Olympiad problems

2003 Baltic Way, 2

Prove that any real solution of $x^3+px+q=0$, where $p,q$ are real numbers, satisfies the inequality $4qx\le p^2$.

2011 Tokio University Entry Examination, 6

Tags: calculus , integration , quadratics , function , algebra , domain , analytic geometry

(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

2014 Postal Coaching, 3

Tags: quadratics , algebra , polynomial , algebra unsolved

Find all real numbers $p$ for which the equation $x^3+3px^2+(4p-1)x+p=0$ has two real roots with difference $1$.

1969 IMO Longlists, 14

Tags: quadratics , algebra , Inequality , roots , polynomial , IMO Longlist , IMO Shortlist

$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and $q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$

2022 South Africa National Olympiad, 2

Tags: quadratics , algebra , system of equations

Find all pairs of real numbers $x$ and $y$ which satisfy the following equations: \begin{align*} x^2 + y^2 - 48x - 29y + 714 & = 0 \\ 2xy - 29x - 48y + 756 & = 0 \end{align*}

2012 ELMO Shortlist, 1

Tags: modular arithmetic , quadratics , algorithm , number theory , number theory proposed

Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square. [i]David Yang, Alex Zhu.[/i]

1959 IMO, 3

Tags: quadratics , trigonometry , algebra , Trigonometric Equations , IMO , IMO 1959

Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ \[ a \cos^2{x}+b \cos{x}+c=0. \] Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$.

2011 Today's Calculation Of Integral, 687

Tags: calculus , integration , quadratics , function , algebra , domain , analytic geometry

(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

2012 AMC 10, 17

Tags: quadratics , number theory , relatively prime , algebra , quadratic formula , AMC

Let $a$ and $b$ be relatively prime integers with $a>b>0$ and $\tfrac{a^3-b^3}{(a-b)^3}=\tfrac{73}{3}$. What is $a-b$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

2006 Estonia Team Selection Test, 1

Tags: algebra , quadratics , trinomial , Integer Polynomial , Integers

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

2004 France Team Selection Test, 1

Tags: quadratics , modular arithmetic , number theory , number theory solved

If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$. Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?

1991 IMO Shortlist, 14

Tags: modular arithmetic , number theory , quadratics , Perfect Square , Discriminant , IMO Shortlist

Let $ a, b, c$ be integers and $ p$ an odd prime number. Prove that if $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ is a perfect square for $ 2p \minus{} 1$ consecutive integer values of $ x,$ then $ p$ divides $ b^2 \minus{} 4ac.$

1979 Romania Team Selection Tests, 4.

Tags: algebra , Polynomials , polynomial , quadratics , inequalities

Give an example of a second degree polynomial $P\in \mathbb{R}[x]$ such that \[\forall x\in \mathbb{R}\text{ with } |x|\leqslant 1: \; \left|P(x)+\frac{1}{x-4}\right| \leqslant 0.01.\] Are there linear polynomials with this property? [i]Octavian Stănășilă[/i]

2003 Putnam, 4

Tags: Putnam , quadratics , function , calculus , derivative , inequalities , absolute value

Suppose that $a, b, c, A, B, C$ are real numbers, $a \not= 0$ and $A \not= 0$, such that \[|ax^2+ bx + c| \le |Ax^2+ Bx + C|\] for all real numbers $x$. Show that \[|b^2- 4ac| \le |B^2- 4AC|\]

PEN H Problems, 39

Tags: quadratics , Diophantine Equations

Let $A, B, C, D, E$ be integers, $B \neq 0$ and $F=AD^{2}-BCD+B^{2}E \neq 0$. Prove that the number $N$ of pairs of integers $(x, y)$ such that \[Ax^{2}+Bxy+Cx+Dy+E=0,\] satisfies $N \le 2 d( \vert F \vert )$, where $d(n)$ denotes the number of positive divisors of positive integer $n$.

1957 AMC 12/AHSME, 2

Tags: quadratics

In the equation $ 2x^2 \minus{} hx \plus{} 2k \equal{} 0$, the sum of the roots is $ 4$ and the product of the roots is $ \minus{}3$. Then $ h$ and $ k$ have the values, respectively: $ \textbf{(A)}\ 8\text{ and }{\minus{}6} \qquad \textbf{(B)}\ 4\text{ and }{\minus{}3}\qquad \textbf{(C)}\ {\minus{}3}\text{ and }4\qquad \textbf{(D)}\ {\minus{}3}\text{ and }8\qquad \textbf{(E)}\ 8\text{ and }{\minus{}3}$

1955 AMC 12/AHSME, 27

Tags: quadratics

If $ r$ and $ s$ are the roots of $ x^2\minus{}px\plus{}q\equal{}0$, then $ r^2\plus{}s^2$ equals: $ \textbf{(A)}\ p^2\plus{}2q \qquad \textbf{(B)}\ p^2\minus{}2q \qquad \textbf{(C)}\ p^2\plus{}q^2 \qquad \textbf{(D)}\ p^2\minus{}q^2 \qquad \textbf{(E)}\ p^2$

2004 Junior Balkan MO, 3

Tags: quadratics , modular arithmetic , number theory proposed , number theory

If the positive integers $x$ and $y$ are such that $3x + 4y$ and $4x + 3y$ are both perfect squares, prove that both $x$ and $y$ are both divisible with $7$.

2024 All-Russian Olympiad Regional Round, 9.2

Tags: quadratics , function graphing , analytic geometry , conics , parabola , geometry , trapezoid

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the diagonals of all such trapezoids share a common point.

2020 Indonesia MO, 2

Tags: quadratics , algebra , polynomial , 2020 , Indonesia MO , Indonesian MO

Problem 2. Let $P(x) = ax^2 + bx + c$ where $a, b, c$ are real numbers. If $$P(a) = bc, \hspace{0.5cm} P(b) = ac, \hspace{0.5cm} P(c) = ab$$ then prove that $$(a - b)(b - c)(c - a)(a + b + c) = 0.$$

2005 Poland - Second Round, 1

Tags: algebra , polynomial , modular arithmetic , quadratics , number theory proposed , number theory

The polynomial $W(x)=x^2+ax+b$ with integer coefficients has the following property: for every prime number $p$ there is an integer $k$ such that both $W(k)$ and $W(k+1)$ are divisible by $p$. Show that there is an integer $m$ such that $W(m)=W(m+1)=0$.

2013 Hitotsubashi University Entrance Examination, 4

Tags: quadratics , geometry proposed , geometry

Let $t$ be a positive constant. Given two points $A(2t,\ 2t,\ 0),\ B(0,\ 0,\ t)$ in a space with the origin $O$. Suppose mobile points $P$ in such way that $\overrightarrow{OP}\cdot \overrightarrow{AP}+\overrightarrow{OP}\cdot \overrightarrow{BP}+\overrightarrow{AP}\cdot \overrightarrow{BP}=3.$ Find the value of $t$ such that the maximum value of $OP$ is 3.

2013 Bundeswettbewerb Mathematik, 4

Tags: modular arithmetic , quadratics , floor function , combinatorics proposed , combinatorics

Two players $A$ and $B$ play the following game taking alternate moves. In each move, a player writes one digit on the blackboard. Each new digit is written either to the right or left of the sequence of digits already written on the blackboard. Suppose that $A$ begins the game and initially the blackboard was empty. $B$ wins the game if ,after some move of $B$, the sequence of digits written in the blackboard represents a perfect square. Prove that $A$ can prevent $B$ from winning.

1999 Brazil Team Selection Test, Problem 5

Tags: quadratics , number theory unsolved , number theory

(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove that $n$ is a power of $2$; (b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such that $2^n-1$ divides $m^2 + 9$.

1984 IMO Longlists, 38

Tags: function , induction , quadratics , algebra , domain , algebra unsolved

Determine all continuous functions $f: \mathbb R \to \mathbb R$ such that \[f(x + y)f(x - y) = (f(x)f(y))^2, \quad \forall(x, y) \in\mathbb{R}^2.\]