Olympiad problems

2021 Moldova Team Selection Test, 1

Tags: algebra , polynomial

Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$, $\rho_2$, $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$. Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$, $C(\rho_2, 0)$, $D(\rho_3, 0)$. If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$, find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained. [i]Brazitikos Silouanos, Greece[/i]

2017 QEDMO 15th, 5

Tags: polynomial , algebra

For which natural numbers $n$ can the polynomial $f (x) = x^n + x^{n-1} +...+ x + 1$ as write $f (x) = g (h (x))$, where $g$ and $h$ should be real polynomials of degrees greater than $1$?

2017 Romanian Masters In Mathematics, 2

Tags: algebra , polynomial

Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k\le n$ and $k+1$ distinct integers $x_1,x_2,\cdots ,x_{k+1}$ such that \[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\]. [i]Note.[/i] A polynomial is [i]monic[/i] if the coefficient of the highest power is one.

1995 AIME Problems, 2

Tags: algebra , logarithm , polynomial , modular arithmetic , quadratic , vieta , search

Find the last three digits of the product of the positive roots of \[ \sqrt{1995}x^{\log_{1995}x}=x^2. \]

2019 USAMO, 6

Tags: polynomial , weird problem , the saddest

Find all polynomials $P$ with real coefficients such that $$\frac{P(x)}{yz}+\frac{P(y)}{zx}+\frac{P(z)}{xy}=P(x-y)+P(y-z)+P(z-x)$$ holds for all nonzero real numbers $x,y,z$ satisfying $2xyz=x+y+z$. [i]Proposed by Titu Andreescu and Gabriel Dospinescu[/i]

1979 Vietnam National Olympiad, 2

Tags: polynomial , algebra

Find all real numbers $a, b, c$ such that $x^3 + ax^2 + bx + c$ has three real roots $\alpha, \beta,\gamma$ (not necessarily all distinct) and the equation $x^3 + \alpha^3 x^2 + \beta^3 x + \gamma^3$ has roots $\alpha^3, \beta^3,\gamma^3$ .

1974 IMO Longlists, 45

Tags: polynomial , inequality , distance , minimization , optimization

The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$

2012 ELMO Shortlist, 7

Tags: function , algebra , polynomial

Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant. [i]Victor Wang.[/i]

2010 IMO Shortlist, 2

Tags: polynomial , algebra , inequalities

Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that \[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\] [i]Proposed by Nazar Serdyuk, Ukraine[/i]

2017 Harvard-MIT Mathematics Tournament, 1

Tags: polynomial , algebra

Let $P(x)$, $Q(x)$ be nonconstant polynomials with real number coefficients. Prove that if \[\lfloor P(y) \rfloor = \lfloor Q(y) \rfloor\] for all real numbers $y$, then $P(x) = Q(x)$ for all real numbers $x$.

2012 Belarus Team Selection Test, 3

Tags: algebra , polynomial , inequalities

Given a polynomial $P(x)$ with positive real coefficients. Prove that $P(1)P(xy) \ge P(x)P(y)$ for all $x\ge1, y \ge 1$. (K. Gorodnin)

1987 Greece Junior Math Olympiad, 3

Tags: algebra , polynomial

Find real $a,b$ such that polynomial $P(x)=x^{n+1}+ax+b$ to be divisible by $(x-1)^2$. Then find the quotient $P(x):(x-1)^2 , n\in \mathbb{N}^*$

2012 District Olympiad, 2

Tags: algebra , polynomial , superior algebra , quadratic

Let $(A,+,\cdot)$ a 9 elements ring. Prove that the following assertions are equivalent: (a) For any $x\in A\backslash\{0\}$ there are two numbers $a\in \{-1,0,1\}$ and $b\in \{-1,1\}$ such that $x^2+ax+b=0$. (b) $(A,+,\cdot)$ is a field.

2020 Ecuador NMO (OMEC), 4

Tags: polynomial , algebra

Find all polynomials $P(x)$ such that, for all real numbers $x, y, z$ that satisfy $x+ y +z =0$, $$P(x) +P(y) +P(z)=0$$

2010 Contests, 3

Tags: polynomial , algebra

Find all two-variable polynomials $p(x,y)$ such that for each $a,b,c\in\mathbb R$: \[p(ab,c^2+1)+p(bc,a^2+1)+p(ca,b^2+1)=0\]

2011 IMC, 3

Tags: polynomial , algebra , superior algebra

Let $p$ be a prime number. Call a positive integer $n$ interesting if \[x^n-1=(x^p-x+1)f(x)+pg(x)\] for some polynomials $f$ and $g$ with integer coefficients. a) Prove that the number $p^p-1$ is interesting. b) For which $p$ is $p^p-1$ the minimal interesting number?

2011 Saudi Arabia BMO TST, 2

Tags: algebra , polynomial

Let $n$ be a positive integer. Prove that all roots of the equation $$x(x + 2) (x + 4 )... (x + 2n) + (x +1) (x + 3 )... (x + 2n - 1) = 0$$ are real and irrational.

2009 Indonesia TST, 1

Tags: number theory , algebra , quadratic , polynomial

Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.

1994 Poland - Second Round, 1

Tags: polynomial , algebra , polynomial division

Find all real polynomials $P(x)$ of degree $5$ such that $(x-1)^3| P(x)+1$ and $(x+1)^3| P(x)-1$.

1992 IMO Shortlist, 9

Tags: algebra , polynomial , functional equation , iteration

Let $ f(x)$ be a polynomial with rational coefficients and $ \alpha$ be a real number such that \[ \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}.\] Prove that for each $ n \geq 1,$ \[ \left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992},\] where $ f^{n}(x) \equal{} f(f(\cdots f(x))),$ and $ n$ is a positive integer.

2009 Italy TST, 1

Tags: algebra , polynomial

Let $n$ be an even positive integer. An $n$-degree monic polynomial $P(x)$ has $n$ real roots (not necessarily distinct). Suppose $y$ is a positive real number such that for any real number $t<y$, we have $P(t)>0$. Prove that \[P(0)^{\frac{1}{n}}-P(y)^{\frac{1}{n}}\ge y.\]

2007 All-Russian Olympiad, 1

Tags: polynomial , algebra , quadratic

Given reals numbers $a$, $b$, $c$. Prove that at least one of three equations $x^{2}+(a-b)x+(b-c)=0$, $x^{2}+(b-c)x+(c-a)=0$, $x^{2}+(c-a)x+(a-b)=0$ has a real root. [i]O. Podlipsky[/i]

2015 IFYM, Sozopol, 7

Tags: polynomial , algebra

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial.

2008 Harvard-MIT Mathematics Tournament, 10

Tags: floor function , polynomial , function , algebra

Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.

2009 India IMO Training Camp, 5

Tags: algebra , complex number , polynomial

Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients. We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that $ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$. Prove that there exists $ a,b,c\in\mathbb{C}$ such that $ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.