Found problems: 236
1982 Vietnam National Olympiad, 1
Determine a quadric polynomial with intergral coefficients whose roots are $\cos 72^{\circ}$ and $\cos 144^{\circ}.$
2013 Iran MO (3rd Round), 4
Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation:
\[x^2 - (n^2 +1)y^2 = n^2.\]
(25 points)
2002 AMC 12/AHSME, 6
Suppose that $ a$ and $ b$ are are nonzero real numbers, and that the equation $ x^2\plus{}ax\plus{}b\equal{}0$ has solutions $ a$ and $ b$. Then the pair $ (a,b)$ is
$ \textbf{(A)}\ (\minus{}2,1) \qquad
\textbf{(B)}\ (\minus{}1,2) \qquad
\textbf{(C)}\ (1,\minus{}2) \qquad
\textbf{(D)}\ (2,\minus{}1) \qquad
\textbf{(E)}\ (4,4)$
1985 IberoAmerican, 3
Find all the roots $ r_{1}$, $ r_{2}$, $ r_{3}$ y $ r_{4}$ of the equation $ 4x^{4}\minus{}ax^{3}\plus{}bx^{2}\minus{}cx\plus{}5 \equal{} 0$, knowing that they are real, positive and that:
\[ \frac{r_{1}}{2}\plus{}\frac{r_{2}}{4}\plus{}\frac{r_{3}}{5}\plus{}\frac{r_{4}}{8}\equal{} 1.\]
1950 AMC 12/AHSME, 3
The sum of the roots of the equation $ 4x^2\plus{}5\minus{}8x\equal{}0$ is equal to:
$\textbf{(A)}\ 8 \qquad
\textbf{(B)}\ -5 \qquad
\textbf{(C)}\ -\dfrac{5}{4} \qquad
\textbf{(D)}\ -2 \qquad
\textbf{(E)}\ \text{None of these}$
2018-2019 Winter SDPC, 1
Let $r_1$, $r_2$, $r_3$ be the distinct real roots of $x^3-2019x^2-2020x+2021=0$. Prove that $r_1^3+r_2^3+r_3^3$ is an integer multiple of $3$.
PEN A Problems, 4
If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.
1954 AMC 12/AHSME, 41
The sum of all the roots of $ 4x^3\minus{}8x^2\minus{}63x\minus{}9\equal{}0$ is:
$ \textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \minus{}8 \qquad
\textbf{(D)}\ \minus{}2 \qquad
\textbf{(E)}\ 0$
2011 NIMO Problems, 9
The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$.
[i]Proposed by Eugene Chen
[/i]
2010 AMC 10, 19
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$
2009 CHKMO, 2
Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.
1967 AMC 12/AHSME, 17
If $r_1$ and $r_2$ are the distinct real roots of $x^2+px+8=0$, then it must follow that:
$\textbf{(A)}\ |r_1+r_2|>4\sqrt{2}\qquad
\textbf{(B)}\ |r_1|>3 \; \text{or} \; |r_2| >3 \\
\textbf{(C)}\ |r_1|>2 \; \text{and} \; |r_2|>2\qquad
\textbf{(D)}\ r_1<0 \; \text{and} \; r_2<0\qquad
\textbf{(E)}\ |r_1+r_2|<4\sqrt{2}$
2010 Brazil National Olympiad, 1
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.
2003 All-Russian Olympiad, 1
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
1998 All-Russian Olympiad, 1
The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$. Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc.
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$
1977 AMC 12/AHSME, 23
If the solutions of the equation $x^2+px+q=0$ are the cubes of the solutions of the equation $x^2+mx+n=0$, then
$\textbf{(A) }p=m^3+3mn\qquad\textbf{(B) }p=m^3-3mn\qquad$
$\textbf{(C) }p+q=m^3\qquad\textbf{(D) }\left(\frac{m}{n}\right)^2=\frac{p}{q}\qquad \textbf{(E) }\text{none of these}$
1959 AMC 12/AHSME, 44
The roots of $x^2+bx+c=0$ are both real and greater than $1$. Let $s=b+c+1$. Then $s:$
$ \textbf{(A)}\ \text{may be less than zero}\qquad\textbf{(B)}\ \text{may be equal to zero}\qquad$ $\textbf{(C)}\ \text{must be greater than zero}\qquad\textbf{(D)}\ \text{must be less than zero}\qquad $
$\textbf{(E)}\text{ must be between -1 and +1} $
2007 Irish Math Olympiad, 1
Let $ r,s,$ and $ t$ be the roots of the cubic polynomial: $ p(x)\equal{}x^3\minus{}2007x\plus{}2002.$
Determine the value of: $ \frac{r\minus{}1}{r\plus{}1}\plus{}\frac{s\minus{}1}{s\plus{}1}\plus{}\frac{t\minus{}1}{t\plus{}1}$.
1954 AMC 12/AHSME, 25
The two roots of the equation $ a(b\minus{}c)x^2\plus{}b(c\minus{}a)x\plus{}c(a\minus{}b)\equal{}0$ are $ 1$ and:
$ \textbf{(A)}\ \frac{b(c\minus{}a)}{a(b\minus{}c)} \qquad
\textbf{(B)}\ \frac{a(b\minus{}c)}{c(a\minus{}b)} \qquad
\textbf{(C)}\ \frac{a(b\minus{}c)}{b(c\minus{}a)} \qquad
\textbf{(D)}\ \frac{c(a\minus{}b)}{a(b\minus{}c)} \qquad
\textbf{(E)}\ \frac{c(a\minus{}b)}{b(c\minus{}a)}$
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$
2009 International Zhautykov Olympiad, 1
On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y \equal{} x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y \equal{} 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i \equal{} 1,2,3,4$.
Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.
1996 Canada National Olympiad, 1
If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 - x - 1 = 0$, compute $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$.
2017 Azerbaijan JBMO TST, 2
Let $x,y,z$ be 3 different real numbers not equal to $0$ that satisfiying
$x^2-xy=y^2-yz=z^2-zx$.
Find all the values of $\frac{x}{z}+\frac{y}{x}+\frac{z}{y}$ and $(x+y+z)^3+9xyz$.
2020 HK IMO Preliminary Selection Contest, 11
Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$.