Olympiad problems

1974 Polish MO Finals, 3

Let $r$ be a natural number. Prove that the quadratic trinomial $x^2 - rx- 1$ does not divide any nonzero polynomial whose coefficients are integers with absolute values less than $r$.

1962 Czech and Slovak Olympiad III A, 1

Tags: quadratic polynomial , perfect square , prime , algebra

Determine all integers $x$ such that $2x^2-x-36$ is a perfect square of a prime.

2009 Puerto Rico Team Selection Test, 4

Tags: algebra , quadratic polynomial , trinomial

Find all integers $ b$ and $ c$ such that the equation $ x^2 - bx + c = 0$ has two real roots $ x_1, x_2$ satisfying $ x_1^2 + x_2^2 = 5$.

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 $x>0,x_1>x_2$.

2005 Junior Balkan Team Selection Tests - Moldova, 8

Tags: trinomial , quadratic , quadratic polynomial , algebra , function , inequalities

The families of second degree functions $f_m, g_m: R\to R, $ are considered , $f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1$, $g_m (x) = m^2x^2 + mx - 1$, where $m$ is a real nonzero parameter. Show that, for any function $h$ of the second degree with the property that $g_m (x) \le h (x) \le f_m (x)$ for any real $x$, there exists $\lambda \in [0, 1]$ which verifies the condition $h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)$, whatever real $x$ is.

2016 Saudi Arabia GMO TST, 1

Tags: algebra , quadratic polynomial , quadratic

Let $f (x) = x^2 + ax + b$ be a quadratic function with real coefficients $a, b$. It is given that the equation $f (f (x)) = 0$ has $4$ distinct real roots and the sum of $2$ roots among these roots is equal to $-1$. Prove that $b \le -\frac14$

1955 Czech and Slovak Olympiad III A, 4

Tags: equation , quadratic polynomial

Given that $a,b,c$ are distinct real numbers, show that the equation \[\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}=0\] has a real root.

2000 Czech And Slovak Olympiad IIIA, 4

Tags: algebra , polynomial , quadratic polynomial , trinomial , quadratic

For which quadratic polynomials $f(x)$ does there exist a quadratic polynomial $g(x)$ such that the equations $g(f(x)) = 0$ and $f(x)g(x) = 0$ have the same roots, which are mutually distinct and form an arithmetic progression?

2003 Switzerland Team Selection Test, 7

Tags: polynomial , trinomial , quadratic polynomial , algebra

Find all polynomials $Q(x)= ax^2+bx+c$ with integer coefficients for which there exist three different prime numbers $p_1, p_2, p_3$ such that $|Q(p_1)| = |Q(p_2)| = |Q(p_3)| = 11$.

1954 Czech and Slovak Olympiad III A, 1

Tags: quadratic polynomial , equation , parameter , parameterization

Solve the equation $$ax^2+2(a-1)x+a-5=0$$ in real numbers with respect to (real) parametr $a$.

1964 Czech and Slovak Olympiad III A, 3

Tags: algebra , parameterization , quadratic , quadratic polynomial

Determine all values of parameter $\alpha\in [0,2\pi]$ such that the equation $$(2\cos\alpha-1)x^2+4x+4\cos\alpha+2=0$$ has 1) a positive root $x_1$, 2) if a second root $x_2$ exists and if $x_2\neq x_1$, the $x_2\leq 0$.

2013 Saudi Arabia Pre-TST, 2.2

Tags: algebra , max , quadratic polynomial

The quadratic equation $ax^2 + bx + c = 0$ has its roots in the interval $[0, 1]$. Find the maximum of $\frac{(a - b)(2a - b)}{a(a - b + c)}$.

1969 Czech and Slovak Olympiad III A, 1

Tags: algebra , equation , quadratic polynomial , rational

Find all rational numbers $x,y$ such that \[\left(x+y\sqrt5\right)^2=7+3\sqrt5.\]

2016 Saudi Arabia BMO TST, 1

Tags: algebra , polynomial , quadratic polynomial , quadratic

Let $P_i(x) = x^2 + b_i x + c_i , i = 1,2, ..., n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \le i < j \le n , i, j \in N$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.