Found problems: 236
2013 AIME Problems, 12
Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.
1996 Taiwan National Olympiad, 4
Show that for any real numbers $a_{3},a_{4},...,a_{85}$, not all the roots of the equation $a_{85}x^{85}+a_{84}x^{84}+...+a_{3}x^{3}+3x^{2}+2x+1=0$ are real.
2021 Science ON grade X, 3
Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies
$$|a^{2021}-N|<2^{-1000}.$$
[i] (Vlad Robu) [/i]
1973 USAMO, 4
Determine all roots, real or complex, of the system of simultaneous equations
\begin{align*} x+y+z &= 3, \\
x^2+y^2+z^2 &= 3, \\
x^3+y^3+z^3 &= 3.\end{align*}
2008 China National Olympiad, 3
Find all triples $(p,q,n)$ that satisfy
\[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\]
where $p,q$ are odd primes and $n$ is an positive integer.
2015 AMC 10, 23
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$?
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
1959 AMC 12/AHSME, 27
Which one of the following is [i] not [/i] true for the equation \[ix^2-x+2i=0,\] where $i=\sqrt{-1}$?
$ \textbf{(A)}\ \text{The sum of the roots is 2} \qquad$
$\textbf{(B)}\ \text{The discriminant is 9}\qquad$
$\textbf{(C)}\ \text{The roots are imaginary}\qquad$
$\textbf{(D)}\ \text{The roots can be found using the quadratic formula}\qquad$
$\textbf{(E)}\ \text{The roots can be found by factoring, using imaginary numbers} $
PEN A Problems, 112
Prove that there exist infinitely many pairs $(a, b)$ of relatively prime positive integers such that \[\frac{a^{2}-5}{b}\;\; \text{and}\;\; \frac{b^{2}-5}{a}\] are both positive integers.
2014 Harvard-MIT Mathematics Tournament, 4
Let $b$ and $c$ be real numbers and define the polynomial $P(x)=x^2+bx+c$. Suppose that $P(P(1))=P(P(2))=0$, and that $P(1) \neq P(2)$. Find $P(0)$.
2010 AMC 10, 21
The polynomial $ x^3\minus{}ax^2\plus{}bx\minus{}2010$ has three positive integer zeros. What is the smallest possible value of $ a$?
$ \textbf{(A)}\ 78 \qquad
\textbf{(B)}\ 88 \qquad
\textbf{(C)}\ 98 \qquad
\textbf{(D)}\ 108 \qquad
\textbf{(E)}\ 118$
2005 International Zhautykov Olympiad, 3
Find all prime numbers $ p,q < 2005$ such that $ q | p^{2} \plus{} 8$ and $ p|q^{2} \plus{} 8.$
2008 ITest, 79
Done with her new problems, Wendy takes a break from math. Still without any fresh reading material, she feels a bit antsy. She starts to feel annoyed that Michael's loose papers clutter the family van. Several of them are ripped, and bits of paper litter the floor. Tired of trying to get Michael to clean up after himself, Wendy spends a couple of minutes putting Michael's loose papers in the trash. "That seems fair to me," confirms Hannah encouragingly.
While collecting Michael's scraps, Wendy comes across a corner of a piece of paper with part of a math problem written on it. There is a monic polynomial of degree $n$, with real coefficients. The first two terms after $x^n$ are $a_{n-1}x^{n-1}$ and $a_{n-2}x^{n-2}$, but the rest of the polynomial is cut off where Michael's page is ripped. Wendy barely makes out a little of Michael's scribbling, showing that $a_{n-1}=-a_{n-2}$. Wendy deciphers the goal of the problem, which is to find the sum of the squares of the roots of the polynomial. Wendy knows neither the value of $n$, nor the value of $a_{n-1}$, but still she finds a [greatest] lower bound for the answer to the problem. Find the absolute value of that lower bound.
1960 AMC 12/AHSME, 1
If $2$ is a solution (root) of $x^3+hx+10=0$, then $h$ equals:
$ \textbf{(A) }10\qquad\textbf{(B) }9 \qquad\textbf{(C) }2\qquad\textbf{(D) }-2\qquad\textbf{(E) }-9 $
2007 Today's Calculation Of Integral, 241
1.Let $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta )$ are $ x$ coordinates of the intersection points of a parabola $ y \equal{} ax^2 \plus{} bx \plus{} c\ (a\neq 0)$ and the line $ y \equal{} ux \plus{} v$.
Prove that the area of the region bounded by these graphs is $ \boxed{\frac {|a|}{6}(\beta \minus{} \alpha )^3}$.
2. Let $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta )$ are $ x$ coordinates of the intersection points of parabolas $ y \equal{} ax^2 \plus{} bx \plus{} c$ and $ y \equal{} px^2 \plus{} qx \plus{} r\ (ap\neq 0)$.
Prove that the area of the region bounded by these graphs is $ \boxed{\frac {|a \minus{} p|}{6}(\beta \minus{} \alpha )^3}$.
2013 Iran Team Selection Test, 5
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?
[i]Proposed by Mahan Malihi[/i]
2013 Stanford Mathematics Tournament, 23
Let $a$ and $b$ be the solutions to $x^2-7x+17=0$. Compute $a^4+b^4$.
1965 AMC 12/AHSME, 19
If $ x^4 \plus{} 4x^3 \plus{} 6px^2 \plus{} 4qx \plus{} r$ is exactly divisible by $ x^3 \plus{} 3x^2 \plus{} 9x \plus{} 3$, the value of $ (p \plus{} q)r$ is:
$ \textbf{(A)}\ \minus{} 18 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 45 \qquad$
2005 AIME Problems, 8
The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
2012 European Mathematical Cup, 3
Are there positive real numbers $x$, $y$ and $z$ such that
$ x^4 + y^4 + z^4 = 13\text{,} $
$ x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,} $
$ x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?} $
[i]Proposed by Matko Ljulj.[/i]
2010 BMO TST, 2
Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2\minus{}ax\plus{}1\equal{}0$ we build the sequence with $ S_{n}\equal{}x_{1}^n \plus{} x_{2}^n$.
[b]a)[/b]Prove that the sequence $ \frac{S_{n}}{S_{n\plus{}1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing.
[b]b)[/b]Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1$
2008 Federal Competition For Advanced Students, Part 2, 2
(a) Does there exist a polynomial $ P(x)$ with coefficients in integers, such that $ P(d) \equal{} \frac{2008}{d}$ holds for all positive divisors of $ 2008$?
(b) For which positive integers $ n$ does a polynomial $ P(x)$ with coefficients in integers exists, such that $ P(d) \equal{} \frac{n}{d}$ holds for all positive divisors of $ n$?
2010 AMC 12/AHSME, 17
Equiangular hexagon $ ABCDEF$ has side lengths $ AB \equal{} CD \equal{} EF \equal{} 1$ and $ BC \equal{} DE \equal{} FA \equal{} r$. The area of $ \triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $ r$?
$ \textbf{(A)}\ \frac {4\sqrt {3}}{3} \qquad
\textbf{(B)}\ \frac {10}{3} \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ \frac {17}{4} \qquad
\textbf{(E)}\ 6$
2013 AMC 12/AHSME, 22
Let $m>1$ and $n>1$ be integers. Suppose that the product of the solutions for $x$ of the equation
\[8(\log_n x)(\log_m x) - 7 \log_n x - 6 \log_m x - 2013 = 0\]
is the smallest possible integer. What is $m+n$?
${ \textbf{(A)}\ 12\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 48\qquad\textbf{(E)}\ 272 $
1997 Brazil National Olympiad, 4
Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$, where $F_n$ is the Fibonacci sequence
($F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$)
Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$
1978 IMO Longlists, 25
Consider a polynomial $P(x) = ax^2 + bx + c$ with $a > 0$ that has two real roots $x_1, x_2$. Prove that the absolute values of both roots are less than or equal to $1$ if and only if $a + b + c \ge 0, a -b + c \ge 0$, and $a - c \ge 0$.