Olympiad problems

1941 Moscow Mathematical Olympiad, 080

How many roots does equation $\sin x = \frac{x}{100}$ have?

1954 Moscow Mathematical Olympiad, 285

Tags: algebra , quadratic , trinomial , absolute value , root

The absolute values of all roots of the quadratic equation $x^2+Ax+B = 0$ and $x^2+Cx+D = 0$ are less then $1$. Prove that so are absolute values of the roots of the quadratic equation $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.

2000 Estonia National Olympiad, 3

Tags: polynomial , root , integer , algebra

Find all values of $a$ for which the equation $x^3 - x + a = 0$ has three different integer solutions.

1999 Bosnia and Herzegovina Team Selection Test, 6

Tags: algebra , polynomial , root , imaginary

It is given polynomial $$P(x)=x^4+3x^3+3x+p, (p \in \mathbb{R})$$ $a)$ Find $p$ such that there exists polynomial with imaginary root $x_1$ such that $\mid x_1 \mid =1$ and $2Re(x_1)=\frac{1}{2}\left(\sqrt{17}-3\right)$ $b)$ Find all other roots of polynomial $P$ $c)$ Prove that does not exist positive integer $n$ such that $x_1^n=1$

1958 February Putnam, A1

Tags: root , polynomial

If $a_0 , a_1 ,\ldots, a_n$ are real number satisfying $$ \frac{a_0 }{1} + \frac{a_1 }{2} + \ldots + \frac{a_n }{n+1}=0,$$ show that the equation $a_n x^n + \ldots +a_1 x+a_0 =0$ has at least one real root.

1947 Putnam, B5

Tags: integer polynomial , root

Let $a,b,c,d$ be distinct integers such that $$(x-a)(x-b)(x-c)(x-d) -4=0$$ has an integer root $r.$ Show that $4r=a+b+c+d.$

2008 Postal Coaching, 4

Tags: algebra , polynomial , root

Find all real numbers$p, q$ for which the polynomial equation $P(x) = x^4 - \frac{8p^2}{q}x^3 + 4qx^2 - 3px + p^2 = 0$ has four positive roots.

2006 German National Olympiad, 5

Tags: inequalities , absolute value , polynomial , root

Let $x \neq 0$ be a real number satisfying $ax^2+bx+c=0$ with $a,b,c \in \mathbb{Z}$ obeying $|a|+|b|+|c| > 1$. Then prove \[ |x| \geq \frac{1}{|a|+|b|+|c|-1}. \]

1982 IMO Shortlist, 7

Tags: algebra , polynomial , root , cubic

Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$

1988 IMO, 1

Tags: algebra , polynomial , vieta , inequality , root

Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

2011 VTRMC, Problem 7

Tags: algebra , root , polynomial

Let $P(x)=x^{100}+20x^{99}+198x^{98}+a_{97}x^{97}+\ldots+a_1x+1$ be a polynomial where the $a_i~(1\le i\le97)$ are real numbers. Prove that the equation $P(x)=0$ has at least one nonreal root.

2015 Caucasus Mathematical Olympiad, 2

Tags: algebra , root , trinomial

Let $a$ and $b$ be arbitrary distinct numbers. Prove that the equation $(x +a) (x+b)=2x+a+b$ has two different roots.

2025 VJIMC, 4

Tags: complex analysis , holomorphic , taylor series , root , function

Let $D = \{z\in \mathbb{C}: |z| < 1\}$ be the open unit disk in the complex plane and let $f : D \to D$ be a holomorphic function such that $\lim_{|z|\to 1}|f(z)| = 1$. Let the Taylor series of $f$ be $f(z) = \sum_{n=0}^{\infty} a_nz^n$. Prove that the number of zeroes of $f$ (counted with multiplicities) equals $\sum_{n=0}^{\infty} n|a_n|^2$.

2015 Germany Team Selection Test, 1

Tags: algebra , polynomial , real number , integer , root

Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties: - For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$. - There is a real number $\xi$ with $P(\xi)=0$.

1991 IMO Shortlist, 21

Tags: algebra , polynomial , functional equation , root

Let $ f(x)$ be a monic polynomial of degree $ 1991$ with integer coefficients. Define $ g(x) \equal{} f^2(x) \minus{} 9.$ Show that the number of distinct integer solutions of $ g(x) \equal{} 0$ cannot exceed $ 1995.$

1989 IMO Longlists, 8

Tags: algebra , polynomial , root , equation , diophantine equation

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

2006 IMO Shortlist, 4

Tags: algebra , polynomial , root

Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.

2024 Rioplatense Mathematical Olympiad, 3

Tags: algebra , rioplatense , root

Given a set $S$ of integers, an allowed operation consists of the following three steps: $\bullet$ Choose a positive integer $n$. $\bullet$ Choose $n+1$ elements $a_0, a_1, \dots, a_n \in S$, not necessarily distinct. $\bullet$ Add to the set $S$ all the integer roots of the polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0$. Beto must choose an initial set $S$ and perform several allowed operations, so that at the end of the process $S$ contains among its elements the integers $1, 2, 3, \dots, 2023, 2024$. Determine the smallest $k$ for which there exists an initial set $S$ with $k$ elements that allows Beto to achieve his objective.

1971 IMO Longlists, 31

Tags: equation , root , polynomial , algebra

Determine whether there exist distinct real numbers $a, b, c, t$ for which: [i](i)[/i] the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$ [i](ii)[/i] the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$ [i](iii)[/i] the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$

Kvant 2023, M2738

Tags: algebra , root

The real numbers $a_1,a_2,a_3$ and $b{}$ are given. The equation \[(x-a_1)(x-a_2)(x-a_3)=b\]has three distinct real roots, $c_1,c_2,c_3.$ Determine the roots of the equation \[(x+c_1)(x+c_2)(x+c_3)=b.\][i]Proposed by A. Antropov and K. Sukhov[/i]

2005 IMO Shortlist, 1

Tags: algebra , polynomial , root , quadratic , coefficient

Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0 \] where each of $c_0,c_1,\ldots,c_{n-1}$ is equal to $1$ or $-1$.

1973 Putnam, A4

Tags: function , root , real analysis

How many zeroes does the function $f(x)=2^x -1 -x^2 $ have on the real line?

2014 India PRMO, 17

Tags: algebra , integer , root , minimum

For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$?

2019 Romania National Olympiad, 4

Tags: complex number , polynomial , root , superior algebra

Let $n \geq 3$ and $a_1,a_2,...,a_n$ be complex numbers different from $0$ with $|a_i| < 1$ for all $i \in \{1,2,...,n-1 \}.$ If the coefficients of $f = \prod_{i=1}^n (X-a_i)$ are integers, prove that $\textbf{a)}$ The numbers $a_1,a_2,...,a_n$ are distinct. $\textbf{b)}$ If $a_j^2 = a_ia_k,$ then $i=j=k.$

1988 IMO Shortlist, 16

Tags: algebra , polynomial , vieta , inequality , root

Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.