Olympiad problems

2014 Israel National Olympiad, 7

Find one real value of $x$ satisfying $\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$.

1941 Putnam, A4

Tags: root , polynomial

Let the roots $a,b,c$ of $$f(x)=x^3 +p x^2 + qx+r$$ be real, and let $a\leq b\leq c$. Prove that $f'(x)$ has a root in the interval $\left[\frac{b+c}{2}, \frac{b+2c}{3}\right]$. What will be the form of $f(x)$ if the root in question falls at either end of the interval?

2021 German National Olympiad, 1

Tags: quadratic , quadratic equation , root , algebra

Determine all real numbers $a,b,c$ and $d$ with the following property: The numbers $a$ and $b$ are distinct roots of $2x^2-3cx+8d$ and the numbers $c$ and $d$ are distinct roots of $2x^2-3ax+8b$.

2014 India PRMO, 9

Tags: root , trinomial , algebra

Natural numbers $k, l,p$ and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$ then $a +\frac1b$ and $b + \frac1a$ are the roots of $x^2 -px + q = 0$. What is the sum of all possible values of $q$?

2006 German National Olympiad, 5

Tags: root , absolute value , polynomial , inequalities

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 \[ |x| \geq \frac{1}{|a|+|b|+|c|-1}. \]

2008 Postal Coaching, 4

Tags: root , polynomial , algebra

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.

1985 IMO Longlists, 92

Tags: polynomial , root , algorithm , algebra

Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.

1949 Putnam, A5

Tags: root

How many roots of the equation $z^6 +6z +10=0$ lie in each quadrant of the complex plane?

2017 Hanoi Open Mathematics Competitions, 10

Tags: integer , root , algebra , trinomial

Find all non-negative integers $a, b, c$ such that the roots of equations: $\begin{cases}x^2 - 2ax + b = 0 \\ x^2- 2bx + c = 0 \\ x^2 - 2cx + a = 0 \end{cases}$ are non-negative integers.

1976 IMO Longlists, 11

Tags: algebra , recurrence relation , root , chebyshev polynomial , polynomial

Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.

1954 Moscow Mathematical Olympiad, 285

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

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 $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.

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 $\sum_{n=0}^{\infty} n|a_n|^2$.

2021 Hong Kong TST, 2

Tags: polynomial , algebra , root

Let $f(x)$ be a polynomial with rational coefficients, and let $\alpha$ be a real number. If \[\alpha^3-2019\alpha=(f(\alpha))^3-2019f(\alpha)=2021,\] prove that $(f^n(\alpha))^3-2019f^n(\alpha)=2021$ for any positive integer $n$. (Here, we define $f^n(x)=\underbrace{f(f(f\cdots f}_{n\text{ times}}(x)\cdots ))$.)

1966 IMO Longlists, 35

Tags: irrational number , polynomial , algebra , root

Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.

1966 IMO Shortlist, 48

Tags: number theory , root , parametric equation , equation , algebra

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

1967 IMO Shortlist, 2

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

The equation \[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\] is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$

2013 Hanoi Open Mathematics Competitions, 12

Tags: quadratic , root , inequalities , algebra

If $f(x) = ax^2 + bx + c$ satisfies the condition $|f(x)| < 1; \forall x \in [-1, 1]$, prove that the equation $f(x) = 2x^2 - 1$ has two real roots.

1963 Czech and Slovak Olympiad III A, 4

Tags: real root , quadratic polynomial , root , quadratic , algebra

Consider two quadratic equations \begin{align*}x^2+ax+b&=0, \\ x^2+cx+d&=0,\end{align*} with real coefficients. Find necessary and sufficient conditions such that the first equation has (real) roots $x,x_1,$ the second $x,x_2$ and $x>0,x_1>x_2$.

2016 Saudi Arabia BMO TST, 1

Tags: polynomial , algebra , root

Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$.

1970 Czech and Slovak Olympiad III A, 3

Tags: parameterization , root , inequalities , algebra

Let $p>0$ be a given parameter. Determine all real $x$ such that \[\frac{1}{\,x+\sqrt{p-x^2\,}\,}+\frac{1}{\,x-\sqrt{p-x^2\,}\,}\ge\frac{1}{\,p\,}.\]

1982 IMO Longlists, 16

Tags: algebra , root , cubic , polynomial

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 $p(1)+p(-1)-2(1+p(0)).$

1973 IMO, 3

Tags: polynomial , algebra , root , minimum value , coefficient

Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.

1953 Putnam, B5

Tags: root , polynomial

Show that the roots of $x^4 +ax^3 +bx^2 +cx +d$, if suitably numbered, satisfy the relation $\frac{r_1 }{r_2 } = \frac{ r_3 }{r _4},$ provided $a^2 d=c^2 \ne 0.$

1967 IMO Longlists, 48

Tags: equation , root , exponential equation , algebra

Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$

2015 Germany Team Selection Test, 1

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

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 $P(\xi)=0$.