Olympiad problems

2009 Vietnam Team Selection Test, 3

Let a, b be positive integers. a, b and a.b are not perfect squares. Prove that at most one of following equations $ ax^2 \minus{} by^2 \equal{} 1$ and $ ax^2 \minus{} by^2 \equal{} \minus{} 1$ has solutions in positive integers.

2007 All-Russian Olympiad Regional Round, 11.2

Tags: polynomial , quadratic , algebra

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]$.

2005 All-Russian Olympiad, 2

Tags: calculus , integration , quadratic , algebra

Find the number of subsets $A\subset M=\{2^0,\,2^1,\,2^2,\dots,2^{2005}\}$ such that equation $x^2-S(A)x+S(B)=0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B=M\setminus A$ ($A$ and $B$ are not empty).

2006 China Northern MO, 4

Tags: quadratic , function , algebra

Given a function $f(x)=x^{2}+ax+b$ with $a,b \in R$, if there exists a real number $m$ such that $\left| f(m) \right| \leq \frac{1}{4}$ and $\left| f(m+1) \right| \leq \frac{1}{4}$, then find the maximum and minimum of the value of $\Delta=a^{2}-4b$.

1998 USAMTS Problems, 2

Tags: quadratic , arithmetic series , algebra , quadratic formula

There are inﬁnitely many ordered pairs $(m,n)$ of positive integers for which the sum \[ m + ( m + 1) + ( m + 2) +... + ( n - 1 )+n\] is equal to the product $mn$. The four pairs with the smallest values of $m$ are $(1, 1), (3, 6), (15, 35),$ and $(85, 204)$. Find three more $(m, n)$ pairs.

1991 AIME Problems, 14

Tags: trigonometry , symmetry , quadratic , geometry , circumcircle

A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{AB}$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$.

2009 USAMO, 4

Tags: inequalities , induction , quadratic , floor function

For $ n\geq2$ let $ a_1, a_2, \ldots a_n$ be positive real numbers such that \[ (a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n)\left(\frac {1}{a_1} \plus{} \frac {1}{a_2} \plus{} \cdots \plus{} \frac {1}{a_n}\right) \leq \left(n \plus{} \frac {1}{2}\right)^2. \] Prove that $ \max(a_1, a_2, \ldots, a_n)\leq 4\min(a_1, a_2, \ldots, a_n)$.

1999 India Regional Mathematical Olympiad, 6

Tags: quadratic

Find all solutions in integers $m,n$ of the equation \[ (m-n)^2 = \frac{4mn}{ m+n-1}. \]

1997 AIME Problems, 12

Tags: function , limit , quadratic , linear algebra

The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$. where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$. Find the unique number that is not in the range of $f$.

2009 Hong Kong TST, 5

Tags: inequalities , quadratic , function , algebra

Let $ a,b,c$ be the three sides of a triangle. Determine all possible values of $ \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca}$

1999 Brazil Team Selection Test, Problem 5

Tags: quadratic , number theory

(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove that $n$ is a power of $2$; (b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such that $2^n-1$ divides $m^2 + 9$.

1991 AIME Problems, 13

Tags: probability , floor function , inequalities , quadratic , complementary counting

A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $1/2$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?

2010 Hanoi Open Mathematics Competitions, 6

Tags: number theory , quadratic , integer

Let $a,b$ be the roots of the equation $x^2-px+q = 0$ and let $c, d$ be the roots of the equation $x^2 - rx + s = 0$, where $p, q, r,s$ are some positive real numbers. Suppose that $M =\frac{2(abc + bcd + cda + dab)}{p^2 + q^2 + r^2 + s^2}$ is an integer. Determine $a, b, c, d$.

2010 Contests, 2

Tags: polynomial , quadratic , algebra

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 $P(x_0)<Q(x_0)$.

PEN B Problems, 1

Tags: primitive root , quadratic , modular arithmetic

Let $n$ be a positive integer. Show that there are infinitely many primes $p$ such that the smallest positive primitive root of $p$ is greater than $n$.

2013 Hitotsubashi University Entrance Examination, 4

Tags: quadratic , geometry

Let $t$ be a positive constant. Given two points $A(2t,\ 2t,\ 0),\ B(0,\ 0,\ t)$ in a space with the origin $O$. Suppose mobile points $P$ in such way that $\overrightarrow{OP}\cdot \overrightarrow{AP}+\overrightarrow{OP}\cdot \overrightarrow{BP}+\overrightarrow{AP}\cdot \overrightarrow{BP}=3.$ Find the value of $t$ such that the maximum value of $OP$ is 3.

1992 Taiwan National Olympiad, 1

Tags: trigonometry , quadratic , geometry

Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$. The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\hat{BAC}<\frac{\pi}{8}$ then $S_{ABC}<2S_{A'B'C'}$.

2008 Saint Petersburg Mathematical Olympiad, 5

Tags: modular arithmetic , quadratic , number theory

Given are distinct natural numbers $a$, $b$, and $c$. Prove that \[ \gcd(ab+1, ac+1, bc+1)\le \frac{a+b+c}{3}\]

2023 All-Russian Olympiad, 1

Tags: trigonometry , quadratic

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

1990 AIME Problems, 4

Tags: quadratic

Find the positive solution to \[ \frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0 \]

2005 District Olympiad, 4

Tags: geometry , circumcircle , incenter , ratio , quadratic , angle bisector , perpendicular bisector

In the triangle $ABC$ let $AD$ be the interior angle bisector of $\angle ACB$, where $D\in AB$. The circumcenter of the triangle $ABC$ coincides with the incenter of the triangle $BCD$. Prove that $AC^2 = AD\cdot AB$.

2010 Contests, 1

Tags: quadratic , number theory

Solve in the integers the diophantine equation $$x^4-6x^2+1 = 7 \cdot 2^y.$$

1980 AMC 12/AHSME, 14

Tags: function , quadratic , algebra , system of equations

If the function $f$ is defined by \[ f(x)=\frac{cx}{2x+3} , ~~~x\neq -\frac 32 , \] satisfies $x=f(f(x))$ for all real numbers $x$ except $-\frac 32$, then $c$ is $\text{(A)} \ -3 \qquad \text{(B)} \ - \frac{3}{2} \qquad \text{(C)} \ \frac{3}{2} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$

1984 AMC 12/AHSME, 21

Tags: quadratic , algebra

The number of triples $(a,b,c)$ of positive integers which satisfy the simultaneous equations \begin{align*} ab+bc &= 44,\\ ac+bc &= 23, \end{align*} is $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }4$

1973 Putnam, B5

Tags: quadratic , algebra , rational function , 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 $x+\frac{1}{x}.$