Found problems: 1132
2012 Stanford Mathematics Tournament, 2
Find all real values of $x$ such that $(\frac{1}{5}(x^2-10x+26))^{x^2-6x+5}=1$
2005 Balkan MO, 2
Find all primes $p$ such that $p^2-p+1$ is a perfect cube.
1974 AMC 12/AHSME, 10
What is the smallest integral value of $k$ such that
\[ 2x(kx-4)-x^2+6=0 \]
has no real roots?
$ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $
2018 Turkey Team Selection Test, 1
Prove that, for all integers $a, b$, there exists a positive integer $n$, such that the number $n^2+an+b$ has at least $2018$ different prime divisors.
2023 Auckland Mathematical Olympiad, 5
There are $11$ quadratic equations on the board, where each coefficient is replaced by a star. Initially, each of them looks like this
$$\star x^2 + \star x + \star= 0.$$
Two players are playing a game making alternating moves. In one move each ofthem replaces one star with a real nonzero number.
The first player tries to make as many equations as possible without roots and the second player tries to make the number of equations without roots as small as possible.
What is the maximal number of equations without roots that the fi rst player can achieve if the second player plays to her best? Describe the strategies of both players.
2010 Hanoi Open Mathematics Competitions, 7
Determine all positive integer $a$ such that the equation $2x^2 - 30x + a = 0$ has two prime roots, i.e. both roots are prime numbers.
2024 JHMT HS, 13
Compute the largest nonnegative integer $T \leq 30$ that is the remainder when $T^2 + 4$ is divided by $31$.
2022 AMC 12/AHSME, 4
For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?
$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$
2008 AMC 10, 25
A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$?
[asy]unitsize(4mm);
defaultpen(linewidth(.8)+fontsize(8));
draw(Circle((0,0),4));
path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle;
draw(mat);
draw(rotate(60)*mat);
draw(rotate(120)*mat);
draw(rotate(180)*mat);
draw(rotate(240)*mat);
draw(rotate(300)*mat);
label("$x$",(-2.687,0),E);
label("$1$",(-3.187,1.5513),S);[/asy]$ \textbf{(A)}\ 2\sqrt {5} \minus{} \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} \minus{} \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 \plus{} 2\sqrt {3}}{2}$
2006 National Olympiad First Round, 11
What is the sum of the real roots of the equation $4x^4-3x^2+7x-3=0$?
$
\textbf{(A)}\ -1
\qquad\textbf{(B)}\ -2
\qquad\textbf{(C)}\ -3
\qquad\textbf{(D)}\ -4
\qquad\textbf{(E)}\ \text {None of above}
$
2010 All-Russian Olympiad, 1
Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.
2008 AIME Problems, 14
Let $ \overline{AB}$ be a diameter of circle $ \omega$. Extend $ \overline{AB}$ through $ A$ to $ C$. Point $ T$ lies on $ \omega$ so that line $ CT$ is tangent to $ \omega$. Point $ P$ is the foot of the perpendicular from $ A$ to line $ CT$. Suppose $ AB \equal{} 18$, and let $ m$ denote the maximum possible length of segment $ BP$. Find $ m^{2}$.
2014 AIME Problems, 10
Let $z$ be a complex number with $|z| = 2014$. Let $P$ be the polygon in the complex plane whose vertices are $z$ and every $w$ such that $\tfrac{1}{z+w} = \tfrac{1}{z} + \tfrac{1}{w}$. Then the area enclosed by $P$ can be written in the form $n\sqrt{3},$ where $n$ is an integer. Find the remainder when $n$ is divided by $1000$.
2008 ITest, 76
During the car ride home, Michael looks back at his recent math exams. A problem on Michael's calculus mid-term gets him starting thinking about a particular quadratic, \[x^2-sx+p,\] with roots $r_1$ and $r_2$. He notices that \[r_1+r_2=r_1^2+r_2^2=r_1^3+r_2^3=\cdots=r_1^{2007}+r_2^{2007}.\] He wonders how often this is the case, and begins exploring other quantities associated with the roots of such a quadratic. He sets out to compute the greatest possible value of \[\dfrac1{r_1^{2008}}+\dfrac1{r_2^{2008}}.\] Help Michael by computing this maximum.
2014 AIME Problems, 14
Let $m$ be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\] There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$. Find $a+b+c$.
2002 AMC 10, 6
For how many positive integers $ n$ is $ n^2\minus{}3n\plus{}2$ a prime number?
$ \textbf{(A)}\ \text{none} \qquad
\textbf{(B)}\ \text{one} \qquad
\textbf{(C)}\ \text{two} \qquad
\textbf{(D)}\ \text{more than two, but finitely many}\\
\textbf{(E)}\ \text{infinitely many}$
1987 India National Olympiad, 6
Prove that if coefficients of the quadratic equation $ ax^2\plus{}bx\plus{}c\equal{}0$ are odd integers, then the roots of the equation cannot be rational numbers.
1987 Romania Team Selection Test, 11
Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$. For a given positive integer $n$, $n\geq 2$ consider the system of equations: \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivalent if there exists a real number $\lambda \neq 0$, $x_1=\lambda y_1$, $\ldots$, $x_n= \lambda y_n$. How many nonequivalent solutions does the system have?
[i]Mircea Becheanu[/i]
2010 India IMO Training Camp, 2
Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.
2004 Singapore Team Selection Test, 3
Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying
\[ f\left(\frac {x \plus{} y}{x \minus{} y}\right) \equal{} \frac {f\left(x\right) \plus{} f\left(y\right)}{f\left(x\right) \minus{} f\left(y\right)}
\]
for all $ x \neq y$.
2011 Math Prize For Girls Problems, 18
The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. If $P(3) = 89$, what is the value of $P(10)$?
2013 Iran Team Selection Test, 12
Let $ABCD$ be a cyclic quadrilateral that inscribed in the circle $\omega$.Let $I_{1},I_{2}$ and $r_{1},r_{2}$ be incenters and radii of incircles of triangles $ACD$ and $ABC$,respectively.assume that $r_{1}=r_{2}$. let $\omega'$ be a circle that touches $AB,AD$ and touches $\omega$ at $T$. tangents from $A,T$ to $\omega$ meet at the point $K$.prove that $I_{1},I_{2},K$ lie on a line.
2007 Bulgarian Autumn Math Competition, Problem 10.1
Find all integers $b$ and $c$ for which the equation $x^2-bx+c=0$ has two real roots $x_{1}$ and $x_{2}$ satisfying $x_{1}^2+x_{2}^2=5$.
2002 Germany Team Selection Test, 1
Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121.
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.