Found problems: 29
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$?
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
When writing the product of two three-digit numbers, the multiplication sign was omitted, forming a six-digit number. It turns out that the six-digit number is equal to three times the product.
Find the six-digit number.
How many ordered pairs $(m,n)$ of positive integers are solutions to $\frac{4}{m}+\frac{2}{n}=1$?
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{more than}\ 4 $
Consider the sequence defined by $a_k=\frac 1{k^2+k}$ for $k\ge 1.$ Given that $a_m+a_{m+1}+\cdots+a_{n-1}=1/29,$ for positive integers $m$ and $n$ with $m<n$, find $m+n.$
Find the number of ordered pairs of integers $(m, n)$ that satisfy $20m-10n = mn$.
For how many integers $n$, does the equation system \[\begin{array}{rcl}
2x+3y &=& 7\\
5x + ny &=& n^2
\end{array}\] have a solution over integers?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{None of the preceding}
$
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$?
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
In an $n$-by-$m$ grid, $1$ row and $1$ column are colored blue, the rest of the cells are white. If precisely $\frac{1}{2010}$ of the cells in the grid are blue, how many values are possible for the ordered pair $(n,m)$
How many non-congruent right triangles are there such that the perimeter in $\text{cm}$ and the area in $\text{cm}^2$ are numerically equal?
$\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ \text{infinitely many}$
Suppose $xy-5x+2y=30$, where $x$ and $y$ are positive integers. Find the sum of all possible values of $x$
Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$.
The harmonic mean of two positive numbers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}.$
Find the largest value of $a^b$ such that the positive integers $a,b>1$ satisfy $$a^bb^a+a^b+b^a=5329$$
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 $a+b+c$.
Let $a,b,c$ be real numbers satisfying:
\[ab-a=b+119\]
\[bc-b=c+59\]
\[ca-c=a+71\]
Determine all possible values of $a+b+c$.
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $ 3$ times their perimeters?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form [i]special[/i] if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\
12 \qquad\textbf{(E)}\ 13$
Consider the set of all fractions $\tfrac{x}{y},$ where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\%$?
$ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{infinitely many} $
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form [i]special[/i] if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\
12 \qquad\textbf{(E)}\ 13$
Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$, where $n$ is either $2012$ or $2013$.
Notice that in the fraction $\frac{16}{64}$ we can perform a simplification as $\cancel{\frac{16}{64}}=\frac 14$ obtaining a correct equality. Find all fractions whose numerators and denominators are two-digit positive integers for which such a simplification is correct.
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$
Find all triples $(a, b, c)$ of positive integers with $a\le b\le c$ such that\[\left(1+\dfrac1{a}\right)\left(1+\dfrac1{b}\right)\left(1+\dfrac1{c}\right)=3.\]
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$?
$\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad\textbf{(E) }26$