Olympiad problems

2000 USAMO, 2

Let $S$ be the set of all triangles $ABC$ for which \[ 5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r}, \] where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. Prove that all triangles in $S$ are isosceles and similar to one another.

2007 ITest, 36

Tags: probability , quadratic , number theory , relatively prime

Let $b$ be a real number randomly sepected from the interval $[-17,17]$. Then, $m$ and $n$ are two relatively prime positive integers such that $m/n$ is the probability that the equation \[x^4+25b^2=(4b^2-10b)x^2\] has $\textit{at least}$ two distinct real solutions. Find the value of $m+n$.

2024 Belarus Team Selection Test, 4.2

Tags: quadratic , algebra , polynomial

Let $f(x)=x^2+bx+c$, where $b,c \in \mathbb{R}$ and $b>0$ Do there exist disjoint sets $A$ and $B$, whose union is $[0,1]$ and $f(A)=B$, where $f(X)=\{f(x), x \in X\}$ [i]D. Zmiaikou[/i]

2012 Today's Calculation Of Integral, 798

Tags: calculus , integration , function , derivative , quadratic , analytic geometry , geometry

Denote by $C,\ l$ the graphs of the cubic function $C: y=x^3-3x^2+2x$, the line $l: y=ax$. (1) Find the range of $a$ such that $C$ and $l$ have intersection point other than the origin. (2) Denote $S(a)$ by the area bounded by $C$ and $l$. If $a$ move in the range found in (1), then find the value of $a$ for which $S(a)$ is minimized. 50 points

2011 ELMO Shortlist, 7

Tags: quadratic , search , trigonometry , algebra

Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$. [i]Alex Zhu.[/i]

2010 ELMO Shortlist, 3

Tags: quadratic , diophantine equation , number theory

Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that \begin{align*} a^2 + b^2 + 3 &= 4ab\\ c^2 + d^2 + 3 &= 4cd\\ 4c^3 - 3c &= a \end{align*} [i]Travis Hance.[/i]

2006 USA Team Selection Test, 3

Tags: inequalities , function , quadratic , three variable inequality

Find the least real number $k$ with the following property: if the real numbers $x$, $y$, and $z$ are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\]

2004 Finnish National High School Mathematics Competition, 1

Tags: quadratic , algebra

The equations $x^2 +2ax+b^2 = 0$ and $x^2 +2bx+c^2 = 0$ both have two different real roots. Determine the number of real roots of the equation $x^2 + 2cx + a^2 = 0.$

2024 Auckland Mathematical Olympiad, 11

Tags: algebra , function , quadratic , polynomial

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 $b \neq d$.

2007 Princeton University Math Competition, 3

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

2015 Harvard-MIT Mathematics Tournament, 8

Tags: quadratic , abstract algebra , algebra , polynomial

Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds: [list] [*] there exist polynomials $P$, $Q$ with integer coefficients such that $f(x)^2-(ax+b)=(x^2+1)P(x)+35Q(x)$; [*] or more conceptually, the remainder when (the polynomial) $f(x)^2-(ax+b)$ is divided by (the polynomial) $x^2+1$ is a polynomial with integer coefficients all divisible by $35$. [/list]

2004 France Team Selection Test, 1

Tags: number theory , quadratic , modular arithmetic

If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$. Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?

2014 Contests, 1

Tags: algebra , polynomial , quadratic

Find all the polynomials with real coefficients which satisfy $ (x^2-6x+8)P(x)=(x^2+2x)P(x-2)$ for all $x\in \mathbb{R}$.

2015 AMC 12/AHSME, 20

Tags: geometry , perimeter , quadratic , area of a triangle , heron's formula , algebra , quadratic formula

Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Which of the following numbers is closest to $b$? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

2011 Purple Comet Problems, 9

Tags: algebra , polynomial , quadratic , quadratic formula , sum of roots , product of roots

There are integers $m$ and $n$ so that $9 +\sqrt{11}$ is a root of the polynomial $x^2 + mx + n.$ Find $m + n.$

Oliforum Contest IV 2013, 2

Tags: circumcircle , quadratic , algebra , quadratic formula , power of a point , geometry

Given an acute angled triangle $ABC$ with $M$ being the mid-point of $AB$ and $P$ and $Q$ are the feet of heights from $A$ to $BC$ and $B$ to $AC$ respectively. Show that if the line $AC$ is tangent to the circumcircle of $BMP$ then the line $BC$ is tangent to the circumcircle of $AMQ$.

2004 Vietnam National Olympiad, 2

Tags: inequalities , algebra , polynomial , function , limit , quadratic , real analysis

Let $x$, $y$, $z$ be positive reals satisfying $\left(x+y+z\right)^{3}=32xyz$ Find the minimum and the maximum of $P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}$

2007 Harvard-MIT Mathematics Tournament, 5

Tags: geometry , rectangle , quadratic

A convex quadrilateral is determined by the points of intersection of the curves $x^4+y^4=100$ and $xy=4$; determine its area.

1989 IMO Longlists, 18

Tags: quadratic , combinatorics

There are some boys and girls sitting in an $ n \times n$ quadratic array. We know the number of girls in every column and row and every line parallel to the diagonals of the array. For which $ n$ is this information sufficient to determine the exact positions of the girls in the array? For which seats can we say for sure that a girl sits there or not?

2000 Harvard-MIT Mathematics Tournament, 16

Tags: quadratic

Solve for real $x,y$: $x+y=2$ $x^5+y^5=82$

2001 Federal Competition For Advanced Students, Part 2, 2

Tags: quadratic , number theory , pco

Determine all integers $m$ for which all solutions of the equation $3x^3-3x^2+m = 0$ are rational.

1991 Arnold's Trivium, 87

Tags: calculus , derivative , quadratic , 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 $\epsilon = 0$.

2015 Romania National Olympiad, 2

Tags: function , algebra , polynomial , quadratic

A quadratic function has the property that for any interval of length $ 1, $ the length of its image is at least $ 1. $ Show that for any interval of length $ 2, $ the length of its image is at least $ 4. $

2020 MMATHS, I10

Tags: quadratic

Let $f(x)$ be a quadratic polynomial such that $f(f(1)) = f(-f(-1)) = 0$ and $f(1) \neq -f(-1)$. Suppose furthermore that the quadratic $2f(x)$ has coefficients that are nonzero integers. Find $f(0)$. [i]Proposed by Andrew Wu[/i]

1959 AMC 12/AHSME, 34

Tags: vieta , irrational number , complex number , imaginary numbers , quadratic

Let the roots of $x^2-3x+1=0$ be $r$ and $s$. Then the expression $r^2+s^2$ is: $ \textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}$ $\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}$