instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find the number of positive integers $n$ such that $2^n$ is between $\sqrt{10}$ and $\sqrt{1000}$. | nvidia |
The sum of the squares of the first 100 prime numbers and the sum of the cubes of their reciprocals. | nvidia |
Two numbers are independently selected from the set of positive integers less than or equal to 5. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | nvidia |
Find the smallest positive integer $a$ such that for any positive integer $n$, the remainder of $n^2 + 1$ when divided by $a$ is not equal to 2. | nvidia |
Let $A$ be a set of 17 elements and let $B$ be a set of 25 elements.
Find the number of binary relations from $A$ to $B$ that are not functions from $A$ to $B$. | nvidia |
Find the sum of all positive integers less than 20 that have exactly two distinct prime factors. | nvidia |
Let's call the points $A(2,2)$ and $B(-1,-1)$.
Given that point $C$ is such that $\angle ACB=90^\circ$, and $AB=3\cdot AC$.
Find the distance between points $A$ and $C$. | nvidia |
Determine the smallest positive integer $n$ such that the sum of the greatest common divisor of $n$ and $100$ and the least common multiple of $n$ and $100$ is a multiple of $100$. | nvidia |
What is the smallest square number that can be expressed as the sum of four distinct prime numbers, each with a digit sum of 10? | nvidia |
For how many ordered pairs of positive integers $(a,b),$ with $b<a\le 50,$ are both $\frac ab$ and $\frac{a+2}{b+2}$ integers? | nvidia |
Consider a sequence of integers $a_1, a_2, \ldots, a_{11}$ such that $a_1 + a_2 + \cdots + a_{11} = 77$. What is the minimum number of terms that must be removed so that the remaining sum is divisible by 7? | nvidia |
How many positive integers can be expressed in the form $2^a3^b5^c7^d$, where $a$, $b$, $c$, and $d$ are integers such that $0 \le a < b < c < d \le 10$? | nvidia |
There are $36$ different complex numbers $w$ such that $w^{36}=1$. For how many of these is $w^9$ a real number? | nvidia |
In the coordinate plane, a circle centered at the origin with radius 5 intersects the parabola $y=x^2$. Find the sum of the x-coordinates of the intersection points that lie in the first quadrant. | 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 |
What is the sum of all of the two-digit primes that are greater than 12 but less than 99 and are still prime when their two digits are interchanged? | 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 |
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 value of $k$ for which the function $f(x)=\frac{k}{x^2}+x^2$ attains its minimum value at $x=a$, where $a$ is a fixed positive real number. | nvidia |
A right triangle with integer length sides is inscribed in a circle with radius 25 units, such that the hypotenuse is a diameter of the circle. Find the length of the longest leg. | nvidia |
Let $b_1,$ $b_2,$ $\dots$ be a sequence of real numbers such that
\[b_n = \frac{3b_{n - 1} + n}{4}\]for all $n > 1.$ Find the largest possible value of $b_1$ such that the sequence is strictly increasing. | nvidia |
Find all possible values for the greatest common divisor (GCD) of two positive integers a and b, given that the product of the GCD and the least common multiple (LCM) of a and b is 100, and the sum of the digits of a is twice the sum of the digits of b. | nvidia |
Compute the smallest positive integer $n$ such that $n + i,$ $(n + i)^2,$ and $(n + i)^3$ are the vertices of a triangle in the complex plane whose area is greater than 2015. | nvidia |
There are eight positive real numbers, $a_1,$ $a_2,$ $\dots,$ $a_8,$ such that each number except the last is less than twice the next number: $$a_i<2a_{i+1} \qquad (1\le i\le7).$$ What is the maximum value of $\dfrac{a_1}{a_8}$? | nvidia |
The function $f(x)$ satisfies
\[f(2^x) + xf(2^{-x}) = 1\]for all real numbers $x.$ Find $f(2).$ | nvidia |
Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$. Find $g^{-1}(f(-7))$. | nvidia |
The quadratic equation $x^2 + ax + b = 0$ has solutions $x_1$ and $x_2$ such that $x_1 + x_2 = -a$ and $x_1 x_2 = b$. Determine the sum of the solutions of the equation $(x - x_1)(x - x_2) = 0$ when it is known that the product of the solutions is twice the sum of the solutions. | nvidia |
Find the smallest positive integer that is both an integer power of 11 and is not a palindrome. | nvidia |
My co-worker Erich is very odd. He only likes numbers that are divisible by 5. How many different last digits are possible in numbers that Erich likes? | nvidia |
Find the largest value of $k$ such that the expression $x^3 + kx^2 + x + k$ can be factored as the product of a linear factor and a quadratic factor with integer coefficients. | nvidia |
Find the value(s) of $x$ such that $4xy-6y+3x-4=0$ is true for all values of $y$. | nvidia |
Find the smallest positive integer m such that for every positive integer n the remainder when mn(mn+1) is divided by m+n is m itself. | nvidia |
Problem:
A portion of the graph of a cubic 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);
... | nvidia |
What is the largest positive integer that is divisible by all the positive integers from 1 to 6? | nvidia |
How many positive integers less than 100 have exactly three positive factors? | nvidia |
Find the largest integer $k$ such that $14^k$ is a factor of $21!$. | nvidia |
Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{a} & \frac{1}{b} \\ \frac{1}{c} & \frac{1}{d} \end{pmatrix} \renewcommand{\arraystretch}{1}.\] | nvidia |
The product of two positive integers is 48. The positive difference between these two integers is 4. What is the sum of the two integers? | nvidia |
Two circles of radius 1 are externally tangent at point P. Two points M and N are chosen on the larger arc of each circle, respectively, such that the line segment MN is tangent to both circles and intersects the line joining the centers of the two circles at a point Q. Find the length of QP if $\angle MPN$ measures 12... | nvidia |
Let $a_1, a_2, a_3$ be the three real roots of the equation $x^3+2x^2-x-2=0$. Find $a_2-a_1-a_3$, given that $a_1$ and $a_3$ are the roots of the equation $x^2-2x+1=0$. | nvidia |
Define a sequence of integers by $a_1 = 0$ and
\[a_{n + 1} = a_n^2 + n\]for all $n \ge 1.$ Find the largest positive integer $N$ such that $a_N$ can be expressed as a sum of two perfect squares. | nvidia |
In a triangle, the sum of the lengths of the two shorter sides is equal to the length of the longest side. If the perimeter of the triangle is 24, find the length of the longest side. | nvidia |
In an equilateral triangle with integer side length, the perpendicular bisector of one of the sides intersects the opposite side at a point that divides it into two segments with a length ratio of 2:1. What is the length of the shortest segment? | nvidia |
Solve the equation $\sqrt{2x+1}=\sqrt{x+2}$, where $x$ is a real number. | nvidia |
In the equation
$$kx^2+5x+1=0$$
determine $k$ such that the product of the roots be equal to two. | nvidia |
Find the minimum value of $25^x - 5^x + 2$ over all real numbers $x$. | nvidia |
Determine the number of positive integers between 100 and 1000 (inclusive) that can be expressed as the sum of three consecutive integers. | nvidia |
Let $A$ be a $3 \times 3$ matrix such that the sum of the entries of each row is $2$, and $\det A = -7$. Find $\det (A^{-1})$. | nvidia |
Let $A$ be a set of positive integers such that for any integer $x$, $x$ is in $A$ if and only if $2x$ and $3x$ are also in $A$. Three integers are chosen independently and at random with replacement from the set $A$ and labeled $x_1,x_2,$ and $x_3$ in the order they are chosen. The probability that both $x_1+x_2$ and ... | nvidia |
Given that $x$ is a positive integer less than 100, how many solutions does the congruence $x^2 \equiv 1 \pmod{34}$ have? | nvidia |
Let $f(n)$ be the number of ways to color the squares of a $1$ by $n$ board with red and blue such that no two adjacent squares are colored red. Prove that $f(17)=19448$ and calculate $f(19)$. | nvidia |
Find the smallest positive integer $n$ such that the interval $(n^2, (n+1)^2)$ contains exactly two prime numbers. | nvidia |
A circle and a parabola intersect at four points. Two of these points are given by the coordinates (0,0) and (4,4). Find the equation of the circle such that the sum of the areas of the two triangles formed by the points of intersection and the center of the circle is minimal. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x \neq y$ and $x + y = 120$. Find the maximum possible value of $\frac{x^2y^2}{x^2+y^2}$. | nvidia |
Find the number of positive six-digit integers that are palindromes. | nvidia |
The polynomial $P(x)$, with real coefficients, satisfies the condition:
$$P(x^2+2x+1) \le P(x^2-2x+1)$$
for all real numbers $x.$ Find the product of the roots of $P(x)$. | nvidia |
A fair six-sided die is rolled three times. If the probability that the sum of the three rolls is greater than 14 can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a+b$. | nvidia |
Two fair, eight-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 13? | nvidia |
Find the largest possible value of $n$ such that $n^2$ can be expressed as the difference of two consecutive squares. | nvidia |
Let $\alpha(n)$ be the number of positive divisors of a positive integer $n$. $\mathcal{A}$ is the set of positive integers such that for all elements $m$ in $\mathcal{A}$, we have that $\alpha(m)=6$ and $m<10^{5}$. If $x$ is the smallest element in $\mathcal{A}$, compute the product of the digits of $\alpha(x)$. | nvidia |
How many positive integers $n$ satisfy\[\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?\](Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$
| nvidia |
Compute the sum of all the roots of $(2x+3)(x-4)+(2x+3)(x-6)=0$. | nvidia |
Let $P(x)$ be a polynomial of degree $3$ with real coefficients satisfying the following conditions:
(i) There exist real numbers $r$ and $s$ such that $P(x) = (x-r)^2(x-s)$;
(ii) There exist positive integers $m$ and $n$ such that $P(m) = 0$ and $P(n) = 0$;
(iii) For every positive integer $k$ such that $P(k) \neq ... | nvidia |
Find the sum of all positive integers less than 100 that are perfect cubes of perfect squares. | nvidia |
Given the matrix
\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \]
where $a, b, c, d$ are positive integers such that $ad - bc = 1,$ find the remainder when the determinant of the matrix
\[ \begin{pmatrix} a^{12} & b^{12} \\ c^{12} & d^{12} \end{pmatrix} \]
is divided by the determinant of the matrix
\[ \begin{p... | nvidia |
A square is divided into four rectangles by two horizontal and vertical lines. The side length of the square is a prime number and the areas of all four rectangles are positive integers. What is the least possible area of the square? | nvidia |
Find the value of $k$ such that the sum of the first $k$ terms of the geometric sequence $2,6,18,54,\ldots$ is equal to $2^k$. | nvidia |
Let $a$ and $b$ be positive integers such that $b>1$. How many pairs $(a,b)$ satisfy the equation
\[\dfrac{b^a-1}{b-1} = a^2+1\] | nvidia |
Consider the operation $\oplus$ defined as $a\oplus b=\frac{a}{b}+b$. If $x$ is a positive real number such that $x\oplus x=5$, what is the value of $x\oplus(2x)$? | nvidia |
Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 70$. Express $x$ as a decimal. | nvidia |
How many positive integers less than $500$ can be written as the sum of two positive perfect cubes in two different ways? | nvidia |
How many integers from 1 through 10000 can be written using only the digits 0, 1, 6, 7, and 9, with the condition that no two adjacent digits can be the same? | nvidia |
Let $\alpha$ and $\beta$ be complex numbers such that $\alpha+\beta$, $\alpha^2+\beta^2$, and $\alpha^3+\beta^3$ are all integers. Suppose that $\alpha^4+\beta^4=\alpha^5+\beta^5$. How many possible values of $\alpha+\beta$ are there? | nvidia |
Find the sum of all positive integer values of $a$ such that the equation $ax^2+12x+5=0$ has a repeated root. | nvidia |
Let $x,$ $y,$ $z$ be non-zero complex numbers such that $x^2+y^2+z^2=xy+yz+zx.$ Find all possible values of $\frac{x^3}{y^2z^2}+\frac{y^3}{z^2x^2}+\frac{z^3}{x^2y^2}.$ | nvidia |
The sum of the squares of three consecutive integers is 504. What is the sum of the cubes of these integers? | nvidia |
A circle and a parabola intersect at four points. Prove that these four points form a rectangle whose sides are parallel to the axes. | nvidia |
Find all rational numbers $r$ such that $|r^2 - 2| < \frac{1}{2}$. | nvidia |
What is the sum of the coordinates of the point that must be on the graph of $y=(h^{-1}(x))^2$ if the point $(5,2)$ is on the graph of $y=h(x)$, and $h(x)$ is a one-to-one function? | nvidia |
Determine the value of $a$ such that the equation $x^2+y^2+z^2+ax+by+cz=0$ represents a sphere tangent to the plane $x+y+z=0$. | nvidia |
Let $g$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[g(a^2 + b^2) = g(a) \times g(b)\]for all nonnegative integers $a$ and $b.$
Let $m$ be the number of possible values of $g(64),$ and let $t$ be the product of the possible values of $g(64).$ Find $m + t.$ | nvidia |
How many 4-digit numbers have the property that the sum of the thousands digit and the hundreds digit is equal to the sum of the tens digit and the units digit? | nvidia |
Find the sum of all real solutions to the equation \[\frac{x-2}{x^2+4x+1} = \frac{x-5}{x^2-10x}.\] | nvidia |
Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-4x+4|$ and $y=2\sin x+1$. | nvidia |
Let $m$ and $n$ be two relatively prime positive integers. Find the smallest positive integer $a$ such that $a^2 \equiv 1 \pmod{m}$ and $a^2 \equiv -1 \pmod{n}$. | nvidia |
Circle $C$ has its center at $(3, -4)$ and a radius of $\sqrt{85}$ units. What is the sum of the $x$-coordinates of the two points on circle $C$ that are also on the line $y=-4$? | nvidia |
Find the smallest positive integer $n$ such that the remainder of $\frac{21!}{15^n}$ is less than $15$. | nvidia |
Find the largest positive integer $n$ such that
\[\sin^n x + \cos^n x \ge \frac{1}{n}\]for all real numbers $x.$ | nvidia |
How many different positive integers can be represented as the sum of two distinct members of the set {1, 2, 3, 4, 5, 6}, such that the sum is also a multiple of 3? | nvidia |
What is the sum of all real numbers that are tripled when added to twice their reciprocals? | nvidia |
Find the number of integer values of $a$ for which the inequality $x^2 - 3x + a \ge 0$ has exactly two integer solutions. | nvidia |
The diagonals of a parallelogram intersect at a point that is 3 units away from the centroid of a triangle whose vertices are the midpoints of the sides of the parallelogram. If the lengths of the diagonals are 16 and 12 units, what is the area of the triangle? | nvidia |
What is the smallest positive integer that has exactly 3 distinct positive divisors? | nvidia |
Determine the value of $x$ such that $xx137_{11}$ is divisible by $13_{10}$, where $xx137_{11}$ represents a base-11 number whose first digit is $x$, middle two digits are also $x$, and the last two digits are $1$ and $7$. | nvidia |
Given that the polynomial equation \[x^3 + bx + c = 0,\]has $5-\sqrt{2}$ as a root, and it also has an integer root, what would be the integer value of the third root if the sum of the coefficients of the polynomial ($1+b+c$) equals zero? | nvidia |
A right triangle with legs of length $a$ and $b$ is inscribed in a circle. The center of the circle is equidistant from the vertices of the triangle. Find the radius of the circle in terms of $a$ and $b$. | nvidia |
Find the non-zero continuous functions $f(x)$ that satisfy $f(x+y) = f(x)f(y)$ for all real numbers $x$ and $y$. | nvidia |
Find all $x$ such that $x^2+5x<6$. Express your answer in interval notation. | nvidia |
Find the sum of all the terms of the form $\frac{1}{n(n+1)}$ where n is a positive integer and the sum is carried out from $n=1$ to $n=\infty$. Express your answer as a common fraction. | nvidia |
Find the sum of the proper divisors of $421$ that are less than $\sqrt{421}$. | nvidia |
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