Olympiad problems

2013 Indonesia MO, 5

Let $P$ be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics $P_1, P_2, P_3$ such that: - $P(x) = P_1(x) + P_2(x) + P_3(x)$ - $P_1, P_2, P_3$ have positive leading coefficients and zero discriminants (and hence each has a double root) - The roots of $P_1, P_2, P_3$ are different

2001 Polish MO Finals, 3

Tags: quadratics , pigeonhole principle , combinatorics proposed , combinatorics

Given positive integers $n_1<n_2<...<n_{2000}<10^{100}$. Prove that we can choose from the set $\{n_1,...,n_{2000}\}$ nonempty, disjont sets $A$ and $B$ which have the same number of elements, the same sum and the same sum of squares.

2000 Taiwan National Olympiad, 1

Tags: Euler , function , modular arithmetic , quadratics , number theory , prime numbers , number theory unsolved

Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

2005 AIME Problems, 8

Tags: quadratics , geometry , trapezoid , number theory , relatively prime , Pythagorean Theorem , algebra

Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3$. The radii of $C_1$ and $C_2$ are $4$ and $10$, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2$. Given that the length of the chord is $\frac{m\sqrt{n}}{p}$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p$.

2023 All-Russian Olympiad, 1

Tags: trigonometry , quadratics

If $x\in\mathbb{R}$ satisfy $sin$ $x+tan$ $x\in\mathbb{Q}$, $cos$ $x+cot$ $x\in\mathbb{Q}$ Prove that $sin$ $2x$ is a root of an integral coefficient quadratic function

1992 IMO Shortlist, 2

Tags: quadratics , algebra , functional equation , recurrence relation , IMO Shortlist

Let $ \mathbb{R}^\plus{}$ be the set of all non-negative real numbers. Given two positive real numbers $ a$ and $ b,$ suppose that a mapping $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ satisfies the functional equation: \[ f(f(x)) \plus{} af(x) \equal{} b(a \plus{} b)x.\] Prove that there exists a unique solution of this equation.

2008 AMC 10, 20

Tags: probability , function , quadratics , complementary counting , AMC

The faces of a cubical die are marked with the numbers $ 1$, $ 2$, $ 2$, $ 3$, $ 3$, and $ 4$. The faces of a second cubical die are marked with the numbers $ 1$, $ 3$, $ 4$, $ 5$, $ 6$, and $ 8$. Both dice are thrown. What is the probability that the sum of the two top numbers will be $ 5$, $ 7$, or $ 9$ ? $ \textbf{(A)}\ \frac {5}{18} \qquad \textbf{(B)}\ \frac {7}{18} \qquad \textbf{(C)}\ \frac {11}{18} \qquad \textbf{(D)}\ \frac {3}{4} \qquad \textbf{(E)}\ \frac {8}{9}$

PEN P Problems, 17

Tags: quadratics , Additive Number Theory , pen

Let $p$ be a prime number of the form $4k+1$. Suppose that $r$ is a quadratic residue of $p$ and that $s$ is a quadratic nonresidue of $p$. Show that $p=a^{2}+b^{2}$, where \[a=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-r)}{p}\right), b=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-s)}{p}\right).\] Here, $\left( \frac{k}{p}\right)$ denotes the Legendre Symbol.

1987 IMO Longlists, 69

Tags: number theory , polynomial , prime numbers , quadratics , IMO Shortlist , IMO , IMO 1987

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i] [b][i]Original Formulation[/i][/b] Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. [i]Proposed by Soviet Union. [/i]

2006 Junior Tuymaada Olympiad, 5

Tags: algebra , quadratic trinomial , trinomial , quadratics , constant , Sides of a triangle

The quadratic trinomials $ f $, $ g $ and $ h $ are such that for every real $ x $ the numbers $ f (x) $, $ g (x) $ and $ h (x) $ are the lengths of the sides of some triangles, and the numbers $ f (x) -1 $, $ g (x) -1 $ and $ h (x) -1 $ are not the lengths of the sides of the triangle. Prove that at least of the polynomials $ f + g-h $, $ f + h-g $, $ g + h-f $ is constant.

2004 India National Olympiad, 5

Tags: quadratics , number theory , algebra unsolved , algebra

S is the set of all ($a$, $b$, $c$, $d$, $e$, $f$) where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides abcdef for all members of $S$.

2004 AIME Problems, 12

Tags: geometry , trapezoid , analytic geometry , quadratics , USAMTS , algebra

Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6$, $BC=5=DA$, and $CD=4$. Draw circles of radius 3 centered at $A$ and $B$, and circles of radius 2 centered at $C$ and $D$. A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}p$, where $k$, $m$, $n$, and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p$.

2014 India IMO Training Camp, 1

Tags: quadratics , algebra , quadratic formula , algebra unsolved

Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.

2006 China Second Round Olympiad, 7

Tags: trigonometry , quadratics

Let $f(x)=\sin^4x-\sin x\cos x+cos^4 x$. Find the range of $f(x)$.

PEN H Problems, 34

Tags: quadratics , Diophantine Equations

Are there integers $m$ and $n$ such that $5m^2 -6mn+7n^2 =1985$?

2005 Taiwan TST Round 2, 3

Tags: quadratics , number theory , Sequence , relatively prime , IMO Shortlist

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

2018 CCA Math Bonanza, T10

Tags: quadratics

The irrational number $\alpha>1$ satisfies $\alpha^2-3\alpha-1=0$. Given that there is a fraction $\frac{m}{n}$ such that $n<500$ and $\left|\alpha-\frac{m}{n}\right|<3\cdot10^{-6}$, find $m$. [i]2018 CCA Math Bonanza Team Round #10[/i]

2004 China Team Selection Test, 1

Tags: function , quadratics , algebra unsolved , algebra

Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.

2006 USA Team Selection Test, 3

Tags: inequalities , function , quadratics , three variable inequality

Find the least real number $k$ with the following property: if the real numbers $x$, $y$, and $z$ are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\]

2010 ELMO Shortlist, 3

Tags: quadratics , Diophantine equation , number theory proposed , number theory

Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that \begin{align*} a^2 + b^2 + 3 &= 4ab\\ c^2 + d^2 + 3 &= 4cd\\ 4c^3 - 3c &= a \end{align*} [i]Travis Hance.[/i]

2009 Stanford Mathematics Tournament, 9

Tags: conics , parabola , calculus , algebra , polynomial , derivative , quadratics

Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x=\frac{y^2}{2}$

2004 China Team Selection Test, 1

Tags: function , quadratics , algebra unsolved , algebra

Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.

2008 Czech-Polish-Slovak Match, 3

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

Find all primes $p$ such that the expression \[\binom{p}1^2+\binom{p}2^2+\cdots+\binom{p}{p-1}^2\] is divisible by $p^3$.

2007 Stanford Mathematics Tournament, 9

Tags: quadratics

Find $a^2+b^2$ given that $a, b$ are real and satisfy \[a=b+\frac{1}{a+\frac{1}{b+\frac{1}{a+\cdots}}}; b=a-\frac{1}{b+\frac{1}{a-\frac{1}{b+\cdots}}}\]

2012 Indonesia TST, 3

Tags: quadratics , geometry , rectangle , vector , geometry unsolved

The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$. Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.