Olympiad problems

1979 Brazil National Olympiad, 2

The remainder on dividing the polynomial $p(x)$ by $x^2 - (a+b)x + ab$ (where $a \not = b$) is $mx + n$. Find the coefficients $m, n$ in terms of $a, b$. Find $m, n$ for the case $p(x) = x^{200}$ divided by $x^2 - x - 2$ and show that they are integral.

2010 All-Russian Olympiad Regional Round, 9.8

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

For every positive integer $n$, let $S_n$ be the sum of the first $n$ prime numbers: $S_1 = 2, S_2 = 2 + 3 = 5, S_3 = 2 + 3 + 5 = 10$, etc. Can both $S_n$ and $S_{n+1}$ be perfect squares?

2014 AIME Problems, 5

Tags: function , algebra , polynomial , quadratics , Vieta , email , AMC

Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.

2005 Postal Coaching, 5

Tags: trigonometry , quadratics , geometry , angle bisector , algebra , geometry solved

Characterize all triangles $ABC$ s.t. \[ AI_a : BI_b : CI_c = BC: CA : AB \] where $I_a$ etc. are the corresponding excentres to the vertices $A, B , C$

PEN P Problems, 31

Tags: quadratics , Additive Number Theory

A finite sequence of integers $a_{0}, a_{1}, \cdots, a_{n}$ is called quadratic if for each $i \in \{1,2,\cdots,n \}$ we have the equality $\vert a_{i}-a_{i-1} \vert = i^2$. [list=a] [*] Prove that for any two integers $b$ and $c$, there exists a natural number $n$ and a quadratic sequence with $a_{0}=b$ and $a_{n}=c$. [*] Find the smallest natural number $n$ for which there exists a quadratic sequence with $a_{0}=0$ and $a_{n}=1996$. [/list]

2005 Iran MO (3rd Round), 1

Tags: quadratics , number theory proposed , number theory

Find all $n,p,q\in \mathbb N$ that:\[2^n+n^2=3^p7^q\]

2001 China Western Mathematical Olympiad, 1

Tags: floor function , function , calculus , quadratics , algebra

Find all real numbers $ x$ such that $ \lfloor x^3 \rfloor \equal{} 4x \plus{} 3$.

2011 Tuymaada Olympiad, 4

Tags: Euler , quadratics , number theory unsolved , number theory

Prove that, among $100000$ consecutive $100$-digit positive integers, there is an integer $n$ such that the length of the period of the decimal expansion of $\frac1n$ is greater than $2011$.

PEN H Problems, 3

Tags: quadratics , Diophantine Equations

Does there exist a solution to the equation \[x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65\] in integers with $x, y, z, u, v$ greater than $1998$?

1977 AMC 12/AHSME, 27

Tags: geometry , 3D geometry , sphere , quadratics , Vieta , AMC

There are two spherical balls of different sizes lying in two corners of a rectangular room, each touching two walls and the floor. If there is a point on each ball which is $5$ inches from each wall which that ball touches and $10$ inches from the floor, then the sum of the diameters of the balls is $\textbf{(A) }20\text{ inches}\qquad\textbf{(B) }30\text{ inches}\qquad\textbf{(C) }40\text{ inches}\qquad$ $\textbf{(D) }60\text{ inches}\qquad \textbf{(E) }\text{not determined by the given information}$

2001 National Olympiad First Round, 3

Tags: modular arithmetic , quadratics

How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to square of an integer? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of the preceding} $

2002 Hong kong National Olympiad, 4

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

Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.

2012 ELMO Shortlist, 6

Tags: floor function , algebra , polynomial , Vieta , quadratics , inequalities , number theory proposed

Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

1993 National High School Mathematics League, 7

Tags: complex numbers , imaginary numbers , algebra , quadratics

Equation $(1-\text{i})x^2+(\lambda+\text{i})x+(1+\text{i}\lambda)=0(\lambda\in\mathbb{R})$ has two imaginary roots, then the range value of $\lambda$ is________.

1992 AIME Problems, 8

Tags: quadratics , algebra , polynomial , AMC , AIME , USA(J)MO , USAMO

For any sequence of real numbers $A=(a_1,a_2,a_3,\ldots)$, define $\Delta A$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)$, whose $n^\text{th}$ term is $a_{n+1}-a_n$. Suppose that all of the terms of the sequence $\Delta(\Delta A)$ are $1$, and that $a_{19}=a_{92}=0$. Find $a_1$.

2012 Albania National Olympiad, 2

Tags: analytic geometry , algebra , polynomial , quadratics , algebra unsolved

The trinomial $f(x)$ is such that $(f(x))^3-f(x)=0$ has three real roots. Find the y-coordinate of the vertex of $f(x)$.

2023 All-Russian Olympiad, 1

Tags: quadratics , algebra

Given are two monic quadratics $f(x), g(x)$ such that $f, g, f+g$ have two distinct real roots. Suppose that the difference of the roots of $f$ is equal to the difference of the roots of $g$. Prove that the difference of the roots of $f+g$ is not bigger than the above common difference.

2006 Taiwan National Olympiad, 3

Tags: quadratics , calculus , integration , algebra , algebra proposed

If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$.

PEN C Problems, 3

Tags: quadratics , floor function , modular arithmetic , Quadratic Residues , pen

Let $p$ be an odd prime number. Show that the smallest positive quadratic nonresidue of $p$ is smaller than $\sqrt{p}+1$.

2009 Princeton University Math Competition, 8

Tags: algebra , polynomial , quadratics , geometry

The real numbers $x$, $y$, $z$, and $t$ satisfy the following equation: \[2x^2 + 4xy + 3y^2 - 2xz -2 yz + z^2 + 1 = t + \sqrt{y + z - t} \] Find 100 times the maximum possible value for $t$.

2011 Moldova Team Selection Test, 2

Tags: quadratics , algebra , polynomial , complex numbers , algebra unsolved

Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations: $x+y+4=\frac{12x+11y}{x^2+y^2}$ $y-x+3=\frac{11x-12y}{x^2+y^2}$

1988 India National Olympiad, 5

Tags: quadratics , number theory proposed , number theory

Show that there do not exist any distinct natural numbers $ a$, $ b$, $ c$, $ d$ such that $ a^3\plus{}b^3\equal{}c^3\plus{}d^3$ and $ a\plus{}b\equal{}c\plus{}d$.

2005 Taiwan National Olympiad, 3

Tags: quadratics , calculus , integration , algebra , algebra proposed

If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$.

2007 India National Olympiad, 3

Tags: quadratics , inequalities , number theory unsolved , number theory

Let $ m$ and $ n$ be positive integers such that $ x^2 \minus{} mx \plus{}n \equal{} 0$ has real roots $ \alpha$ and $ \beta$. Prove that $ \alpha$ and $ \beta$ are integers [b]if and only if[/b] $ [m\alpha] \plus{} [m\beta]$ is the square of an integer. (Here $ [x]$ denotes the largest integer not exceeding $ x$)

2007 Princeton University Math Competition, 3

Tags: quadratics

Find all values of $b$ such that the difference between the maximum and minimum values of $f(x) = x^2-2bx-1$ on the interval $[0, 1]$ is $1$.