Found problems: 1148
We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.
Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$.
Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.
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)$
Let $a,~b,$ and $x$ be positive real numbers distinct from one. Then \[4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)\]
$\textbf{(A) }\text{for all values of }a,~b,\text{ and }x\qquad$
$\textbf{(B) }\text{if and only if }a=b^2\qquad$
$\textbf{(C) }\text{if and only if }b=a^2\qquad$
$\textbf{(D) }\text{if and only if }x=ab\qquad$
$ \textbf{(E) }\text{for none of these}$
Find all rational numbers $r$ such that the equation $rx^2 + (r + 1)x + r = 1$ has integer solutions.
Find all pairs of positive integers $ a,b$ such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a \\ a^2+a+1 &\equiv 0 \pmod b . \end{align*}
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]
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]
The discriminant of the equation $ x^2\plus{}2x\sqrt{3}\plus{}3\equal{}0$ is zero. Hence, its roots are:
$ \textbf{(A)}\ \text{real and equal} \qquad
\textbf{(B)}\ \text{rational and equal} \qquad
\textbf{(C)}\ \text{rational and unequal} \\
\textbf{(D)}\ \text{irrational and unequal} \qquad
\textbf{(E)}\ \text{imaginary}$
The roots of the equation $x^2+4x-5 = 0$ are also the roots of the equation $2x^3+9x^2-6x-5 = 0$. What is the third root of the second equation?
Consider the graphs $ y \equal{} Ax^2$ and and $ y^2 \plus{} 3 \equal{} x^2 \plus{} 4y$, where $ A$ is a positive constant and $ x$ and $ y$ are real variables. In how many points do the two graphs intersect?
$ \textbf{(A)}\ \text{exactly } 4 \qquad \textbf{(B)}\ \text{exactly } 2$
$ \textbf{(C)}\ \text{at least } 1, \text{ but the number varies for different positive values of } A$
$ \textbf{(D)}\ 0 \text{ for at least one positive value of } A \qquad \textbf{(E)}\ \text{none of these}$
Given the function $f(x) = x^2$, the sector of $f$ from $a$ to $b$ is defined as the limited region between the
graph of $y = f(x)$ and the straight line segment that joins the points $(a, f(a))$ and $(b, f(b))$. Define the
increasing sequence $x_0$, $x_1, \cdots$ with $x_0 = 0$ and $x_1 = 1$, such that the area of the sector of $f$ from $x_n$ to $x_{n+1}$ is constant for $n \geq 0$. Determine the value of $x_n$ in function of $n$.
If two real numbers $ x$ and $ y$ satisfy the equation $ \frac{x}{y} \equal{} x \minus{} y$, then:
$ \textbf{(A)}\ {x \ge 4}\text{ and }{x \le 0}\qquad \\
\textbf{(B)}\ {y}\text{ can equal }{1}\qquad \\
\textbf{(C)}\ \text{both }{x}\text{ and }{y}\text{ must be irrational}\qquad \\
\textbf{(D)}\ {x}\text{ and }{y}\text{ cannot both be integers}\qquad \\
\textbf{(E)}\ \text{both }{x}\text{ and }{y}\text{ must be rational}$
\[ \begin{array}{c} {x^{2} \plus{} y^{2} \plus{} z^{2} \equal{} 21} \\
{x \plus{} y \plus{} z \plus{} xyz \equal{} \minus{} 3} \\
{x^{2} yz \plus{} y^{2} xz \plus{} z^{2} xy \equal{} \minus{} 40} \end{array}
\]
The number of real triples $ \left(x,y,z\right)$ satisfying above system is
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None}$
Solve the system of equations:
$x^2=\frac{1}{y}+\frac{1}{z}$,
$y^2=\frac{1}{z}+\frac{1}{x}$,
$z^2=\frac{1}{x}+\frac{1}{y}$.
in the real numbers.
The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$.
Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.
Find the real numbers $ m $ which have the property that the equation
$$ x^2-2mx+2m^2=25 $$
has two integer solutions.
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?
A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$.
[i]Proposed by Aaron Lin[/i]
The graph of $ y \equal{} 2x^2 \plus{} 4x \plus{} 3$ has its:
$ \textbf{(A)}\ \text{lowest point at } {(\minus{}1,9)}\qquad
\textbf{(B)}\ \text{lowest point at } {(1,1)}\qquad \\
\textbf{(C)}\ \text{lowest point at } {(\minus{}1,1)}\qquad
\textbf{(D)}\ \text{highest point at } {(\minus{}1,9)}\qquad \\
\textbf{(E)}\ \text{highest point at } {(\minus{}1,1)}$
An ordered quadruple $(y_1,y_2,y_3,y_4)$ is $\textbf{quadratic}$ if there exist real numbers $a$, $b$, and $c$ such that \[y_n=an^2+bn+c\] for $n=1,2,3,4$.
Prove that if $16$ numbers are placed in a $4\times 4$ grid such that all four rows are quadratic and the first three columns are also quadratic then the fourth column must also be quadratic.
[i](We say that a row is quadratic if its entries, in order, are quadratic. We say the same for a column.)[/i]
[asy]
size(100);
defaultpen(linewidth(0.8));
for(int i=0;i<=4;i=i+1)
draw((i,0)--(i,4));
for(int i=0;i<=4;i=i+1)
draw((0,i)--(4,i));
[/asy]
For one root of $ ax^2 \plus{} bx \plus{} c \equal{} 0$ to be double the other, the coefficients $ a,\,b,\,c$ must be related as follows:
$ \textbf{(A)}\ 4b^2 \equal{} 9c\qquad
\textbf{(B)}\ 2b^2 \equal{} 9ac\qquad
\textbf{(C)}\ 2b^2 \equal{} 9a\qquad \\
\textbf{(D)}\ b^2 \minus{} 8ac \equal{} 0\qquad
\textbf{(E)}\ 9b^2 \equal{} 2ac$
Let $p$ be a prime number of the form $4k+1$. Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]
Prove that for any quadratic polynomial $f(x)=x^2+px+q$ with integer coefficients, it is possible to find another polynomial $q(x)=2x^2+rx+s$ with integer coefficients so that \[\{f(x)|x \in \mathbb{Z} \} \cap \{g(x)|x \in \mathbb{Z} \} = \emptyset .\]
Determine $ m$ so that $ 4x^2\minus{}6x\plus{}m$ is divisible by $ x\minus{}3$. The obtained value, $ m$, is an exact divisor of:
$ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 64$