Olympiad problems

1957 AMC 12/AHSME, 10

Tags: analytic geometry , function , conic , parabola , quadratic , derivative , calculus

The graph of $ y \equal{} 2x^2 \plus{} 4x \plus{} 3$ has its: $ \textbf{(A)}\ \text{lowest point at } {(\minus{}1,9)}\qquad \textbf{(B)}\ \text{lowest point at } {(1,1)}\qquad \\ \textbf{(C)}\ \text{lowest point at } {(\minus{}1,1)}\qquad \textbf{(D)}\ \text{highest point at } {(\minus{}1,9)}\qquad \\ \textbf{(E)}\ \text{highest point at } {(\minus{}1,1)}$

2012 ELMO Shortlist, 1

Tags: algorithm , number theory , quadratic , modular arithmetic

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

2014 Online Math Open Problems, 18

Tags: quadratic

Find the number of pairs $(m,n)$ of integers with $-2014\le m,n\le 2014$ such that $x^3+y^3 = m + 3nxy$ has infinitely many integer solutions $(x,y)$. [i]Proposed by Victor Wang[/i]

2007 All-Russian Olympiad Regional Round, 11.2

Tags: algebra , quadratic , polynomial

Two quadratic polynomials $ f_{1},f_{2}$ satisfy $ f_{1}'(x)f_{2}'(x)\geq |f_{1}(x)|\plus{}|f_{2}(x)|\forall x\in\mathbb{R}$ . Prove that $ f_{1}\cdot f_{2}\equal{} g^{2}$ for some $ g\in\mathbb{R}[x]$.

1991 Arnold's Trivium, 18

Tags: matrix , integration , linear algebra , quadratic , gauss , calculus

Calculate \[\int\cdots\int \exp\left(-\sum_{1\le i\le j\le n}x_ix_j\right)dx_1\cdots dx_n\]

2015 India National Olympiad, 6

Tags: number theory , quadratic , pigeonhole principle , modular arithmetic

Show that from a set of $11$ square integers one can select six numbers $a^2,b^2,c^2,d^2,e^2,f^2$ such that $a^2+b^2+c^2 \equiv d^2+e^2+f^2\pmod{12}$.

2004 Vietnam National Olympiad, 3

Tags: calculus , number theory , integration , quadratic , modular arithmetic

Let $ S(n)$ be the sum of decimal digits of a natural number $ n$. Find the least value of $ S(m)$ if $ m$ is an integral multiple of $ 2003$.

1991 Arnold's Trivium, 95

Tags: rotation , algebra , geometry , invariant , quadratic , geometric transformation , polynomial

Decompose the space of homogeneous polynomials of degree $5$ in $(x, y, z)$ into irreducible subspaces invariant with respect to the rotation group $SO(3)$.

1996 AMC 12/AHSME, 25

Tags: inequalities , analytic geometry , search , graphing lines , quadratic , geometry , slope

Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have? $\text{(A)}\ 72 \qquad \text{(B)}\ 73 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 75\qquad \text{(E)}\ 76$

2008 iTest Tournament of Champions, 3

Tags: quadratic

Simon and Garfunkle play in a round-robin golf tournament. Each player is awarded one point for a victory, a half point for a tie, and no points for a loss. Simon beat Garfunkle in the ﬁrst game by a record margin as Garfunkle sent a shot over the bridge and into troubled waters on the ﬁnal hole. Garfunkle went on to score $8$ total victories, but no ties at all. Meanwhile, Simon wound up with exactly $8$ points, including the point for a victory over Garfunkle. Amazingly, every other player at the tournament scored exactly $n$. Find the sum of all possible values of $n$.

PEN P Problems, 31

Tags: quadratic , 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]

2003 India Regional Mathematical Olympiad, 6

Tags: polynomial , algebra , quadratic

Find all real numbers $a$ for which the equation $x^2a- 2x + 1 = 3 |x|$ has exactly three distinct real solutions in $x$.

2013 AMC 12/AHSME, 17

Tags: inequalities , quadratic , am-gm , optimization , algebra

Let $a,b,$ and $c$ be real numbers such that \begin{align*} a+b+c &= 2, \text{ and} \\ a^2+b^2+c^2&= 12 \end{align*} What is the difference between the maximum and minimum possible values of $c$? $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ \frac{10}{3}\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ \frac{16}{3}\qquad\textbf{(E)}\ \frac{20}{3} $

2007 IMC, 1

Tags: polynomial , quadratic , modular arithmetic , algebra

Let $ f$ be a polynomial of degree 2 with integer coefficients. Suppose that $ f(k)$ is divisible by 5 for every integer $ k$. Prove that all coefficients of $ f$ are divisible by 5.

1973 Putnam, B5

Tags: function , algebra , quadratic , rational function

(a) Let $z$ be a solution of the quadratic equation $$az^2 +bz+c=0$$ and let $n$ be a positive integer. Show that $z$ can be expressed as a rational function of $z^n , a,b,c.$ (b) Using (a) or by any other means, express $x$ as a rational function of $x^{3}$ and $x+\frac{1}{x}.$

2007 Princeton University Math Competition, 3

Tags: algebra , calculus , geometry , integration , analytic geometry , quadratic , quadratic formula

For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?

2022 All-Russian Olympiad, 6

Tags: algebra , quadratic

What is the smallest natural number $a$ for which there are numbers $b$ and $c$ such that the quadratic trinomial $ax^2 + bx + c$ has two different positive roots not exceeding $\frac {1}{1000}$?

2010 Today's Calculation Of Integral, 525

Tags: geometry , trigonometry , inequalities , integration , quadratic , calculus , vector

Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

2003 Federal Competition For Advanced Students, Part 1, 3

Tags: algebra , quadratic , polynomial

Given a positive real number $t$, find the number of real solutions $a, b, c, d$ of the system \[a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t.\]

2013 Stanford Mathematics Tournament, 3

Tags: quadratic , algebra , polynomial

Karl likes the number $17$ his favorite polynomials are monic quadratics with integer coefficients such that $17$ is a root of the quadratic and the roots differ by no more than $17$. Compute the sum of the coefficients of all of Karl's favorite polynomials. (A monic quadratic is a quadratic polynomial whose $x^2$ term has a coefficient of $1$.)

2006 Czech and Slovak Olympiad III A, 2

Tags: algebra , quadratic

Let $m,n$ be positive integers such that the equation (in respect of $x$) \[(x+m)(x+n)=x+m+n\] has at least one integer root. Prove that $\frac{1}{2}n<m<2n$.

2011 Mongolia Team Selection Test, 1

Tags: quadratic , ring theory , diophantine equation , modular arithmetic , number theory

Let $A=\{a^2+13b^2 \mid a,b \in\mathbb{Z}, b\neq0\}$. Prove that there a) exist b) exist infinitely many $x,y$ integer pairs such that $x^{13}+y^{13} \in A$ and $x+y \notin A$. (proposed by B. Bayarjargal)

1994 AIME Problems, 7

Tags: system of equations , geometry , quadratic , linear equation , analytic geometry , algebra

For certain ordered pairs $(a,b)$ of real numbers, the system of equations \begin{eqnarray*} && ax+by =1\\ &&x^2+y^2=50\end{eqnarray*} has at least one solution, and each solution is an ordered pair $(x,y)$ of integers. How many such ordered pairs $(a,b)$ are there?

2006 All-Russian Olympiad, 7

Tags: quadratic , quadratic equation , algebra

Given a quadratic trinomial $f\left(x\right)=x^2+ax+b$. Assume that the equation $f\left(f\left(x\right)\right)=0$ has four different real solutions, and that the sum of two of these solutions is $-1$. Prove that $b\leq -\frac14$.

PEN H Problems, 21

Tags: quadratic , diophantine equation

Prove that the equation \[6(6a^{2}+3b^{2}+c^{2}) = 5n^{2}\] has no solutions in integers except $a=b=c=n=0$.