This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 1132

1972 Canada National Olympiad, 7
Tags: quadratics , limit , algebra , binomial theorem
a) Prove that the values of $x$ for which $x=(x^2+1)/198$ lie between $1/198$ and $197.99494949\cdots$. b) Use the result of problem a) to prove that $\sqrt{2}<1.41\overline{421356}$. c) Is it true that $\sqrt{2}<1.41421356$?

2009 Princeton University Math Competition, 2
Tags: algebra , polynomial , quadratics
Given that $P(x)$ is the least degree polynomial with rational coefficients such that \[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.

1991 AIME Problems, 8
Tags: quadratics
For how many real numbers $a$ does the quadratic equation $x^2 + ax + 6a=0$ have only integer roots for $x$?

2020 CCA Math Bonanza, T5
Tags: quadratics
Find all pairs of real numbers $(x,y)$ satisfying both equations \[ 3x^2+3xy+2y^2 =2 \] \[ x^2+2xy+2y^2 =1. \] [i]2020 CCA Math Bonanza Team Round #5[/i]

2000 Junior Balkan MO, 1
Tags: quadratics , algebra , quadratic formula
Let $x$ and $y$ be positive reals such that \[ x^3 + y^3 + (x + y)^3 + 30xy = 2000. \] Show that $x + y = 10$.

1971 AMC 12/AHSME, 18
Tags: ratio , quadratics , AMC
The current in a river is flowing steadily at $3$ miles per hour. A motor boat which travels at a constant rate in still water goes downstream $4$ miles and then returns to its starting point. The trip takes one hour, excluding the time spent in turning the boat around. The ratio of the downstream to the upstream rate is $\textbf{(A) }4:3\qquad\textbf{(B) }3:2\qquad\textbf{(C) }5:3\qquad\textbf{(D) }2:1\qquad \textbf{(E) }5:2$

1998 National Olympiad First Round, 4
Tags: quadratics , function
$ x,y,z\in \mathbb R$, find the minimal value of $ f\left(x,y,z\right) = 2x^{2} + 5y^{2} + 10z^{2} - 2xy - 4yz - 6zx + 3$. $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ -3 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None}$

1999 Harvard-MIT Mathematics Tournament, 1
Tags: function , quadratics , algebra
If $a@b=\dfrac{a^3-b^3}{a-b}$, for how many real values of $a$ does $a@1=0$?

2008 National Olympiad First Round, 18
Tags: quadratics
How many positive integers $n$ are there such that $\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n}}}}$ is an integer? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of the above} $

1962 AMC 12/AHSME, 21
Tags: quadratics
It is given that one root of $ 2x^2 \plus{} rx \plus{} s \equal{} 0$, with $ r$ and $ s$ real numbers, is $ 3\plus{}2i (i \equal{} \sqrt{\minus{}1})$. The value of $ s$ is: $ \textbf{(A)}\ \text{undetermined} \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ \minus{}13 \qquad \textbf{(E)}\ 26$

2018 Istmo Centroamericano MO, 4
Tags: trinomial , quadratics , Perfect Squares , number theory
Let $t$ be an integer. Suppose the equation $$x^2 + (4t - 1) x + 4t^2 = 0$$ has at least one positive integer solution $n$. Show that $n$ is a perfect square.

2009 Middle European Mathematical Olympiad, 6
Tags: trigonometry , quadratics , algebra proposed , algebra
Let $ a$, $ b$, $ c$ be real numbers such that for every two of the equations \[ x^2\plus{}ax\plus{}b\equal{}0, \quad x^2\plus{}bx\plus{}c\equal{}0, \quad x^2\plus{}cx\plus{}a\equal{}0\] there is exactly one real number satisfying both of them. Determine all possible values of $ a^2\plus{}b^2\plus{}c^2$.

2007 IMC, 1
Tags: quadratics , algebra , polynomial , modular arithmetic , IMC , college contests
Let $ f$ be a polynomial of degree 2 with integer coefficients. Suppose that $ f(k)$ is divisible by 5 for every integer $ k$. Prove that all coefficients of $ f$ are divisible by 5.

1965 AMC 12/AHSME, 10
Tags: quadratics , inequalities , limit
The statement $ x^2 \minus{} x \minus{} 6 < 0$ is equivalent to the statement: $ \textbf{(A)}\ \minus{} 2 < x < 3 \qquad \textbf{(B)}\ x > \minus{} 2 \qquad \textbf{(C)}\ x < 3$ $ \textbf{(D)}\ x > 3 \text{ and }x < \minus{} 2 \qquad \textbf{(E)}\ x > 3 \text{ and }x < \minus{} 2$

2012 Tuymaada Olympiad, 4
Tags: quadratics , modular arithmetic , number theory , number theory proposed
Let $p=1601$. Prove that if \[\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n},\] where we only sum over terms with denominators not divisible by $p$ (and the fraction $\dfrac {m} {n}$ is in reduced terms) then $p \mid 2m+n$. [i]Proposed by A. Golovanov[/i]

1990 Vietnam Team Selection Test, 1
Tags: inequalities , quadratics , geometry unsolved , geometry
Let be given a convex polygon $ M_0M_1\ldots M_{2n}$ ($ n\ge 1$), where $ 2n \plus{} 1$ points $ M_0$, $ M_1$, $ \ldots$, $ M_{2n}$ lie on a circle $ (C)$ with diameter $ R$ in an anticlockwise direction. Suppose that there is a point $ A$ inside this convex polygon such that $ \angle M_0AM_1$, $ \angle M_1AM_2$, $ \ldots$, $ \angle M_{2n \minus{} 1}AM_{2n}$, $ \angle M_{2n}AM_0$ are equal. Assume that $ A$ is not coincide with the center of the circle $ (C)$ and $ B$ be a point lies on $ (C)$ such that $ AB$ is perpendicular to the diameter of $ (C)$ passes through $ A$. Prove that \[ \frac {2n \plus{} 1}{\frac {1}{AM_0} \plus{} \frac {1}{AM_1} \plus{} \cdots \plus{} \frac {1}{AM_{2n}}} < AB < \frac {AM_0 \plus{} AM_1 \plus{} \cdots \plus{} AM_{2n}}{2n \plus{} 1} < R \]

1955 AMC 12/AHSME, 32
Tags: quadratics , arithmetic sequence , geometric sequence
If the discriminant of $ ax^2\plus{}2bx\plus{}c\equal{}0$ is zero, then another true statement about $ a$, $ b$, and $ c$ is that: $ \textbf{(A)}\ \text{they form an arithmetic progression} \\ \textbf{(B)}\ \text{they form a geometric progression} \\ \textbf{(C)}\ \text{they are unequal} \\ \textbf{(D)}\ \text{they are all negative numbers} \\ \textbf{(E)}\ \text{only b is negative and a and c are positive}$

2000 Spain Mathematical Olympiad, 1
Tags: algebra , polynomial , parameterization , quadratics , algebra unsolved
Consider the polynomials \[P(x) = x^4 + ax^3 + bx^2 + cx + 1 \quad \text{and} \quad Q(x) = x^4 + cx^3 + bx^2 + ax + 1.\] Find the conditions on the parameters $a, b, $c with $a\neq c$ for which $P(x)$ and $Q(x)$ have two common roots and, in such cases, solve the equations $P(x) = 0$ and $Q(x) = 0.$

1994 Turkey MO (2nd round), 5
Tags: quadratics , calculus , integration , number theory unsolved , number theory
Find the set of all ordered pairs $(s,t)$ of positive integers such that \[t^{2}+1=s(s+1).\]

2004 Vietnam National Olympiad, 2
Tags: inequalities , algebra , polynomial , function , limit , quadratics , real analysis
Let $x$, $y$, $z$ be positive reals satisfying $\left(x+y+z\right)^{3}=32xyz$ Find the minimum and the maximum of $P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}$

2022 Girls in Math at Yale, 9
Tags: Yale , college , quadratics
Suppose that $P(x)$ is a monic quadratic polynomial satisfying $aP(a) = 20P(20) = 22P(22)$ for some integer $a\neq 20, 22$. Find the minimum possible positive value of $P(0)$. [i]Proposed by Andrew Wu[/i] (Note: wording changed from original to specify that $a \neq 20, 22$.)

2013 AIME Problems, 14
Tags: trigonometry , quadratics , ratio , number theory , relatively prime , trig identities , geometric series
For $\pi\leq\theta<2\pi$, let \[ P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots \] and \[ Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\theta-\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta +\ldots \] so that $\tfrac PQ = \tfrac{2\sqrt2}7$. Then $\sin\theta = -\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2013 AMC 10, 11
Tags: quadratics , special factorizations , algebra , quadratic formula , AMC
Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34$. What is $x+y$? $ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $

2010 Greece National Olympiad, 1
Tags: quadratics , number theory proposed , number theory
Solve in the integers the diophantine equation $$x^4-6x^2+1 = 7 \cdot 2^y.$$

2010 Tuymaada Olympiad, 3
Tags: quadratics , algebra , polynomial , Vieta , absolute value , algebra unsolved
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 $f(1)=0$.