Olympiad problems

2017 Bosnia And Herzegovina - Regional Olympiad, 1

If $a$ is real number such that $x_1$ and $x_2$, $x_1\neq x_2$ , are real numbers and roots of equation $x_2-x+a=0$. Prove that $\mid {x_1}^2-{x_2}^2 \mid =1$ iff $\mid {x_1}^3-{x_2}^3 \mid =1$

2016 India PRMO, 9

Tags: algebra , trinomial , roots

Let $a$ and $b$ be the roots of the equation $x^2 + x - 3 = 0$. Find the value of the expression $4b^2 -a^3$.

2006 Polish MO Finals, 3

Tags: algebra , polynomial , roots , quadratics , coefficients , IMO Shortlist

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 $-1$.

2008 Postal Coaching, 4

Tags: algebra , polynomial , roots

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.

2016 Saudi Arabia BMO TST, 1

Tags: algebra , roots , polynomial

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 $Q(x) = x^2 + 1$.

1968 IMO Shortlist, 11

Tags: algebra , equation , roots , IMO Shortlist

Find all solutions $(x_1, x_2, . . . , x_n)$ of the equation \[1 +\frac{1}{x_1} + \frac{x_1+1}{x{}_1x{}_2}+\frac{(x_1+1)(x_2+1)}{x{}_1{}_2x{}_3} +\cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x{}_1x{}_2\cdots x_n} =0\]

1981 Putnam, A5

Tags: Putnam , polynomial , roots

Let $P(x)$ be a polynomial with real coefficients and form the polynomial $$Q(x) = ( x^2 +1) P(x)P'(x) + x(P(x)^2 + P'(x)^2 ).$$ Given that the equation $P(x) = 0$ has $n$ distinct real roots exceeding $1$, prove or disprove that the equation $Q(x)=0$ has at least $2n - 1$ distinct real roots.

2017 India PRMO, 19

Tags: algebra , roots , polynomial

Suppose $1, 2, 3$ are the roots of the equation $x^4 + ax^2 + bx = c$. Find the value of $c$.

2019 Romania National Olympiad, 4

Tags: complex numbers , polynomial , roots , 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 $i=j=k.$

2010 Belarus Team Selection Test, 8.1

Tags: sum of digits , number theory , roots

The function $f : N \to N$ is defined by $f(n) = n + S(n)$, where $S(n)$ is the sum of digits in the decimal representation of positive integer $n$. a) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has no natural roots. b) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has at least two distinct natural roots. (I. Voronovich)

Russian TST 2018, P1

Tags: polynomial , roots , algebra

Let $f(x) = x^2 + 2018x + 1$. Let $f_1(x)=f(x)$ and $f_k(x)=f(f_{k-1}(x))$ for all $k\geqslant 2$. Prove that for any positive integer $n{}$, the equation $f_n(x)=0$ has at least two distinct real roots.

1982 IMO Shortlist, 7

Tags: algebra , polynomial , roots , Cubic , IMO Shortlist , IMO Longlist

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

1991 IMO Shortlist, 21

Tags: algebra , polynomial , functional equation , roots , IMO Shortlist

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.$

2016 Irish Math Olympiad, 3

Tags: algebra , polynomial , roots , Sum

Do there exist four polynomials $P_1(x), P_2(x), P_3(x), P_4(x)$ with real coefficients, such that the sum of any three of them always has a real root, but the sum of any two of them has no real root?

2011 Israel National Olympiad, 2

Tags: algebra , roots , polynomial , sum of roots

Evaluate the sum $\sqrt{1-\frac{1}{2}\cdot\sqrt{1\cdot3}}+\sqrt{2-\frac{1}{2}\cdot\sqrt{3\cdot5}}+\sqrt{3-\frac{1}{2}\cdot\sqrt{5\cdot7}}+\dots+\sqrt{40-\frac{1}{2}\cdot\sqrt{79\cdot81}}$.

2005 IMO Shortlist, 1

Tags: algebra , polynomial , roots , quadratics , coefficients , IMO Shortlist

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 $-1$.

1988 IMO Shortlist, 16

Tags: algebra , polynomial , Vieta , Inequality , roots , IMO , IMO 1988

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.

1949 Putnam, A5

Tags: Putnam , roots

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

1958 February Putnam, A1

Tags: Putnam , roots , 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.

1967 IMO Shortlist, 1

Tags: algebra , equation , roots , IMO Shortlist , IMO Longlist , Exponential equation

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

2006 IMO Shortlist, 4

Tags: algebra , polynomial , roots , IMO , IMO 2006 , IMO Shortlist , Dan Schwarz

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 $Q(t) = t$.