instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find the number of factors of 5 that divides the product of all integers from 1 to 500, inclusive. | nvidia |
Given that $x$ is a positive integer, find the smallest value of $x$ such that $x^3 + 2x^2 - 7x + 1$ is divisible by $x^2 + 3x + 1$. | nvidia |
If $z$ is a complex number such that $|z|=5$, find $z\overline{z}$. (Note that $\overline{z}$ denotes the conjugate of $z$.) | nvidia |
Let $a$ and $b$ be positive integers such that $a+b=91$. Determine the maximum value of $\gcd(a,b)$. | nvidia |
The value of an infinite product $(1+\frac{1}{2})(1-\frac{1}{4})(1+\frac{1}{8})(1-\frac{1}{16})\dotsm$ is equal to $\frac{4}{3}$. Find the sum of the squares of the terms of the sequence $\{\frac{1}{2^{2n+1}}\}$ where $n$ runs from 1 to infinity. | nvidia |
Let $f(n)$ be the number of $n$-digit positive integers that contain exactly one 7, and let $g(n)$ be the number of $n$-digit positive integers that contain no 7's. Find the smallest positive integer $n$ for which $f(n) < g(n)$. | nvidia |
Find the sum of the squares of the roots of the polynomial $x^4 + 3x^2 + 1$. | nvidia |
Three standard dice are rolled. What is the probability that the sum of the numbers on the first two dice is equal to the number on the third die? | nvidia |
The blue curve shown is the graph of the polar equation $r=\cos\frac{\theta}{2}$. Find the area enclosed by the curve.
[asy]
size(150);
real ticklen=3;
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real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15... | nvidia |
In the three-dimensional space, the parallelepiped formed by the points (0,0,0), (a,b,c), (d,e,f), and (a+d, b+e, c+f) has volume V>0. If the sum of the squares of the components of the three vectors is equal to 1, i.e. a^2+b^2+c^2+d^2+e^2+f^2=1, determine the minimal possible value of the square of the length of the m... | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that
$$a^{\frac{1}{p}}\cdot b^{\frac{1}{q}}\cdot c^{\frac{1}{r}}=1,$$
where $p$, $q$, and $r$ are positive integers.
Determine the value of
$$a^{\frac{1}{q}}\cdot b^{\frac{1}{r}}\cdot c^{\frac{1}{p}}$$
in terms of $p$, $q$, and $r$. | nvidia |
Let $\alpha$ be a real number, $\alpha>1$, and let $w$ be a complex number such that $w-\frac{1}{w}=\alpha i$.
Find the argument of $w$. | nvidia |
Let $a_1 , a_2 , \dots$ be a sequence for which $a_1=2$ , $a_2=3$, and $a_n=\frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$? | nvidia |
Find the sum of all the odd integers between $-30$ and $50$, inclusive. | nvidia |
Given that $x^2+px+q=0$ and the roots of the equation are in the ratio of 2:3, if the larger root is 3 more than the smaller root, find the sum of the possible values of $p$. | nvidia |
Compute $1-2+3-4+ \dots -98+99$ . | nvidia |
Find the number of 10-digit positive integers that have all digits either 1 or 2, and have exactly three occurrences of the digit 1, with no two 1's consecutive. | nvidia |
Find the smallest positive integer $n$ such that the polynomial $x^n - 1$ has a factor of the form $x^a + x^b + x^c + 1$, where $a$, $b$, and $c$ are distinct positive integers. | nvidia |
Let $a$ be a real number randomly selected from the interval $[0,6]$.
Then, p and q are two relatively prime positive integers such that p/q is the probability that the equation $x^2-x=a \mod 12$ has more than one solution for x.
Find the value of $p+q$. | nvidia |
In a triangle, the slope of the altitude from a vertex $A$ to the opposite side $\overline{BC}$ is $m$, where $m>0$. Find the value of $k$ so that if $\overline{BC}$ has equation $x=k(y-7)$, then the length of the altitude from $A$ to $\overline{BC}$ is minimized. | nvidia |
A sequence of distinct positive integers $\{a_n\}$ has the property that for every pair of integers $i$ and $j$, the sum of the terms from $a_i$ to $a_j$ (inclusive) is divisible by $i + j$. If $a_1 = 1$, what is the largest possible value of $a_{100}$? | nvidia |
Find the number of positive integers $n \le 1000$ that can be expressed in the form
\[\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor = n\]for some real number $x.$
Determine the smallest positive integer $n$ such that there exist two different real numbers $x$ and $y$ that satisfy the equation
\[\lfloor x... | nvidia |
Consider the function $g(x)$ that satisfies the equation:
\[g(3^x) - xg(3^{-x}) = 2\]
for all real numbers $x.$ Find $g(3).$ | nvidia |
The positive integer divisors of $n^2$ that are less than $n$ but do not divide $n$ can be put into one-to-one correspondence with the positive integer divisors of $n^2$ that are greater than $n$ but do not divide $n$. This is done by the function $d\mapsto \frac{n^2}{d}$.
Now, the number of positive integer divisors ... | nvidia |
What is the probability that the sum of the numbers on two dice rolled simultaneously is less than 7? | nvidia |
The sides of a triangle are given by the equations $y+2$, $2y+3$, and $3y-5$. Find the positive integer values of $y$ such that the triangle is isosceles. | nvidia |
Let $a$, $b$ and $c$ be complex numbers such that $|a|=|b|=|c|=1$ and $|a+b+c|=2$. Find the smallest possible value of $|abc+1|$. | nvidia |
Determine the smallest possible sum $a+b$ if the number $246_a$ can be expressed as $55_b$, where $a$ and $b$ are the bases of the numbers. | nvidia |
Given that $\left(\dfrac{3}{4}\right)^5 = \dfrac{243}{1024}$, find $n$ such that $\left(1-\dfrac{1}{4}\right)^n = \dfrac{243}{1024}$. | nvidia |
Determine the number of positive integers less than or equal to 100 that are not divisible by any of 3, 5, and 7. | nvidia |
Find the number of ways to arrange six distinct objects in a row such that the first and last objects are of the same color, and the two middle objects are of the same color. | nvidia |
Determine the set of values of the parameter $a$ such that the inequality $x^2-ax+2\ge x+1$ is true for all real values of $x$. | nvidia |
On a $4\times 4$ chessboard, four distinct pawns are placed such that each column and row contains exactly one pawn. How many ways are there to move one of the pawns to an adjacent square (horizontally or vertically) so that the resulting configuration still satisfies the condition that each column and row contains exa... | nvidia |
What is the smallest positive integer $n$ such that $3n \equiv 1356 \pmod{22}?$ | nvidia |
Find all real numbers $x$ that satisfy the inequality $\sqrt{x^2+1} + \sqrt{x^2-1} \ge 4$. | nvidia |
Given that $r$ and $s$ are the roots of the equation $x^2+mx+n=0$, and that $r^2+s^2$ and $r^3+s^3$ are the roots of the equation $x^2+px+q=0$, determine the value of $q$ in terms of $m$ and $n$. | nvidia |
Determine the number of non-empty subsets of the set $\{1,2,\ldots,n\}$ such that the sum of the elements in each subset is even. | nvidia |
Find the value of $x$ such that $f(x) = 0$ for the polynomial $f(x) = 2x^4-17x^3+26x^2-24x-60$. | nvidia |
Find the largest three-digit number that is divisible by 7 and has a sum of digits equal to 21. | nvidia |
Find the value of $x$ such that $\sqrt{x + \sqrt{x}} = 7$. | nvidia |
Find the number of positive real solutions of the equation $\left| x^2 - \frac{3}{2} \right| = 5$. | nvidia |
Find a positive integer that is divisible by 15 and has a cube root between 7 and 7.5. | nvidia |
How many distinct positive integers can be expressed as the sum of two different elements from the set $\{2, 3, 5, 7, 11, 13, 17, 19\}$? | nvidia |
$$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & 1 & 7 & 3 & \square_8\\ & & & \square & 2 & 5_8\\ &+ & & & \square & 7_8\\ \cline{2-6} & & 2 & 5 & \square & 2_8\\ \end{array} $$
Determine the value of $\square$ that satisfies this addition problem. | nvidia |
Find the largest value of $x$ that satisfies the inequality $|x-3| + |x-7| \leq 10$. | nvidia |
Consider a complex number $z$ such that $\left|{z-1}\right|=3$. Determine the distance from the origin to the farthest point of the set of complex numbers that satisfy this equation. | nvidia |
The sum of the squares of two positive integers is 392. What is the product of these two integers? | nvidia |
Find the length of the median from the vertex $A$ of a triangle $ABC$ with vertices $A(0,0)$, $B(4,6)$, and $C(6,0)$. | nvidia |
What is the sum of the reciprocals of all the positive integer divisors of 24? | nvidia |
A square grid of 5 rows and 5 columns is filled with numbers from 1 to 25. Each row, column, and diagonal contains each number exactly once. The sum of all products of numbers that are in the same row is denoted by S. Find the maximum possible value of S. | nvidia |
A sphere with radius 3 is centered at the origin. Find all possible values of the distance from the origin to the plane $ax + by + cz = d$ such that the plane is tangent to the sphere. | nvidia |
Determine the largest positive integer $m$ such that $\binom{m}{\left\lfloor m/2 \right\rfloor}$ is a power of 2. | nvidia |
Let $f(x)=\frac{(x-2)(x-5)}{x-3}$ and $g(x)=f(x+5)$.
If the inequality $f(x)\leq 0$ is true if and only if $-3\leq x\leq 3$, determine all real values of $x$ such that $g(x)\leq 0$. | nvidia |
What is the smallest positive integer that has exactly three distinct positive factors and is greater than 10? | nvidia |
A five-digit number is formed using the digits 0 through 9, with each digit used at most once. The number must be divisible by 3 and have the property that the sum of its first three digits is equal to the sum of its last three digits. How many such numbers can be formed? | nvidia |
Let $g(x)$ be a function defined for all positive integers $x$, such that
$$g(x+1)-g(x)=3x+2.$$
Find the smallest possible value of $x$ for which $g(x+100)-g(x)$ is a multiple of 3. | nvidia |
Find the sum of all three-digit positive integers that can be written as the sum of three consecutive integers. | nvidia |
Find the number of terms in the expansion of $(x+y+z)^n$ that contain exactly two of the variables $x, y, z$, where $n$ is a positive integer. | 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 |
What is the sum of all integer values $n$ for which $\binom{20}{n}+\binom{20}{10}=\binom{21}{11}$? | nvidia |
The product of two consecutive positive integers is 204. What is the sum of the two integers? | nvidia |
The graph of $y = F(x)$ is shown in green below. The distance between grid lines is $1$ unit.
[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;
real tickdownbase=... | nvidia |
What is the sum of all positive integer values of $n$ such that $\frac{n+18}{n}$ is an integer? | nvidia |
Find the sum of the squares of all positive integers less than 10 that are not divisible by 3 or 5. | nvidia |
A sequence of positive integers $a_1,a_2,\ldots,a_n$ is said to be $k$-dominant if $\displaystyle\prod_{i=1}^n (a_i+i)^{\frac{1}{i}} > k^n$.
Determine the largest value of $k$ such that no $k$-dominant sequences of length $n$ exist for all sufficiently large $n$. | nvidia |
Let $x$ be a real number such that $x^3+4x=8$. Determine the value of $x^7+64x^2$. | nvidia |
Given that the Fibonacci sequence has the property that the ratio of consecutive terms tends to the golden ratio, find the value of the infinite product \prod_{n=2}^{\infty} \left( \frac{F_{n+1}}{F_n} - \frac{F_{n-1}}{F_n} \right) and determine its convergence. | nvidia |
Find the sum of the squares of the roots of the equation \[x^4 - 5x^3 + 7x^2 + 3x - 4 = 0,\] given that the sum of the roots is 5. | nvidia |
Determine the largest positive integer $n$ such that $11^n - 1$ is divisible by 100. | nvidia |
Find the sum of the eight smallest positive integers $m$ that satisfy the equation $\tau(m) + \tau(m+2) = 9$, where $\tau(m)$ denotes the number of positive integer divisors of $m$ (including $1$ and $m$). | nvidia |
The sum of the elements of a $3\times3$ matrix is equal to 81, while the sum of the squares of the elements is 1185.
Find the determinant of this matrix. | nvidia |
Given an integer n, let S(n) be the sum of the base 10 digits of n. For example, S(1507) = 13. For a particular positive integer n, S(n) = 1274. Which of the following could be the value of S(n+1)? | nvidia |
Determine the number of positive integers $n$ such that $2^n$ has a units digit of 7 when expressed in base 8. | nvidia |
Find the largest integer that is not greater than the sum of the square root of 2 and the square root of 5. | nvidia |
Given a monic polynomial of degree $n$ with real coefficients, where the first two terms after $x^n$ are $a_{n-1}x^{n-1}$ and $a_{n-2}x^{n-2}$, and it is known that $a_{n-1}=a_{n-2}$. The polynomial is otherwise unknown. Find the greatest upper bound for the product of the roots of the polynomial. | nvidia |
A function $f$ has the property that $f(x^2+1)=x^3-x$ for all real numbers $x$. Find the value of $f(2)$. | nvidia |
Find the smallest positive integer $n$ such that the number of prime divisors of $\frac{n!}{n^2}$ is greater than the number of digits in $n^2$. | nvidia |
Let $a$ and $b$ be positive integers such that the rectangle with side lengths $a$ and $b$ can be divided into two rectangles with the same perimeter, one of which has an area of $2a$ and the other an area of $3b$. Determine the minimum value of $a+b$. | nvidia |
Consider the function $f(n)=n^2-n$. Find the number of positive integers less than 100 for which $f(n)$ is a multiple of 7. | nvidia |
Find the smallest positive integer $n$ such that $\lceil\sqrt{n}\rceil-\lfloor\sqrt{n}\rfloor=1$. | nvidia |
Consider a semi-circle that contains an inscribed rectangle and an isosceles triangle, with the center of the circle being the midpoint of the hypotenuse of the triangle. If the perimeter of the rectangle equals 24 inches, and the radius of the semi-circle is 5 inches, find the length of the legs of the triangle in inc... | nvidia |
Problem:
A complex number $a+bi$ is such that its product with its conjugate is equal to 50, and the difference of its real and imaginary parts is 3.
Find the magnitude of this complex number. | nvidia |
What is the smallest positive integer that is a divisor of every number of the form $n^2 + 1$, where $n$ is a positive integer? | nvidia |
Find all real values of $a$ such that the inequality $|x^2 - 4x + 3| \leq a$ has exactly 3 integer solutions. | nvidia |
If $n$ is a positive integer, let $f(n)$ be the sum of the digits of $\frac{1}{5^{{}^n}}$ that are to the right of the decimal point. What is the smallest positive integer $n$ such that $f(n) > 10$? | nvidia |
Find the only positive real number $x$ such that $\displaystyle \frac{x^2-4x-1}{2x+5} = \frac{4x-2}{x^2-4x+4}$ | nvidia |
Determine the absolute value of the complex number $z$, knowing that $\left|z^4\right|=16$ and $\left|z^6\right|=64$. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x + g(y)) = g(x) + g(y) + g(g(x) + y)\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(0),$ and let $t$ be the product of all possible values of $g(0).$ Find $m \times t.$ | nvidia |
Find the value of $a$ such that the equation $x^2+ax+2=0$ has roots that are complex conjugates of each other, and the sum of the squares of the real and imaginary parts of each root is equal to 10. | nvidia |
The sum of the squares of two positive integers is 90. The product of the two integers is 27. What is the sum of the two integers? | nvidia |
Find the sum of all $x$ such that the expression $\frac{x^2-3x+2}{x^2-3x-4}$ is undefined. | nvidia |
Find all real solutions to the equation $x^{\frac{1}{x}} = e^{(1+\frac{1}{x}) \ln(x)}$. | nvidia |
Find the area of the triangle formed by the points $A(2,-3)$, $B(5,6)$, and the intersection point of the perpendicular bisector of the segment $AB$ with the line $y=2x+1$. | nvidia |
Charles has two six-sided die. One of the die is fair, and the other die is biased so that it comes up six with probability $\frac{2}{3}$ and each of the other five sides has probability $\frac{1}{15}$. Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes... | nvidia |
Count the number of ordered pairs of integers $(c,d)$ where $2 \leq c \leq 100$ and $d \geq 0$ such that the polynomial $x^2 - cx + d$ can be factored into the product of two distinct linear factors with integer coefficients. | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that $xyz = \frac{2}{3}.$ Compute the minimum value of $x^2y + y^2z + z^2x.$ | nvidia |
Simplify the expression
$$\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}+\sqrt{a+b+c}},$$
where $a$, $b$, and $c$ are positive integers and $c>\sqrt{b}$.
The answer can be written as $\frac{A\sqrt{B}+C\sqrt{D}}{E}$, where $A$, $B$, $C$, $D$, and $E$ are integers, $E$ is positive, and $B$ and $D$ are not divisible by the square of a... | nvidia |
Find the value of $b$ such that the equation $x^2 + 2bx + 3b = 0$ has exactly one solution, given that one of the solutions is an integer. | nvidia |
Find the smallest positive integer that can be placed in the blank space of the number $34\_\_\,56$, so that the resulting integer is divisible by 12 and all its digits are distinct. | 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 |
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