instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Determine the greatest integer $n$ such that $5^n$ divides $10^{45}$ exactly. | nvidia |
The sum of the digits of a three-digit number is 15. The sum of the digits of the number obtained by reversing the order of the digits of the original number is 18. What is the original number? | nvidia |
Find the number of ways to express $2^4 \cdot 7^9$ as the product of two perfect squares, such that one of the perfect squares is divisible by 7 and the other is not. | nvidia |
Determine the smallest positive integer $n$ such that $2^n\equiv n^2\pmod 5$. | nvidia |
The least positive multiple of $17$ that is greater than $500$ can be expressed as $500+k$, where $k$ is a positive integer. Find the smallest possible value of $k$. | nvidia |
Let's say we have a continuous function $f(x)$ defined on the interval $[a, b]$. The function has a single minimum at $x=c$ and a single maximum at $x=d$. Given that $a < c < d < b$, find the range of all possible values of $y$ such that $f^{-1}(\lfloor{y}\rfloor) \cdot f^{-1}(\lceil{y}\rceil) = k$, where $k$ is a posi... | nvidia |
Given that $a$ and $b$ are positive integers, what is the smallest possible value of $ab$ if the sum of the digits of $ab$ is equal to the sum of the digits of $a$ times the sum of the digits of $b$? | nvidia |
Find all real values of $x$ that satisfy \[\frac{1}{x^2+1} < \frac{1}{x^2-1}.\](Give your answer in interval notation.) | nvidia |
How many positive integers $n$ less than 100 have a corresponding integer $m$ divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? | nvidia |
For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?
$\textbf{(A) }110\qquad\textbf{(B) }191\qquad\textbf{(C) }261\qquad\textbf{(D) }325\qquad\textbf{(E) }425$
| nvidia |
What is the smallest positive integer $n$ such that $7^n$ has the same units digit as $5^n$? | nvidia |
What is the sum of all of the solutions of the equation $\frac{4x}{20}=\frac{5}{x}$? | nvidia |
Consider the equation $ax^2+bx+c=0$, where $a$, $b$, and $c$ are non-zero constants. It is known that the quadratic equation has exactly one solution when the discriminant is equal to zero. Now, suppose that the coefficient $c$ is not constant but depends on $x$ and is given by the function $f(x)=e^{2x}$. Find the valu... | nvidia |
A right triangle contains two circles, one inscribed within it and one circumscribed about it. Find the number of regions into which the sides of the triangle divide the area inside the circumscribed circle but outside the inscribed circle. | nvidia |
Find the sum of all integers from 1 to 100 that are not divisible by either 5 or 8. | nvidia |
Let $a$, $b$ and $c$ be positive real numbers. Determine the maximum possible value of
$a + b + c$
given that
$\ds\frac{a}{1+b^2}+\ds\frac{b}{1+c^2}+\ds\frac{c}{1+a^2}=2$. | nvidia |
In the expansion of $(1+x)^n$ using the binomial theorem, where $n$ is a positive integer and $x$ is a real number, what is the sum of the coefficients of the even-powered terms? | nvidia |
Consider the set of all circles passing through the points $A(0,13)$, $B(13,13)$, $C(13,0)$, and $D(0,0)$. Determine the radius of the circle with the largest area that passes through these four points. | nvidia |
Find the largest constant $C$ so that
\[x^2 + y^2 + 1 \ge C(x + y)\]for all real numbers $x$ and $y.$ | nvidia |
A rectangle is inscribed in a triangle whose sides are 6, 8, and 10. The length of the rectangle's side along the hypotenuse of the triangle is 4. Find the area of the rectangle. | nvidia |
Let $g(n)$ be the product of the proper positive integer divisors of $n$. (Recall that a proper divisor of $n$ is a divisor other than $n$.) For how many values of $n$ does $n$ not divide $g(n)$, given that $2 \le n \le 50$? | nvidia |
What is the number of six-digit positive integers that have at least one pair of consecutive digits that are the same? | nvidia |
Find the greatest integer value of $a$ for which the expression $\frac{3x^2+2x+1}{\sqrt{x^2+ax+6}}$ has a domain of all real numbers. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that
\[x^2y + y^2z + z^2x = xyz.\]Find the minimum value of
\[x^2 + y^2 + z^2.\] | nvidia |
Find the equation of the blue parabola shown on the graph. (Assume that the graph has the standard unit scale.)
[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 tickdownlength=-0.15inch;
r... | nvidia |
Find the minimum value of $9^x - 3^x + 1$ over all real numbers $x.$ | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a+b+c=1$. Find the maximum value of
\[\frac{a^2}{b+c} + \frac{b^2}{a+c} + \frac{c^2}{a+b}.\] | nvidia |
Given a positive integer $n$, let $f(n)$ be the largest integer $k$ such that $\sqrt[n]{k}$ is an integer. Compute the value of $f(1)f(2)f(3)...f(2015)$ modulo 2016. | nvidia |
How many positive integer divisors of $2^{2004}\cdot 3^{2004}\cdot 5^{2004}$ have exactly 4 positive integer divisors? | nvidia |
Consider two numbers, $x$ and $y$, such that their squares are congruent modulo $n$, where $n$ is an arbitrary positive integer.
Determine the smallest positive integer $k$ such that $x^2 \equiv y^2 \pmod{k}$ implies $x \equiv y \pmod{k}$. | nvidia |
The area of the smallest rectangle that will contain a circle of radius 6 is what? | nvidia |
Suppose that $n$ is a positive integer such that in base $7$, then $n$ can be expressed as $\overline{ABC}_7$, and in base $11$, then $n$ can be expressed as $\overline{CBA}_{11}$. Find the largest possible value of $n$ in base $10$. | nvidia |
Let $a$, $b$, and $c$ be three positive real numbers such that $a+b+c=\sqrt[3]{2}$, and $a^2+b^2+c^2=2\sqrt[3]{4}$. Find the number of positive real values of $x$ such that $\sqrt{x-a}$, $\sqrt{x-b}$, and $\sqrt{x-c}$ are all integers. | nvidia |
A number is divisible by $9$ if the sum of its digits is divisible by $9.$ For example, the number $19\,836$ is divisible by $9$ but $19\,825$ is not.
If $D\,767\,E89$ is divisible by $9,$ where $D$ and $E$ each represent a single digit, what is the sum of all possible values of the sum $D+E?$ | nvidia |
If $x$, $y$, and $z$ are digits and $0.xyz$ can be expressed as $\frac{1}{n}$ where $n$ is an integer such that $0<n\le12$, then what is the smallest possible value of $x+y+z$? | nvidia |
Let $A$ be a set of integers from 1 to 6. The six elements of $A$ are divided into three groups of two elements each. What is the number of ways to divide the elements into groups such that the sum of the products of the two elements in each group is greater than 75? | nvidia |
A function $f(x)$ is defined for all positive integers such that $f(1) = 7$, and for all positive integers $x$, $f(x + 1) = f(x) + xf(x)$. Find the smallest possible value of $6f(2)f(3) - 12$. | nvidia |
Let $f(x)$ be a polynomial with integer coefficients. Suppose that for some positive integer $n$, the sum of the coefficients of $f(x^n)$ is $-1$, while the sum of the coefficients of $f(-x^n)$ is $1$. What is the smallest possible value of $n$? | nvidia |
How many even three digit numbers are there such that the sum of the tens and units digits is 10? | nvidia |
Find all values of $y$ that satisfy \[\frac{y^3}{y^2+2} \le \frac{3}{y+2} + \frac{2}{y-1}.\] | nvidia |
Count the number of three-digit numbers that do not have any digits repeated consecutively. | nvidia |
Find the value of the parameter $a$ such that the polynomial $2x^3+ax^2-4x+16$ is exactly divisible by $x+2.$ | nvidia |
Given that $f(x)$ is a function that maps positive real numbers to real numbers, and the functional equation
\[xf(y) - yf(x) = f \left( \frac{x}{y} \right)\]
holds for all positive real numbers $x$ and $y.$ Let $g(x)$ be a function defined as $g(x) = xf(x)$. Prove that $g(x)$ is a constant function and find the const... | nvidia |
Find a two-digit number with a tens digit greater than the units digit by 3, such that the sum of its digits is divisible by 5. | nvidia |
A rectangle is inscribed in the region bounded by the curves $y = x^2$ and $y = \sqrt{x}$. The sides of the rectangle are parallel to the axes, and one of its vertices lies on the point of intersection of the two curves. Find the maximum area of the rectangle. | nvidia |
Find the smallest positive four-digit number that is divisible by each of the four smallest prime numbers. | nvidia |
Find the number of positive integers less than 200 that are divisible by 2, 3, or 5, but not all three. | nvidia |
Find the area of the triangle with vertices $(0,7,10),$ $(-1,6,6),$ and $(-4,9,6).$ | nvidia |
What is the smallest positive integer with exactly 12 positive integer divisors? | nvidia |
Find all real numbers $x$ which satisfy \[\frac{x-1}{x-3} \ge 2.\](Give your answer in interval notation.) | nvidia |
Find the only positive real number $x$ for which $\displaystyle \frac{x-4}{9} = \frac{4}{x-9}$. | nvidia |
Find the smallest positive integer N such that when N is multiplied by 27720, the product modulo 54321 is equal to 1. | nvidia |
Given a function $g : \mathbb{R} \to \mathbb{R},$ and a positive real number $c$, let the function satisfy the following equation:
\[g(g(x)) + cg(x) + x = g(x + g(c))\]
for all real numbers $x.$
Let $m$ be the maximum possible value of $c$ for which there exist at least two possible functions $g(x)$, and let $t$ be ... | nvidia |
The roots of the polynomial $x^3 + bx^2 + cx + d$ are all real. The coefficients $b$, $c$, and $d$ are randomly and uniformly chosen from the interval $[-1, 1]$. Find the probability that the roots of the polynomial are all positive. | nvidia |
The value of $x$ is a positive real number. The sum of the distances between the points $(x,x^2)$, $(-x,-x^2)$ and the origin is $20$. Determine $x$. | nvidia |
Find the absolute value of the difference of single-digit integers $A$ and $B$ such that $$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & & B& A & B_7\\ & & & \mathbf{5} & \mathbf{3} & A_7\\& & + & A & \mathbf{2} & \mathbf{1_7}\\ \cline{2-6} & & A & \mathbf{3} & \mathbf{2} & \mathbf{3_7} \\ \end{array} $$Express your ans... | nvidia |
Find the smallest integer $n$ such that when $\frac{22}{777}$ is expressed as a decimal, the digits from the $n$th position onwards form a repeating block of length less than 20. | nvidia |
What is the smallest integer $n$ such that $\frac{7}{17} - \frac{4}{51}$ can be expressed as a sum of $n$ distinct unit fractions (fractions with a numerator of 1)? | nvidia |
Find the smallest positive integer that has exactly 15 positive divisors, with the additional constraint that the sum of its divisors is a perfect square. | nvidia |
Given a circle centered at the origin with radius 5, find the range of values of x such that the distance between the point (x, 2) and the center of the circle is less than or equal to the radius of the circle. | nvidia |
Find the smallest positive integer $m$ such that $2m \equiv 1 \pmod{17}$. | nvidia |
What is the smallest positive number that is a perfect square and $10$ more than a number that can be expressed as the sum of two distinct prime numbers? | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 1.$ Find the minimum value of $x + y + z.$ | nvidia |
Determine the smallest positive integer n such that the sum of the digits of n! is divisible by 4. | nvidia |
The function $g(x) = \log_2(x-1)$ has a restricted domain. Determine the largest real number $x$ in the domain of $g(g(x))$ that is less than 10. | nvidia |
The area of a triangle with a fixed base length of 24 is to be maximized. The sum of the lengths of the two legs of the triangle is 36. What is the maximum possible area of the triangle? | nvidia |
A positive integer $m$ is chosen such that it has exactly four positive divisors and is the product of two distinct prime numbers. A positive integer $n$ is chosen such that it is the smallest number with exactly six positive divisors and is the product of three distinct prime numbers. Find $m+n$. | nvidia |
Let $a$, $b$, $c$ be integers such that $a^2+b^2+c^2=2ab+2ac+2bc+9$. Prove that $abc$ is divisible by $7$. | nvidia |
Prove that for any angle $x$, the following inequality holds:
\[\frac{\cos^2 x}{\sin x + 1} + \frac{\sin^2 x}{\cos x + 1} \geq 1.\] | nvidia |
The equation $x^2+y^2=1$ describes a circle centered at the origin with radius $1$. There are two lines that pass through the point $(1,0)$ and are tangent to this circle. These two lines intersect at a point $P$ that lies outside the circle. Find the shortest distance from the point $P$ to the line $x=1$. | nvidia |
Let's say we have a positive integer n such that the sum of the squares of its digits is equal to 100. How many distinct four-digit numbers n are there? | nvidia |
Determine the value of the constant $b$ such that the expression $9x^2 - 24x + b$ is a perfect square trinomial. | nvidia |
Let $f(x)$ be a polynomial of degree $4$ such that the value of $f(x)$ is constant and nonzero for all $x$ in the set $\{1, 2, 3, \ldots, 8\}$. Determine the largest possible number of real roots of the equation $f(x) = 0$. | nvidia |
Count the number of positive integers less than 100 that have only prime digits. | nvidia |
Find the largest positive integer $n$ such that there exists a solution to the equation
\[\left| x^n - x^2 + \frac{1}{2016} \right| = \frac{1}{2016}.\] | nvidia |
Consider the infinite geometric series $1 + x + x^2 + \cdots$. Determine $x$ such that the sum of every other term (starting from the second) is equal to twice the sum of the remaining terms. | nvidia |
Find the number of integers in the range from 1 to 1000 that are not divisible by any of the prime numbers 2, 3, 5, or 7. | nvidia |
Determine the sum of all single-digit replacements for $z$ such that the number ${24{,}z38}$ is divisible by 6. | nvidia |
A square and a circle are inscribed in the same regular hexagon. The ratio of the area of the square to the area of the circle is 1.5. Determine the side length of the hexagon. | nvidia |
Given that $x$ is real and $x^3+\frac{1}{x^3}=52$, find $x+\frac{1}{x}$. | nvidia |
A rectangular prism has a volume of 120 cubic units. If the dimensions of the prism are such that the product of the lengths of two of the sides is 16 units, find the length of the third side. | nvidia |
A right circular cone is inscribed in a right circular cylinder such that the radius of the base of the cone is one-third of the radius of the base of the cylinder, and the height of the cone is equal to the height of the cylinder. What is the ratio of the volume of the cone to the volume of the cylinder? | nvidia |
The sum of three different numbers is 75. The two larger numbers differ by 5 and the two smaller numbers differ by 2. What is the value of the largest number? | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xy) + f(xz) - f(x) f(yz) \ge 1\]for all real numbers $x,$ $y,$ and $z.$ | nvidia |
Find the largest four-digit integer that is divisible by the product of its digits. | nvidia |
Find the smallest positive integer n such that $n^2\equiv n+1\pmod{7}$.
Express n in the form $n=a\cos \frac{2\pi k}{7}+b\sin \frac{2\pi k}{7}$, where a, b, and k are integers. | nvidia |
Let $a$ and $b$ be positive integers such that $a>b$. Prove that the greatest common divisor of $a^b-b^a$ and $a^a-b^b$ is equal to $a-b$. | nvidia |
A bookshelf contains 12 books, and each book has a unique color. The books are arranged on the shelf in a specific order. Given that the colors of the books are red, blue, green, yellow, orange, purple, pink, black, white, gray, brown, and turquoise, and that the books are arranged such that no two books of the same co... | nvidia |
Let $m$ and $n$ be positive integers such that $\gcd(m,n) = 15$. Determine the smallest possible value of $\gcd(24m,30n)$ that is divisible by $3$. | nvidia |
Find all complex numbers $z$ such that the number obtained by reading the digits of $z$ backwards (in the same base) is a multiple of $z$. | nvidia |
Find the product of all positive integers $n$ such that $2^{n-1}<n^2<2^n$. | nvidia |
Let $a$ and $b$ be the roots of the equation $2x^2-5x+3=0$. Find the sum of the squares of the numbers $a+b$, $ab$, and $1/(a^2+b^2)$. | nvidia |
Find the value of $x$ for which the expression $\frac{x}{|x|}$ is equal to $\frac{2x-3}{|2x-3|}$ and which has the largest absolute value. | nvidia |
If $a$ and $b$ are positive integers such that $\gcd(a,b) = 24$, then what is the smallest possible value of $\gcd(8a,12b)$? | nvidia |
Find the value of $a$ that satisfies the equation $293_{a}+468_{a}=73B_{a}$, where $B_{a}=11_{10}$. | nvidia |
Let
$f(x)=\ds\sqrt [3]{\log _x{(x+1)}}+\ds\sqrt [3]{\log _{x+1}{(x+2)}}+\cdots +\ds\sqrt [3]{\log _{x+n-1}{(x+n)}}$
where $n$ is a positive integer and $x>0$.
Find $n$ for which the function $f(x)$ has a relative extremum. | nvidia |
Find the value of $x$ such that the equation
\[3x^2-10x+5=0\]
holds true, where the coefficients of the quadratic equation are determined by the system of equations:
\[3a^2-2b^2=1\]
\[2a^3+b^3=5\] | nvidia |
Let $a$ and $b$ be two complex numbers such that $|a| = |b| = 1$. If $a^n + b^n = 1$ for all positive integers $n$, determine the maximum value of $|a + b|$. | nvidia |
The sum of the first 2020 positive odd integers and the sum of their reciprocals are equal. Find the minimum value of the sum of a pair of these odd integers, one of which is a multiple of the other. | nvidia |
Determine the number of different arrangements of the digits in the decimal representation of 3326 that are divisible by 2. | nvidia |
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