Olympiad problems

1999 Baltic Way, 4

For all positive real numbers $x$ and $y$ let \[f(x,y)=\min\left( x,\frac{y}{x^2+y^2}\right) \] Show that there exist $x_0$ and $y_0$ such that $f(x, y)\le f(x_0, y_0)$ for all positive $x$ and $y$, and find $f(x_0,y_0)$.

2005 Today's Calculation Of Integral, 27

Tags: calculus , integration , trigonometry , quadratics , algebra , calculus computations

Let $f(x)=t\sin x+(1-t)\cos x\ (0\leqq t\leqq 1)$. Find the maximum and minimum value of the following $P(t)$. \[P(t)=\left\{\int_0^{\frac{\pi}{2}} e^x f(x) dx \right\}\left\{\int_0^{\frac{\pi}{2}} e^{-x} f(x)dx \right\}\]

2000 Baltic Way, 17

Tags: Vieta , algebra , polynomial , quadratics , system of equations , algebra proposed

Find all real solutions to the following system of equations: \[\begin{cases} x+y+z+t=5\\xy+yz+zt+tx=4\\xyz+yzt+ztx+txy=3\\xyzt=-1\end{cases}\]

2007 AIME Problems, 15

Tags: geometry , ratio , trigonometry , quadratics , AMC , AIME , USA(J)MO

Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA=5$ and $CD=2$. Point $E$ lies on side $CA$ such that $\angle DEF = 60^\circ$. The area of triangle $DEF$ is $14\sqrt{3}$. The two possible values of the length of side $AB$ are $p \pm q\sqrt{r}$, where $p$ and $q$ are rational, and $r$ is an integer not divisible by the square of a prime. Find $r$.

1997 Turkey Junior National Olympiad, 1

Tags: quadratics , algebra , quadratic formula

Solve the equation $\sqrt {a-\sqrt{a+x}}=x$ in real numbers in terms of the real number $a>1$.

2011 AMC 12/AHSME, 21

Tags: quadratics , algebra , difference of squares , special factorizations , AMC

The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x-y|$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70 $

2007 AMC 12/AHSME, 23

Tags: logarithms , geometry , quadratics

Square $ ABCD$ has area $ 36,$ and $ \overline{AB}$ is parallel to the x-axis. Vertices $ A,$ $ B,$ and $ C$ are on the graphs of $ y \equal{} \log_{a}x,$ $ y \equal{} 2\log_{a}x,$ and $ y \equal{} 3\log_{a}x,$ respectively. What is $ a?$ $ \textbf{(A)}\ \sqrt [6]{3}\qquad \textbf{(B)}\ \sqrt {3}\qquad \textbf{(C)}\ \sqrt [3]{6}\qquad \textbf{(D)}\ \sqrt {6}\qquad \textbf{(E)}\ 6$

2013 All-Russian Olympiad, 2

Tags: quadratics , algebra , polynomial , arithmetic sequence , algebra proposed

Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Basil could have called out?

2011 Morocco National Olympiad, 2

Tags: quadratics , algebra unsolved , algebra

Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$ ($p$ and $q$ are two real parameters).

2005 India Regional Mathematical Olympiad, 7

Tags: quadratics

Let $a,b,c$ be three positive real numbers such that $a+ b +c =1$. Let $\lambda = min \{ a^3 + a^2bc , b^3 + b^2 ac , c^3 + ab c^2 \}$ Prove that the roots of $x^2 + x + 4 \lambda = 0$ are real.

1998 Turkey MO (2nd round), 1

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

Find all positive integers $x$ and $n$ such that ${{x}^{3}}+3367={{2}^{n}}$.

1969 IMO Shortlist, 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 $x^2 - ax + b = 0$.

2007 All-Russian Olympiad, 1

Tags: algebra , polynomial , quadratics , algebra proposed

Given reals numbers $a$, $b$, $c$. Prove that at least one of three equations $x^{2}+(a-b)x+(b-c)=0$, $x^{2}+(b-c)x+(c-a)=0$, $x^{2}+(c-a)x+(a-b)=0$ has a real root. [i]O. Podlipsky[/i]

2008 iTest Tournament of Champions, 3

Tags: quadratics

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 $n$.

2013 Stars Of Mathematics, 3

Tags: quadratics , number theory proposed , number theory

Consider the sequence $(a^n + 1)_{n\geq 1}$, with $a>1$ a fixed integer. i) Prove there exist infinitely many primes, each dividing some term of the sequence. ii) Prove there exist infinitely many primes, none dividing any term of the sequence. [i](Dan Schwarz)[/i]

2014 Contests, 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 $1$.

2007 Princeton University Math Competition, 10

Tags: algebra , polynomial , quadratics , Rational Root Theorem

Find the real root of $x^5+5x^3+5x-1$. Hint: Let $x = u+k/u$.

2006 AIME Problems, 5

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

When rolling a certain unfair six-sided die with faces numbered $1, 2, 3, 4, 5$, and $6$, the probability of obtaining face $F$ is greater than $\frac{1}{6}$, the probability of obtaining the face opposite is less than $\frac{1}{6}$, the probability of obtaining any one of the other four faces is $\frac{1}{6}$, and the sum of the numbers on opposite faces is $7$. When two such dice are rolled, the probability of obtaining a sum of $7$ is $\frac{47}{288}$. Given that the probability of obtaining face $F$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

1971 AMC 12/AHSME, 19

Tags: conics , ellipse , quadratics , algebra , quadratic formula , AMC

If the line $y=mx+1$ intersects the ellipse $x^2+4y^2=1$ exactly once, then the value of $m^2$ is $\textbf{(A) }\textstyle\frac{1}{2}\qquad\textbf{(B) }\frac{2}{3}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{5}\qquad \textbf{(E) }\frac{5}{6}$

1960 AMC 12/AHSME, 31

Tags: quadratics , AMC

For $x^2+2x+5$ to be a factor of $x^4+px^2+q$, the values of $p$ and $q$ must be, respectively: $ \textbf{(A)}\ -2, 5\qquad\textbf{(B)}\ 5, 25\qquad\textbf{(C)}\ 10, 20\qquad\textbf{(D)}\ 6, 25\qquad\textbf{(E)}\ 14, 25 $

2006 Junior Balkan Team Selection Tests - Moldova, 4

Tags: quadratics , algebra , algebra proposed

Determine all real solutions of the equation: \[{ \frac{x^{2}}{x-1}+\sqrt{x-1}+\frac{\sqrt{x-1}}{x^{2}}}=\frac{x-1}{x^{2}}+\frac{1}{\sqrt{x-1}}+\frac{x^{2}}{\sqrt{x-1}} . \]

1952 AMC 12/AHSME, 13

Tags: quadratics , function , conics , parabola , calculus

The function $ x^2 \plus{} px \plus{} q$ with $ p$ and $ q$ greater than zero has its minimum value when: $ \textbf{(A)}\ x \equal{} \minus{} p \qquad\textbf{(B)}\ x \equal{} \frac {p}{2} \qquad\textbf{(C)}\ x \equal{} \minus{} 2p \qquad\textbf{(D)}\ x \equal{} \frac {p^2}{4q} \qquad\textbf{(E)}\ x \equal{} \frac { \minus{} p}{2}$

1994 Turkey Team Selection Test, 3

Tags: algebra , polynomial , Vieta , quadratics , number theory proposed , number theory

Find all integer pairs $(a,b)$ such that $a\cdot b$ divides $a^2+b^2+3$.

1980 AMC 12/AHSME, 24

Tags: algebra , polynomial , quadratics , Rational Root Theorem

For some real number $r$, the polynomial $8x^3-4x^2-42x+45$ is divisible by $(x-r)^2$. Which of the following numbers is closest to $r$? $\text{(A)} \ 1.22 \qquad \text{(B)} \ 1.32 \qquad \text{(C)} \ 1.42 \qquad \text{(D)} \ 1.52 \qquad \text{(E)} \ 1.62$

2015 India National Olympiad, 6

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

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 $a^2+b^2+c^2 \equiv d^2+e^2+f^2\pmod{12}$.