has no integer roots.

2005 Czech-Polish-Slovak Match, 4

Tags: polynomial , algebra , combinatorics , floor function

We distribute $n\ge1$ labelled balls among nine persons $A,B,C, \dots , I$. How many ways are there to do this so that $A$ gets the same number of balls as $B,C,D$ and $E$ together?

2019 German National Olympiad, 4

Tags: polynomial , algebra , number theory

Show that for each non-negative integer $n$ there are unique non-negative integers $x$ and $y$ such that we have \[n=\frac{(x+y)^2+3x+y}{2}.\]

1999 India National Olympiad, 3

Tags: quadratic , algebra , number theory , polynomial , integration , calculus

Show that there do not exist polynomials $p(x)$ and $q(x)$ each having integer coefficients and of degree greater than or equal to 1 such that \[ p(x)q(x) = x^5 +2x +1 . \]

2008 Greece Team Selection Test, 1

Tags: divisibility , polynomial , algebra

Find all possible values of $a\in \mathbb{R}$ and $n\in \mathbb{N^*}$ such that $f(x)=(x-1)^n+(x-2)^{2n+1}+(1-x^2)^{2n+1}+a$ is divisible by $\phi (x)=x^2-x+1$

1999 Hungary-Israel Binational, 1

Tags: polynomial , algebra

$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$, for all $ n \in N$. Let $ r_n$ be the average of $ g_n(x)$'s roots. If $ r_{19}\equal{}99$, find $ r_{99}$.

2012 Tuymaada Olympiad, 4

Tags: quadratic , polynomial , algebra , calculus , modular arithmetic

Let $p=4k+3$ be a prime. Prove that if \[\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n}\] (where the fraction $\dfrac {m} {n}$ is in reduced terms), then $p \mid 2m-n$. [i]Proposed by A. Golovanov[/i]

2008 Tournament Of Towns, 3

Tags: sum , root , inequalities , polynomial , algebra

A polynomial $x^n + a_1x^{n-1} + a_2x^{n-2} +... + a_{n-2}x^2 + a_{n-1}x + a_n$ has $n$ distinct real roots $x_1, x_2,...,x_n$, where $n > 1$. The polynomial $nx^{n-1}+ (n - 1)a_1x^{n-2} + (n - 2)a_2x^{n-3} + ...+ 2a_{n-2}x + a_{n-1}$ has roots $y_1, y_2,..., y_{n_1}$. Prove that $\frac{x^2_1+ x^2_2+ ...+ x^2_n}{n}>\frac{y^2_1 + y^2_2 + ...+ y^2_{n-1}}{n - 1}$

2015 Spain Mathematical Olympiad, 1

Tags: polynomial , algebra

On the graph of a polynomial with integer coefficients, two points are chosen with integer coordinates. Prove that if the distance between them is an integer, then the segment that connects them is parallel to the horizontal axis.

2012 Albania National Olympiad, 2

Tags: quadratic , polynomial , analytic geometry , algebra

The trinomial $f(x)$ is such that $(f(x))^3-f(x)=0$ has three real roots. Find the y-coordinate of the vertex of $f(x)$.

2010 Estonia Team Selection Test, 5

Tags: trigonometry , polynomial , algebra

Let $P(x, y)$ be a non-constant homogeneous polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for every real number $t$. Prove that there exists a positive integer $k$ such that $P(x, y) = (x^2 + y^2)^k$.

2015 Vietnam National Olympiad, 1

Tags: induction , polynomial , algebra

Let ${\left\{ {f(x)} \right\}}$ be a sequence of polynomial, where ${f_0}(x) = 2$, ${f_1}(x) = 3x$, and ${f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)$ $(n \ge 2)$ Determine the value of $n$ such that ${f_n}(x)$ is divisible by $x^3-x^2+x$.

2003 Belarusian National Olympiad, 2

Tags: perfect square , polynomial , algebra

Let $P(x) =(x+1)^p (x-3)^q=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}x+a_n$ where $p$ and $q$ are positive integers a) Given $a_1=a_2$, prove that $3n$ is a perfect square. b) Prove that there exist infinitely many pairs $(p, q)$ of positive integers p and q such that the equality $a_1=a_2$ is valid for the polynomial $P(x)$. (D. Bazylev)

2008 Mongolia Team Selection Test, 3

Tags: polynomial , algebra

Given positive integers $ m,n > 1$. Prove that the equation $ (x \plus{} 1)^n \plus{} (x \plus{} 2)^n \plus{} ... \plus{} (x \plus{} m)^n \equal{} (y \plus{} 1)^{2n} \plus{} (y \plus{} 2)^{2n} \plus{} ... \plus{} (y \plus{} m)^{2n}$ has finitely number of solutions $ x,y \in N$

1993 Greece National Olympiad, 4

Tags: symmetry , matrix , inequalities , linear algebra , polynomial , algebra

How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

2018 CCA Math Bonanza, I7

Tags: polynomial , algebra

Find all values of $a$ such that the two polynomials \[x^2+ax-1\qquad\text{and}\qquad x^2-x+a\] share at least 1 root. [i]2018 CCA Math Bonanza Individual Round #7[/i]

1995 Czech And Slovak Olympiad IIIA, 6

Tags: sidelenghts , parameter , algebra , polynomial

Find all real parameters $p$ for which the equation $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$ has three distinct real roots which are sides of a right triangle.

2015 Nordic, 3

Tags: number theory , polynomial , algebra

Let $n > 1$ and $p(x)=x^n+a_{n-1}x^{n-1} +...+a_0$ be a polynomial with $n$ real roots (counted with multiplicity). Let the polynomial $q$ be defined by $$q(x) = \prod_{j=1}^{2015} p(x + j)$$. We know that $p(2015) = 2015$. Prove that $q$ has at least $1970$ different roots $r_1, ..., r_{1970}$ such that $|r_j| < 2015$ for all $ j = 1, ..., 1970$.

2018 Hanoi Open Mathematics Competitions, 5

Tags: polynomial , algebra

Let $f$ be a polynomial such that, for all real number $x$, $f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019$. Compute $f(2018)$.

1983 AMC 12/AHSME, 20

Tags: trigonometry , algebra , polynomial

If $\tan{\alpha}$ and $\tan{\beta}$ are the roots of $x^2 - px + q = 0$, and $\cot{\alpha}$ and $\cot{\beta}$ are the roots of $x^2 - rx + s = 0$, then $rs$ is necessarily $\text{(A)} \ pq \qquad \text{(B)} \ \frac{1}{pq} \qquad \text{(C)} \ \frac{p}{q^2} \qquad \text{(D)} \ \frac{q}{p^2} \qquad \text{(E)} \ \frac{p}{q}$

2024 Brazil Cono Sur TST, 4

Tags: translation , polynomial

In the cartesian plane, consider the subset of all the points with both integer coordinates. Prove that it is possible to choose a finite non-empty subset $S$ of these points in such a way that any line $l$ that forms an angle of $90^{\circ},0^{\circ},135^{\circ}$ or $45^{\circ}$ with the positive horizontal semi-axis intersects $S$ at exactly $2024$ points or at no points.

2020 BMT Fall, 17

Tags: polynomial , algebra

Let $T$ be the answer to question $16$. Compute the number of distinct real roots of the polynomial $x^4 + 6x^3 +\frac{T}{2}x^2 + 6x + 1$.

2020 AMC 12/AHSME, 17

Tags: algebra , polynomial

How many polynomials of the form $x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$, where $a$, $b$, $c$, and $d$ are real numbers, have the property that whenever $r$ is a root, so is $\frac{-1+i\sqrt{3}}{2} \cdot r$? (Note that $i=\sqrt{-1}$) $\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$

1999 AMC 12/AHSME, 12

Tags: algebra , polynomial , function

What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $ y \equal{} p(x)$ and $ y \equal{} q(x)$, each with leading coefficient $ 1$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$