Olympiad problems

2010 Contests, 2

Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.

2013 Romanian Masters In Mathematics, 1

Tags: floor function , modular arithmetic , quadratics , number theory

For a positive integer $a$, define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$. Let $y_n=2^{x_n}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1,\ldots,y_k$ are all prime.

1991 Arnold's Trivium, 87

Tags: calculus , derivative , quadratics , vector , linear algebra , matrix , function

Find the derivatives of the lengths of the semiaxes of the ellipsoid $x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy$ with respect to $\epsilon$ at $\epsilon = 0$.

2007 AIME Problems, 14

Tags: algebra , polynomial , quadratics , function , probability , functional equation , AMC

Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$

2003 AMC 10, 5

Tags: Vieta , quadratics , algebra , polynomial

Let $ d$ and $ e$ denote the solutions of $ 2x^2\plus{}3x\minus{}5\equal{}0$. What is the value of $ (d\minus{}1)(e\minus{}1)$? $ \textbf{(A)}\ \minus{}\frac{5}{2} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

2007 China Team Selection Test, 1

Tags: inequalities , trigonometry , quadratics , function , Hi

$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$ prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$

2003 Purple Comet Problems, 13

Tags: algebra , polynomial , function , quadratics

Let $P(x)$ be a polynomial such that, when divided by $x - 2$, the remainder is $3$ and, when divided by $x - 3$, the remainder is $2$. If, when divided by $(x - 2)(x - 3)$, the remainder is $ax + b$, find $a^2 + b^2$.

2011 AIME Problems, 15

Tags: algebra , polynomial , probability , floor function , AMC , AIME , quadratics

Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.

2006 AMC 10, 8

Tags: quadratics , conics , parabola , AMC

A parabola with equation $ y \equal{} x^2 \plus{} bx \plus{} c$ passes through the points $ (2,3)$ and $ (4,3)$. What is $ c$? $ \textbf{(A) } 2 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11$

PEN D Problems, 10

Tags: modular arithmetic , quadratics , Congruences

Let $p$ be a prime number of the form $4k+1$. Suppose that $2p+1$ is prime. Show that there is no $k \in \mathbb{N}$ with $k<2p$ and $2^k \equiv 1 \; \pmod{2p+1}$.

2001 Tuymaada Olympiad, 3

Tags: quadratics , algebra , polynomial , function , algebra proposed

Do there exist quadratic trinomials $P, \ \ Q, \ \ R$ such that for every integers $x$ and $y$ an integer $z$ exists satisfying $P(x)+Q(y)=R(z)?$ [i]Proposed by A. Golovanov[/i]

2007 Junior Balkan MO, 4

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

Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.

PEN A Problems, 64

Tags: quadratics , Divisibility Theory

The last digit of the number $x^2 +xy+y^2$ is zero (where $x$ and $y$ are positive integers). Prove that two last digits of this numbers are zeros.

2015 India National Olympiad, 5

Tags: geometry , trigonometry , vector , parallelogram , analytic geometry , quadratics , geometry unsolved

Let $ABCD$ be a convex quadrilateral.Let diagonals $AC$ and $BD$ intersect at $P$. Let $PE,PF,PG$ and $PH$ are altitudes from $P$ on the side $AB,BC,CD$ and $DA$ respectively. Show that $ABCD$ has a incircle if and only if $\frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH}.$

2014 Online Math Open Problems, 18

Tags: Online Math Open , quadratics

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 Federal Competition For Advanced Students, Part 1, 1

Tags: quadratics , number theory unsolved , number theory

In a quadratic table with $ 2007$ rows and $ 2007$ columns is an odd number written in each field. For $ 1\leq i\leq2007$ is $ Z_i$ the sum of the numbers in the $ i$-th row and for $ 1\leq j\leq2007$ is $ S_j$ the sum of the numbers in the $ j$-th column. $ A$ is the product of all $ Z_i$ and $ B$ the product of all $ S_j$. Show that $ A\plus{}B\neq0$

2004 Iran Team Selection Test, 1

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

Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]

1990 Baltic Way, 13

Tags: quadratics

Show that the equation $x^2-7y^2 = 1$ has infinitely many solutions in natural numbers.

2006 Romania Team Selection Test, 2

Tags: quadratics , real analysis , complex numbers , algebra , polynomial , sum of roots , number theory proposed

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

2013 North Korea Team Selection Test, 3

Tags: modular arithmetic , quadratics , number theory , North Korea , TST , Divisibility

Find all $ a, b, c \in \mathbb{Z} $, $ c \ge 0 $ such that $ a^n + 2^n | b^n + c $ for all positive integers $ n $ where $ 2ab $ is non-square.

2010 AMC 12/AHSME, 15

Tags: probability , quadratics , algebra , quadratic formula , AMC

A coin is altered so that the probability that it lands on heads is less than $ \frac {1}{2}$ and when the coin is flipped four times, the probability of an equal number of heads and tails is $ \frac {1}{6}$. What is the probability that the coin lands on heads? $ \textbf{(A)}\ \frac {\sqrt {15} \minus{} 3}{6}\qquad \textbf{(B)}\ \frac {6 \minus{} \sqrt {6\sqrt {6} \plus{} 2}}{12}\qquad \textbf{(C)}\ \frac {\sqrt {2} \minus{} 1}{2}\qquad \textbf{(D)}\ \frac {3 \minus{} \sqrt {3}}{6}\qquad \textbf{(E)}\ \frac {\sqrt {3} \minus{} 1}{2}$

2016 Indonesia TST, 2

Tags: invariant , symmetry , quadratics , algebra , polynomial , Vieta , number theory solved

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

2003 CHKMO, 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 $\{x\}=x-[x]$.

1969 IMO Longlists, 44

Tags: geometry , circumcircle , radius , quadratics , IMO Shortlist , IMO Longlist

$(MON 5)$ Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation $x^2 - ax + b = 0$.

2016 Philippine MO, 1

Tags: algebra , invariant , Discriminant , quadratics

The operations below can be applied on any expression of the form $ax^2+bx+c$. $(\text{I})$ If $c \neq 0$, replace $a$ by $4a-\frac{3}{c}$ and $c$ by $\frac{c}{4}$. $(\text{II})$ If $a \neq 0$, replace $a$ by $-\frac{a}{2}$ and $c$ by $-2c+\frac{3}{a}$. $(\text{III}_t)$ Replace $x$ by $x-t$, where $t$ is an integer. (Different values of $t$ can be used.) Is it possible to transform $x^2-x-6$ into each of the following by applying some sequence of the above operations? $(\text{a})$ $5x^2+5x-1$ $(\text{b})$ $x^2+6x+2$