Olympiad problems

2012 Dutch BxMO/EGMO TST, 1

Tags: trinomial , quadratic , root , algebra , polynomial

Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?

1994 AMC 12/AHSME, 20

Tags: quadratic , geometric sequence , ratio , arithmetic sequence

Suppose $x,y,z$ is a geometric sequence with common ratio $r$ and $x \neq y$. If $x, 2y, 3z$ is an arithmetic sequence, then $r$ is $ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4$

1985 AMC 12/AHSME, 19

Tags: quadratic , special factorizations , function

Consider the graphs $ y \equal{} Ax^2$ and and $ y^2 \plus{} 3 \equal{} x^2 \plus{} 4y$, where $ A$ is a positive constant and $ x$ and $ y$ are real variables. In how many points do the two graphs intersect? $ \textbf{(A)}\ \text{exactly } 4 \qquad \textbf{(B)}\ \text{exactly } 2$ $ \textbf{(C)}\ \text{at least } 1, \text{ but the number varies for different positive values of } A$ $ \textbf{(D)}\ 0 \text{ for at least one positive value of } A \qquad \textbf{(E)}\ \text{none of these}$

1993 All-Russian Olympiad, 3

Tags: quadratic , algebra

Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$?

2024 Auckland Mathematical Olympiad, 11

Tags: quadratic , function , polynomial , algebra

It is known that for quadratic polynomials $P(x)=x^2+ax+b$ and $Q(x)=x^2+cx+d$ the equation $P(Q(x))=Q(P(x))$ does not have real roots. Prove that $b \neq d$.

2010 Tuymaada Olympiad, 1

Tags: quadratic , integration , calculus , algebra , polynomial

Baron Münchausen boasts that he knows a remarkable quadratic triniomial with positive coefficients. The trinomial has an integral root; if all of its coefficients are increased by $1$, the resulting trinomial also has an integral root; and if all of its coefficients are also increased by $1$, the new trinomial, too, has an integral root. Can this be true?

2021 Taiwan Mathematics Olympiad, 2.

Tags: arithmetic sequence , quadratic , number theory

Find all integers $n=2k+1>1$ so that there exists a permutation $a_0, a_1,\ldots,a_{k}$ of $0, 1, \ldots, k$ such that \[a_1^2-a_0^2\equiv a_2^2-a_1^2\equiv \cdots\equiv a_{k}^2-a_{k-1}^2\pmod n.\] [i]Proposed by usjl[/i]

1991 AIME Problems, 13

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

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?

1971 Bundeswettbewerb Mathematik, 4

Tags: quadratic , modular arithmetic

Let $P$ and $Q$ be two horizontal neighbouring squares on a $n \times n$ chess board, $P$ on the left and $Q$ on the right. On the left square $P$ there is a stone that shall be moved around the board. The following moves are allowed: 1) move it one square upwards 2) move it one square to the right 3) move it one square down and one square to the left (diagonal movement) Example: you can get from $e5$ to $f5$, $e6$ and $d4$. Show that for no $n$ there is tour visting every square exactly once and ending in $Q$.

2005 All-Russian Olympiad, 2

Tags: quadratic , algebra , integration , calculus

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).

2013 Romanian Masters In Mathematics, 1

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

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.

2003 Moldova Team Selection Test, 1

Tags: quadratic , diophantine equation , relatively prime , number theory

Let $ n\in N^*$. A permutation $ (a_1,a_2,...,a_n)$ of the numbers $ (1,2,...,n)$ is called [i]quadratic [/i] iff at least one of the numbers $ a_1,a_1\plus{}a_2,...,a_1\plus{}a_2\plus{}a\plus{}...\plus{}a_n$ is a perfect square. Find the greatest natural number $ n\leq 2003$, such that every permutation of $ (1,2,...,n)$ is quadratic.

2009 USAMTS Problems, 2

Tags: algebra , quadratic , quadratic formula

Let $a, b, c, d$ be four real numbers such that \begin{align*}a + b + c + d &= 8, \\ ab + ac + ad + bc + bd + cd &= 12.\end{align*} Find the greatest possible value of $d$.

1975 Canada National Olympiad, 4

Tags: algebra , quadratic , floor function , ratio , calculus , integration , geometric sequence

For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.

1950 AMC 12/AHSME, 13

Tags: quadratic

The roots of $ (x^2\minus{}3x\plus{}2)(x)(x\minus{}4)\equal{}0$ are: $\textbf{(A)}\ 4\qquad \textbf{(B)}\ 0\text{ and }4 \qquad \textbf{(C)}\ 1\text{ and }2 \qquad \textbf{(D)}\ 0,1,2\text{ and }4\qquad \textbf{(E)}\ 1,2\text{ and }4$

1963 AMC 12/AHSME, 28

Tags: quadratic

Given the equation $3x^2 - 4x + k = 0$ with real roots. The value of $k$ for which the product of the roots of the equation is a maximum is: $\textbf{(A)}\ \dfrac{16}{9} \qquad \textbf{(B)}\ \dfrac{16}{3}\qquad \textbf{(C)}\ \dfrac{4}{9} \qquad \textbf{(D)}\ \dfrac{4}{3} \qquad \textbf{(E)}\ -\dfrac{4}{3}$

2015 AMC 10, 23

Tags: quadratic , vieta , sfft , function , polynomial , algebra

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

2010 Contests, 1

Tags: quadratic , quadratic formula , algebra

Compute \[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}}\]

2010 Tuymaada Olympiad, 3

Tags: quadratic , vieta , absolute value , polynomial , algebra

Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself). Show that $f(1)=0$.

2008 IberoAmerican, 3

Tags: quadratic , polynomial , algebra , number theory

Let $ P(x) \equal{} x^3 \plus{} mx \plus{} n$ be an integer polynomial satisfying that if $ P(x) \minus{} P(y)$ is divisible by 107, then $ x \minus{} y$ is divisible by 107 as well, where $ x$ and $ y$ are integers. Prove that 107 divides $ m$.

2006 Taiwan National Olympiad, 3

Tags: quadratic , algebra , calculus , integration

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 $p$.

2005 MOP Homework, 6

Tags: algebra , quadratic , system of equations

Solve the system of equations: $x^2=\frac{1}{y}+\frac{1}{z}$, $y^2=\frac{1}{z}+\frac{1}{x}$, $z^2=\frac{1}{x}+\frac{1}{y}$. in the real numbers.

2014 National Olympiad First Round, 11

Tags: quadratic

What is the product of real numbers $a$ which make $x^2+ax+1$ a negative integer for only one real number $x$? $ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ -2 \qquad\textbf{(C)}\ -4 \qquad\textbf{(D)}\ -6 \qquad\textbf{(E)}\ -8 $

2008 AIME Problems, 10

Tags: quadratic , trapezoid , trigonometry , inequalities , geometry

Let $ ABCD$ be an isosceles trapezoid with $ \overline{AD}\parallel{}\overline{BC}$ whose angle at the longer base $ \overline{AD}$ is $ \dfrac{\pi}{3}$. The diagonals have length $ 10\sqrt {21}$, and point $ E$ is at distances $ 10\sqrt {7}$ and $ 30\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \overline{AD}$. The distance $ EF$ can be expressed in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

1999 Flanders Math Olympiad, 3

Tags: quadratic

Determine all $f: \mathbb{R}\rightarrow\mathbb{R}$ for which \[ 2\cdot f(x)-g(x)=f(y)-y \textrm{ and } f(x)\cdot g(x) \geq x+1. \]