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The number of solutions of $3^{2x}-2^{2y}=55$, in which $x$ and $y$ are integers, is:
How many integers $n$ satisfy the equation $\left\lfloor \frac{n^2}{6} \right\rfloor = 13$? | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) - y) = g(x) + g(g(y) - g(-x)) + |x|\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(0),$ and let $t$ be the sum of all possible values of $g(0).$ Find $m \times t.$ | nvidia |
What is the smallest positive integer $n$ such that exactly half of the integers from 1 to $n$ are divisible by 3? | nvidia |
A positive integer $n$ has the property that $n^2+2n+1$ is a perfect square. Find the smallest value of $n$ that satisfies this condition. | nvidia |
A portion of the graph of a quadratic function $f(x)$ is shown below.
Let $g(x)=-f(x)$ and $h(x)=f(-x)$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=g(x)$ intersect, and $b$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, then what is $10a+b$?
[asy]
size(150);
real ... | nvidia |
Find the sum of all positive integral values of $n$ such that $n^2-29n+240=q$ for some perfect square number $q$. Note that there is at least one such $n$. | nvidia |
Find the number in the set $\{121, 169, 225, 289, 361\}$ that has the most divisors. | nvidia |
Let $f(x) = x^x$. If $n$ is a positive integer such that $f(f(n)) = f(6^n)$, find $n$. | nvidia |
Let $T$ be the set of all possible orderings of the numbers $1$ through $5$, so $T$ contains $5!$ elements. For each ordering $t$ in $T$, define the function $f_t(x)$ so that $f_t(x)$ is equal to the number of elements in the ordering $t$ that are less than or equal to $x$. Find $$\sum_{t \in T} f_t(3.7).$$ | nvidia |
Let $x$ and $y$ be two positive real numbers such that $x^2+y^2=1$ and $xy=a$. If $x$ and $y$ are inversely proportional, find the value of $a$. | nvidia |
If $m$ and $n$ are positive integers such that $\gcd(m,n) = 12$, then what is the smallest possible value of $\gcd(10m,15n)$? | nvidia |
A salad bar offers eight ingredients. What is the greatest number of five-ingredient salads that can be made such that no two salads have the same ingredient combination? | nvidia |
Given that $y$ is a positive real number, find $y$ in the equation $\frac{30}{50}=\sqrt{\frac{50}{y}}$. | nvidia |
Let $\mathbf{A}$ be a matrix whose inverse is
$$\mathbf{A}^{-1}=\begin{pmatrix} 1&-2\\-1&3\end{pmatrix}.$$
Find the sum of the elements of the second row of $\mathbf{A}$ if the sum of the elements of the first row is $-1$. | nvidia |
There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? | nvidia |
Find the value of $x$ that satisfies the equation:
$$\frac{x^3 \cdot x^{-5}}{(x^2)^{-2}} \cdot x^{-2} = \frac{1}{x^2}$$ | nvidia |
Find the area of the triangle with vertices $(2,3),$ $(-1,2),$ and $(4,1).$ | nvidia |
Let $m$ and $n$ be positive integers such that $\gcd(m, n)=8$, $\gcd(m, 2n+1)=21$, and $\gcd(2m-1, n)=15$. Find the smallest possible value of $\gcd(m+1, n+1)$. | nvidia |
In the equation of a circle $(x+2)^2 + (y-3)^2 = 25$, find the sum of the $x$-coordinates of the two points where the circle intersects the $x$-axis and the $y$-coordinates of the two points where the circle intersects the $y$-axis. | nvidia |
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ For example, $\tau (1)=1$ and $\tau(6) =4.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 2005$ with $S(n)$ odd, and let $b$ denote ... | nvidia |
Let $S$ be the set of all positive integers whose digits add up to 11, and let $T$ be the set of all positive integers whose digits add up to 12. What is the smallest positive integer that is a multiple of every element in $S$ and $T$? | nvidia |
Find the number of quadrilaterals with positive area whose vertices are points on a 5x5 grid, with coordinates ranging from (0,0) to (4,4), such that no three vertices are collinear. | nvidia |
For how many positive integers $n$ less than or equal to 1000 is $$(\cos t+i\sin t)^n=\cos nt+i\sin nt$$true for all real $t$? | nvidia |
Consider the expression $a^{2n} \cdot a^{3n}$ where $a$ is a positive integer. If the product of this expression is equal to $2^{15}$, find the value of $n$. | nvidia |
If $a$ and $b$ are positive integers such that $ab = 2^3 \cdot 3^4 \cdot 5^2$, find the smallest possible value of $a^3 + b^3$. | nvidia |
In triangle $ABC$, the lengths of sides $AB$ and $AC$ are 10 cm and 15 cm respectively. The area of triangle $ABC$ is $45\mbox{cm}^2$. If point $D$ is chosen inside the triangle so that the area of triangle $ABD$ is half the area of triangle $ABC$, find the length of segment $AD$. | nvidia |
Let $f(n)$ be the base-10 logarithm of the sum of the elements of the $n$th row in Pascal's triangle. Express $\frac{f(n)}{\log_{10} 2}$ in terms of $n$. Recall that Pascal's triangle begins
\begin{tabular}{rccccccccc}
$n=0$:& & & & & 1\\\noalign{\smallskip\smallskip}
$n=1$:& & & & 1 & & 1\\\noalign{\smallskip\smallsk... | nvidia |
A polynomial with integer coefficients is of the form
\[x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0.\]
It is known that the polynomial has a double root at $x = r$ and a simple root at $x = s$, and that $r$ and $s$ are both integers. Furthermore, the product of the coefficients of the polynomial is equal to $3240$. Enter a... | nvidia |
The length of the longer leg of a right triangle is equal to twice the length of the shorter leg minus the length of the hypotenuse. Find the greatest possible area of the triangle. | nvidia |
Let $f(z) = \ds\prod_{k=0}^{n-1} \left( z-e^{2\pi ik/n} \right)$, where $n$ is a positive integer.
For how many positive integers $n \le 1000$ does there exist a complex number $z$ such that $f(z)=f(1/z)$ and $|z|=1$? | nvidia |
What is the greatest integer $n$ such that the number $n!$ contains the same number of zeros at the end as it contains nonzero digits? | nvidia |
Find the sum of all integers $m$ such that $\binom{17}{6} + \binom{17}{7} = \binom{18}{m}$. | nvidia |
Let $\omega$ be a complex number satisfying $\omega^3 = 1$ and $\omega \neq 1.$ Let $n$ be the smallest positive integer such that there exist real numbers $r$ and $s$ satisfying
\[(r + s \omega)^n = (r + s \overline{\omega})^n.\]Determine $n.$ | nvidia |
What is the smallest positive integer $n$ such that $531n \equiv 1067n \pmod{24}?$ | nvidia |
Find the sum of all $x$ that satisfy the equation $\frac{-9x}{x^2-1} = \frac{2x}{x+1} - \frac{6}{x-1}.$ | nvidia |
A constant $c$ is such that the equation $5x^2+3x+c=0$ has two solutions, and the equation $x^2+3x+c=0$ has only one solution.
Find $c$. | nvidia |
In the equation $$a^2-kx+1=0$$ determine $k$ such that the sum of the roots is equal to the product of the roots. | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $a^2b^2c^2=1$. Find the maximum value of $\frac{1}{a}+\frac{1}{2b}+\frac{1}{3c}$. | nvidia |
Find the smallest three-digit palindrome number such that the sum of its digits is a prime number, and when the digits are rearranged in ascending order, they form a composite number. | nvidia |
Consider the function $g(x)$ that satisfies
\[g(x + y) = g(x) + g(y)\]for all real numbers $x$ and $y.$ Given that $g(1) = 3$ and $g(2) = 6,$ find $g(5).$ | nvidia |
A circle with center $(2,5)$ and radius $3$ intersects the $y$-axis at two points. Find the midpoint of the segment connecting these two points. | nvidia |
Let $m$ be the number of positive integers less than $1000$ that are a multiple of $2$ but not $4$, and let $n$ be the number of positive integers less than $1000$ that are a multiple of $6$ but not $12$. Compute $(m-n)^2$. | nvidia |
A square is inscribed in a circle with radius $r$ such that the sides of the square form a 45-degree angle with the radius. Find the area of the square in terms of $r$. Express your answer in simplest radical form. | nvidia |
Find the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=72$. | nvidia |
Let $f$ be a function defined on the set of all real numbers. For each real number $x$, let $f^*(x)$ be the greatest integer less than or equal to $f(x)$. Suppose $f(1) = 1$ and for every real number $x$, $f^*(x + 1) = f^*(x) + 2$. Compute the value of $f(5)$. | nvidia |
Let $f(x)$ be a function such that
$$f(f(f(x)))=x$$
for all $x$.
If the point $(4,7)$ is on the graph of $y=f(x)$, then there is one point that must be on the graph of $y=f(f(x))$. What is the sum of the coordinates of that point? | nvidia |
What is the sum of all the three-digit positive integers that are divisible by 7? | nvidia |
Given that $a$ and $b$ are positive integers with $\text{lcm}(a,b) = 42$, what is the smallest possible value of $\text{lcm}(a+7,b+7)$? | nvidia |
Let $S$ be the set of all real numbers $x$ such that $x^2 + 3x + 2 = 0$. Define a function $g$ on the set $S$ by $g(x) = \log (x + 1)$. Compute the sum of all values of $g(x)$ for $x$ in $S$. | nvidia |
Let $f(x)$ be a polynomial with positive integer coefficients such that $f(2)=10$ and for any prime $p$, if $p$ divides $f(n)$ then $p$ also divides $n$. Find the smallest possible value of $f(100)$. | nvidia |
Determine the largest number that is less than 500, is a multiple of 5, and can be expressed as the sum of three consecutive integers. | nvidia |
Find the sum of the least and greatest elements in the set of the first ten positive integers that are congruent to 1 mod 4 and greater than 200. | nvidia |
A set $\mathcal{T}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{T},$ the geometric mean of the set of values obtained by deleting $x$ from $\mathcal{T}$ is an integer. Given that 2 belongs to $\mathcal{T}$ and that 2010 is the largest element of $\mathcal{T},$ what is th... | nvidia |
Let $P_n$ be the product of the factorials of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $P_n$ is a perfect square. | nvidia |
Find the number of distinct positive integers that can be formed by multiplying two distinct prime factors of the number $4^{13} \cdot 3^7 \cdot 7^5$. | nvidia |
What is the sum of the positive integer divisors of 24, excluding 1 and 24 itself, such that the divisors are not multiples of 3? | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
Determine the range of values for the real number $a$ such that the inequality
\[\frac{ax+1}{x^2+2x+1} > \frac{3x+1}{x^2+x+1}\]holds for all real numbers $x$ satisfying $x < 0$. | nvidia |
The blue graph shown represents the function $f(x) = x^3 + bx^2 + cx + d$. Find $d$. (The grid lines are spaced one unit apart.)
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdown... | nvidia |
The hyperbola $y^2 - 4x^2 = 1$ intersects the graph of $y = x^2 + 1$ at the points $C$ and $D$. The area of the triangle formed by the points $C$, $D$, and the origin can be written as $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are positive integers that do not share any factors other than one. Find the value of $m + n$. | nvidia |
Let $n$ be a positive integer. A sequence $a_1, a_2, \ldots, a_n$ of positive integers is called "gcd-sorted" if $\gcd(a_i, a_{i+1}) < \gcd(a_{i+1}, a_{i+2})$ for all $1 \leq i \leq n-2$. For each $n$, determine the minimum value of $a_n$. | nvidia |
The sum of the squares of three consecutive positive integers is 7809. What is the sum of the three integers? | nvidia |
There are 3 complex numbers $a+bi$, $c+di$, and $e+fi$. If $b=1$, $e=-a-c$, and the sum of the numbers is $-i$, find $d+f$. | nvidia |
Find the sum of the squares of the divisors of 36. | nvidia |
Find the smallest positive integer $n$ such that the decimal representation of $\frac{1}{n}$ has a block of digits that repeats exactly 5 times before the decimal terminates. | nvidia |
In the triangle $ABC$, $A=(0,0)$, $B=(6,0)$, and $C=(x,y)$.
Find $x$ and $y$ such that
$AC$ is perpendicular to $BC$,
and
the midpoint of $BC$ lies on the line $3x+5y=20$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $\sqrt{x} + \sqrt{y} = 5$ and $x + y = 23$. Find the value of $\sqrt{x \cdot y}$. | nvidia |
What is the greatest integer less than or equal to\[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?\]
$\textbf{(A) }80\qquad \textbf{(B) }81 \qquad \textbf{(C) }96 \qquad \textbf{(D) }97 \qquad \textbf{(E) }625\qquad$
| nvidia |
Find the sum of the reciprocals of the positive divisors of 24, except 1 and 24 themselves, if you are given that the sum of the reciprocals of the positive divisors of 6 (again excluding 1 and 6) is $\frac{1}{2}$. | nvidia |
What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions? | nvidia |
Let $x$ and $y$ be real numbers such that $3x + 2y \le 7$ and $2x + 4y \le 8.$ Find the largest possible value of $x + y.$ | nvidia |
In a certain city, the addresses of buildings are encoded using a sequence of three digits (0-9), with the condition that no two adjacent digits can be the same. How many different addresses are possible? | nvidia |
A sequence of triangles is constructed in the following manner:
1. Start with an isosceles triangle with base length 1 and vertex angle 60 degrees.
2. Draw an altitude from the vertex of the triangle to the base, dividing the base into two equal segments.
3. Draw two isosceles triangles with base length 1/2 and vertex... | nvidia |
Let $w$ be a complex number such that $w^3=64+64i$. Find the modulus of $w$. | nvidia |
Find the matrix $A$ such that $A^2=\begin{pmatrix}1 & 3\\0 & 1\end{pmatrix}$. | nvidia |
Find the smallest number whose sum of the squares of its proper divisors is greater than the number itself. | nvidia |
Determine the number of integer solutions $(x, y)$ such that the equation $3x + 5y = 30$ holds true and $1 \leq x, y \leq 10$. | nvidia |
Find the smallest positive integer $N$ with the following property: of the three numbers $N$, $N+1$, and $N+2$, one of them is divisible by $2^2$, one of them is divisible by $3^2$, one is divisible by $5^2$, and one is divisible by $7^2$. | nvidia |
Find the product of all the roots of $x^3-2x^2-5x+1=0$, given that one of the roots is a complex number. | nvidia |
Consider the equation $x^2 + px + 16 = 0$. Find the value of $p$ such that the equation has exactly one solution. | nvidia |
Consider a regular $n$-gon, which is a polygon with $n$ sides of equal length and $n$ internal angles of equal size. For a given number $n$, determine the number of non-similar regular $n$-gons that can be constructed using a specific set of rules: each internal angle must be less than 180 degrees, no three vertices ar... | nvidia |
Find the median of the set of positive divisors of 72. | nvidia |
Find the value of $a$ such that the roots of the quadratic $x^2 - 2ax + a^2 + 1$ are of the form $r \pm si$, where $r$ and $s$ are integers and $s$ is a perfect square. | nvidia |
Compute the number of ways to arrange the letters in the word "COMPUTER" such that the vowels appear together. | nvidia |
Let $g$ be a function defined on the positive integers, such that
\[g(x+y) = g(x) \cdot g(y)\]for all positive integers $x$ and $y.$ Given $g(2) = 3$ and $g(6) = 27,$ find $g(12).$ | nvidia |
Find a nonzero $q$ such that $qx^2-15x+6=0$ has only one solution. | nvidia |
A point (x,y) is chosen at random inside the triangle with vertices (0,0), (0,1), and (1,0). Determine the probability that $x+y<\frac{1}{2}$ | nvidia |
The sum of the squares of the solutions to the equation $\sqrt{3x^2+5x-1}=\sqrt{x^2+3}$ is equal to 2. Determine the average of the coefficients of the equation. | nvidia |
Let $a^2=\frac{16}{44}$ and $b^2=\frac{(2+\sqrt{5})^2}{11}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a+b)^3$ can be expressed in the simplified form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$? | nvidia |
Consider a solid right circular cylinder with radius $2$ and height $3$. The volume of the set of points that are inside or within two units of the cylinder is $\frac{a + b\pi}{c}$, where $a, b,$ and $c$ are positive integers, and $b$ and $c$ are relatively prime. Find $a + b + c$. | nvidia |
A bookstore has a shelf with 11 books, each with a different title. How many different arrangements of the books can be made such that the two books with the longest titles are next to each other? | nvidia |
Find all values of $a$ for which the points $(0,0,0),$ $(1,a,0),$ $(0,1,a),$ and $(a,0,1)$ are coplanar. | nvidia |
The number 25 can be written as the sum of squares of two different positive integers in two ways, namely $25=3^2+4^2$ and $25=5^2+0^2$.
What is the largest such integer that can be expressed in at least two ways? | nvidia |
What is the maximum possible number of different integer heights that a triangle with integer side lengths can have, if the perimeter of the triangle is fixed at 20 units? | nvidia |
The number 504 is the product of three consecutive positive integers and can also be expressed as the sum of the cubes of three consecutive positive integers. Find the sum of those six integers. | nvidia |
Given that $x,$ $y,$ and $z$ are positive integers such that $x^2 + y^2 + z^2 = 36,$ find the maximum value of
\[\frac{xy}{x + y} + \frac{xz}{x + z} + \frac{yz}{y + z}.\] | nvidia |
Determine all real numbers $x$ for which $x(x-1)(x-2)(x-3) \ge 0$. Express your answer in interval notation. | nvidia |
Determine the value of $k$ such that the equation $\frac{x^2}{3} - \frac{y^2}{4} = k$ has exactly two distinct rational solutions. | nvidia |
Find the number of different positive integers that can be represented as the sum of two distinct members of the set $\{1, 2, 3, \ldots, 14, 15, 16\}$ modulo 17. | nvidia |
Find the smallest positive integer $n$ such that $\gcd(n, 15) = 1$ and $n$ is a multiple of 13. | nvidia |
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