instruction_seed stringlengths 30 7.99k | source stringclasses 1
value |
|---|---|
Consider the equation $z^6 + z^4 + z^2 + 1 = 0.$ Determine the smallest positive integer $n$ such that all the solutions of this equation are $n^{\text{th}}$ roots of unity. | nvidia |
Let $z$ be a complex number such that the difference between the arguments of $z^3$ and $z$ is $2\pi/3$, and the difference between the arguments of $z^5$ and $z$ is $4\pi/5$. Find all possible values of $z$. | nvidia |
Find the smallest positive integer n for which the product of two numbers of the form $2^n+1$ and $2^n-1$ is not a perfect square. | nvidia |
Given that $$(a+b+c)(ab+ac+bc)=a^2+b^2+c^2$$and that $$a^3+b^3+c^3=3abc$$for real numbers $a$, $b$, and $c$, what is the value of $a+b+c$? | nvidia |
Let $g$ be a function taking the positive integers to the positive integers, such that
(i) $g$ is increasing (i.e. $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^2 + n^2 = (m + n)^2,$ then $g(m) = n$ or $g(n) = m.$
Find t... | nvidia |
Let $g(x)$ be a function defined for all real numbers $x$ such that $g(x) + g(1 - x) = 2$ and $g(x) = 0$ for $x \in [-1, 0]$. Find the smallest positive integer $n$ such that $g\left(\frac{1}{n}\right) = g\left(\frac{n - 1}{n}\right)$. | nvidia |
If $0.75= \frac{3y-2x}{144-5x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers. | nvidia |
Let $\mathbf{A}$ be a $2 \times 2$ matrix, with real entries, such that $\mathbf{A}^3 = \mathbf{0}.$ Find the number of different possible matrices that $\mathbf{A}^2$ can be. If you think the answer is infinite, then enter "infinite". | nvidia |
Find the maximum value of $10^x - 100^x,$ over all real numbers $x.$ | nvidia |
Find the area of the rectangle with integer sides such that the
length is three times as large as the width, and the perimeter
of the rectangle is equal to the perimeter of a square with
side length 24 cm. | nvidia |
Find the number of perfect squares that can be expressed as the sum of two consecutive prime numbers less than 20. | nvidia |
A polynomial function $P(x)$ with real coefficients can be expressed as the sum of two square roots
\[P(x) = \sqrt{2x-1} + \sqrt{x^2-2x}.\]
Determine the degree of $P(x)$ and the smallest $x$ value for which $P(x)$ is defined. | nvidia |
Given that $a$, $b$, and $c$ are nonzero real numbers such that $a^3+b^3+c^3=27abc$, find the value of $\left|\frac{(a+b+c)}{(a-b)(b-c)(c-a)}\right|$. | nvidia |
Let $f(x)$ be a real function such that
$f(x+1)=f(x)+f(2x)$ for all $x\neq 0$ and $f\left(\frac{1}{2}\right)=\frac{1}{2}$.
Find $f(10)$. | nvidia |
Let $x$ be a positive real number such that the decimal representation of $\sqrt{x}$ starts with the digits 197. Find the smallest possible value of $x$. | nvidia |
When the digits in the number $2005$ are reversed we obtain the number $5002,$ and $5002 = a \cdot b \cdot c$, such that $a$, $b$ and $c$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$ and $p_3$ such that $p_1 + p_2 + p_3 = a+b+c$? | nvidia |
Given a vector $\mathbf{a}$ such that $\|\mathbf{a}\| = 1$ and $\mathbf{a} \cdot \mathbf{b} = \|\mathbf{b}\|^2$ for all vectors $\mathbf{b}$, determine the possible values of $\mathbf{a} \cdot \mathbf{a}$. | nvidia |
Find the minimum value of the expression $x^2+y^2-8x-6y+22$ for real $x$ and $y$. | nvidia |
Given positive integers $x$ and $y$ such that $x\neq y$ and $\frac{1}{x} + \frac{1}{y} = \frac{1}{12}$, find the largest possible value of $xy$. | nvidia |
A game is played with two fair dice, one with numbers 1, 2, 3, 4, 5, and 6, and the other with numbers 0, 2, 4, 6, 8, and 10. A player rolls both dice and calculates the sum of the numbers shown. What is the probability that the sum is greater than the product of the numbers shown? | nvidia |
Determine the sum of the cubes of the roots of the equation
$$x^4-7x^3+5x^2+x+1=0,$$
given that the sum of the fourth powers of the roots is 1111. | nvidia |
A triangle has three sides of the following side lengths: $8$, $12$, and $y^3$. What are all of the positive integer values of $y$ such that the triangle exists? Separate your answers using commas and express them in increasing order. | nvidia |
The remainder upon division of a polynomial by $x^2+3x+2$ can be written in the form $ax+b$, where $a$ and $b$ are constants. Find all possible ordered pairs $(a,b)$ such that when the polynomial $x^2+3x+2$ is multiplied by $x+1$, the remainder becomes zero. | nvidia |
Find all real numbers $x$ such that
\[\frac{a^x + b^x}{c^x + d^x} = \frac{e}{f}.\]
Enter all the solutions, separated by commas.
Note: $a$, $b$, $c$, $d$, $e$, and $f$ are positive real numbers, and $\frac{e}{f}$ is a rational number. | nvidia |
In an equilateral triangle $ABC$ with side length 6, $M$ is a point on $BC$ such that $AM=4$ and $MB=3$. Find the area of triangle $ABM$. | nvidia |
Find all real numbers $x$ such that the sum of the cubes of the roots of the equation $x^3 + ax + b = 0$ is equal to $16$. Determine $a$ and $b$. | nvidia |
Let $a$ be a real number such that the inequality $\left| \frac{x}{x^2 + a} \right| < 1$ has no real solutions. What is the least possible value of $a$? | nvidia |
Let $a$, $b$, $c$ and $d$ be positive integers such that
$$\begin{array}{rcl}
a+b+c+d&\equiv&0\pmod{5}\\
a+b-c-d&\equiv&1\pmod{5}\\
a-b+c-d&\equiv&2\pmod{5}\\
-a+b+c-d&\equiv&3\pmod{5}.
\end{array}$$
Determine the smallest possible value of $a+b+c+d$. | nvidia |
Find the number of integers between 23 and 145 that are not divisible by 5 or 7. | nvidia |
In triangle $ABC$, the bisectors of $\angle A$ and $\angle B$ meet at point $D$. The lengths of the sides $AB$, $BC$, and $CA$ are $a$, $b$, and $c$, respectively. Suppose that point $P$ is on the bisector of $\angle A$ such that $PB = PD = a$. Find the length of $CP$. | nvidia |
There are integers $b,c$ for which both roots of the polynomial $x^2-x-1$ are also roots of the polynomial $x^5-bx-c$. Determine the product $bc$. | nvidia |
Find the largest value of m such that 7x^2+mx+90 can be factored as the product of two linear factors with integer coefficients. | nvidia |
Consider the function $f(x) = \frac{x^3-2x^2-x+2}{x^4+3x^3-4x-12}$. Let $a$ be the number of local maximum points on the graph of $f$, $b$ be the number of local minimum points, $c$ be the number of inflection points, and $d$ be the number of horizontal asymptotes. Find $a+b+2c+3d$. | nvidia |
Determine the value of $c$ that satisfies the equation $214_{c}+143_{c}=357_{c}$. | nvidia |
Compute the least positive integer $n$ such that the expression \[ \left\lfloor \dfrac{(n+2)^3}{(n+1)(n+2)} - \dfrac {n^3}{(n+1)(n+2)} \right\rfloor + \left\lfloor \dfrac {n^3}{(n+1)(n+2)} - \dfrac {(n-2)^3}{(n+1)n} \right\rfloor \]evaluates to a perfect square. | nvidia |
The function
$$f(x)=x^4-2ax^3+3x^2+bx-1$$
has two multiple roots. Determine the sum of the possible values of $a$. | nvidia |
Given that $\log_{10}2=0.3010$ and $\log_{10}3=0.4771$, compute the integer part of $\log_{10}(2^{51})+\log_{10}(3^{43})$. | nvidia |
If $a$,$b$, and $c$ are positive real numbers such that $a(b+c) = 152$, $b(c+a) = 162$, and $c(a+b) = 170$, then find $abc.$ | nvidia |
Consider the set of all polynomials of degree 3 with real coefficients that have all roots equal to -1 or 1 and have the leading coefficient equal to 1.
Find the number of polynomials in this set, which have at least one root equal to -1. | nvidia |
Let $a$ be a real number, $a > 0,$ and let $x$ and $y$ be positive real numbers such that
\[x + y + \frac{1}{x} + \frac{1}{y} = a.\]Find $xy.$ | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 1.$ Find the minimum value of
\[\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}.\] | nvidia |
Find the value of $x$ in the geometric sequence $x, x^2, 3x^3$, given that the sum of the first two terms is 25. | nvidia |
A positive integer is called "balanced" if the product of its digits at even positions (counting from the right) equals the product of its digits at odd positions. Find the number of balanced integers with 5 digits. | nvidia |
Let $a,$ $b,$ $c$ be a three-term arithmetic series where all the terms are positive, such that $abc = 64.$ Find the smallest possible value of $b.$ | nvidia |
Two positive integers, $a$ and $b$, are such that the product of their reciprocals plus the sum of their reciprocals is equal to 1. Given that both $a$ and $b$ are less than 10 and relatively prime, determine the product of $a$ and $b$. | nvidia |
Find the largest positive integer $k$ such that $k^2$ is a factor of $1225 - 1$. | nvidia |
Let $p(x)$ be a polynomial such that the equation $\frac{(x-3)(p(x)-2)}{(x+2)(x-1)} = 3$ has infinitely many solutions for $x.$ For this polynomial $p(x),$ it turns out that there are only finitely many values of $x$ which are not solutions to the equation. Find the product of these values of $x.$ | nvidia |
Let $a,$ $b,$ and $c$ be positive integers such that $a+b+c=18$. Find the maximum value of $\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ac}{a+c}$. | nvidia |
Determine the sum of all values of $x$ such that $x^3 = -8$. | nvidia |
Find the integer $k$ that satisfies $0\le k<29$ and $$-500\equiv k\pmod{29}~?$$ | nvidia |
A bookshelf has two types of books: fiction and non-fiction. The number of fiction books, in boxes, is at least 8 boxes more than one-third the number of non-fiction books, and is no more than three times the number of non-fiction books. Find the least number of boxes of non-fiction books that can be placed on the book... | nvidia |
Let $n$ be the least positive multiple of 21 that is greater than 380.
Thus, the number is
$$21\left\lfloor \frac{380}{21}\right\rfloor +21$$ | nvidia |
The integers $A$ and $B$ are chosen such that
$$\frac{A}{x+2}+\frac{B}{x^2-9x}=\frac{x^2-3x+7}{x^3-7x^2-18x}$$
for all real values of $x$ except $-2$, $0$, and $9$. Find $B/A$. | nvidia |
Given a planar, closed curve $C$, define the function $f(P)$ for a point $P$ in the plane as the sum of the distances from $P$ to the two farthest points on $C$. Let $M$ be the smallest possible value of $f(P)$ over all points $P$ in the plane. Prove that if $C$ is a triangle, then $M$ is equal to the perimeter of the ... | nvidia |
Let $f(x)$ be a function such that $f(0)=1$ and $f(xy)=f(x)f(y)-2f(x+y)$ for all real numbers $x$ and $y$. Find $f(1992)$. | nvidia |
What is the smallest positive integer $n$ such that $n^3-13n+19$ is a prime number? | nvidia |
A sequence of five consecutive positive integers has a product that is divisible by 24. What is the smallest possible value of the largest integer in this sequence? | nvidia |
Find the sum of all divisors of 240 that are multiples of 3. | nvidia |
Given that the point $(4,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=3f(4x)+5$. What is the sum of the coordinates of that point? | nvidia |
In the equation $x^2 - 6ax + a^2 = 0$, determine the value of $a$ such that the sum of the squares of the roots is equal to 8.5. | nvidia |
Call a positive integer $n$ "doubly-oddball" if there exist positive integers $a$, $b$, and $c$ such that $a^b - c^2 = n$ and both $a$ and $c$ are odd. How many doubly-oddball integers are less than $1000$? | nvidia |
How many positive integers are there such that their sum is less than 20 and the sum of their squares is less than 100? | nvidia |
The vertices of a cube are labeled with positive integers such that each vertex is adjacent to exactly three others with which it shares a common factor greater than 1. Find the smallest possible sum of the numbers on the vertices adjacent to the vertex labeled 15. | nvidia |
Find the number of divisors of 315 that are also perfect squares. | nvidia |
Find the sum of all the even factors of $180$, knowing that the prime factorization of $180$ is $2^2 \cdot 3^2 \cdot 5$. | nvidia |
Let $a_1, a_2,..., a_n$ be a sequence of positive integers such that the sum of the first $k$ terms is a perfect square for each $k$. If $a_1 = 1$ and $a_2 = 8$, find $a_3$. | nvidia |
Given positive integers $a$ and $b$, simplify the expression $\frac{a^{\log_{b} a}}{b^{\log_{a} b}}$. | nvidia |
Determine the number of three-digit numbers that are multiples of 5 and have a tens digit that is twice the units digit. | nvidia |
A bookshelf has 7 shelves, and each shelf can hold a maximum of 5 books. If the bookshelf is currently empty, what is the maximum number of books that can be placed on it such that no two books of the same color are on the same shelf? | nvidia |
The legs $PQ$ and $PR$ of a right triangle $PQR$ have lengths which are positive whole numbers. Given that $PQ = k \cdot PR$ for some positive integer $k$, and that the length of the hypotenuse $QR$ is 15, what are all the possible values of $k$? | nvidia |
A rectangle is inscribed in a triangle with base 12 cm and height 8 cm, such that one side of the rectangle lies on the base of the triangle and the other side is parallel to the height. If the area of the rectangle is 24 cm^2, find the perimeter of the rectangle. | nvidia |
Find all real numbers $x$ such that $x$ is the geometric mean of the set $\{2x-1, x+1, 3\}$ and $x+2$ is the geometric mean of the set $\{2x+2, x+4, 4\}$. | nvidia |
Find the value of the product
\[\binom{100}{1} \cdot \binom{99}{3} \cdot \binom{98}{5} \cdots \binom{2}{99}.\] | nvidia |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | nvidia |
Let A(x)=\sum_{n=0}^\infty c_n x^n be a function given by its power series about the point x=0. Suppose that for every positive number r the function f(x)=A(r/x) satisfies lim_{x\to0}f(x)/x^3=0. Determine the least integer n such that c_n may be nonzero. | nvidia |
Find all values of $x$ such that $\log_3(\sqrt{x})=\frac{1}{2}(\log_3(x)+\log_3(5))$. | nvidia |
Let $f(x)$ be a function such that $f(2x) = f(x) + 1$ for all positive integers $x$. If $f(1) = 3$, find the value of $f(32)$. | nvidia |
Let $p$ and $q$ be real numbers such that $p^2 + q^2 = 4.$ Determine the set of all possible values of $pq(p + q).$ | nvidia |
Find the sum of all possible values of $x$ that satisfy the equation $|x-2|+|x-5|=3$. | nvidia |
Find the smallest four-digit integer whose digits are distinct and form an arithmetic sequence, with the additional constraint that the sum of the digits is a perfect square. | nvidia |
Let $f(n)$ be the sum of the proper positive integer divisors of $n$. For how many values of $n$ is $f(n)$ a perfect square, given that $2 \le n \le 100$? | nvidia |
The region $R$ is defined by the set of all points $(x,y,z)$ such that $|x+y| \le 2,$ $|x+z| \le 3,$ and $|y+z| \le 4.$ Find the volume of $R.$ | nvidia |
In the equation
$$x^2 + y^2 = 9$$
find the function $f(x)$ such that the boundary of the region on the $xy$-plane, where $x \ge 0$, can be expressed in the form $y = f(x)$. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $\dfrac{ab+ac+bc}{a+b+c}=1$. If $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3.25$, what is the value of $\dfrac{1}{abc}$? | nvidia |
How many of the positive integers can be expressed as the sum of two squares of integers? | nvidia |
A deck of 52 cards is shuffled and 5 cards are drawn. It is given that there is at least one pair of cards of the same rank, but no three cards of the same rank. If the cards that are not part of the pair are replaced with new cards from the remaining deck, what is the probability that at least three of the five cards ... | nvidia |
Suppose the parabola defined by the equation $y=x^2+bx+c$ has vertex $(h,k)$ where $h$ and $k$ are integers. Find the number of possible values of $b$ for which there are three points on the parabola where both coordinates are integers. | nvidia |
Find the smallest positive integer n for which there exists a polynomial f(x) with integer coefficients and degree 3 such that f(n) = n and f(x) ≠ x for any integer x ≠ n. | nvidia |
Count the number of permutations of the numbers 1 through 5 such that the sum of any two consecutive numbers is odd. | nvidia |
Compute the product of all possible values of $x$ such that $\sqrt{x+1} + \sqrt{x-1} = \sqrt{4x+1}$. | nvidia |
Let $A$ be a symmetric $2\times2$ matrix. Assume that $A$ is orthogonal. Determine its trace.
(Remember that the trace of a square matrix is the sum of the elements on its main diagonal) | nvidia |
The sum of the cubes of two numbers is 126, and the product of the two numbers is 10. What is the value of the sum of the two numbers? | nvidia |
What is the positive integer $N$ for which $$ 22^2 \times 55^2 = 10^2 \times N^2 \ ? $$ | nvidia |
How many positive integers less than 60 have an even number of positive divisors? | nvidia |
Let $x$ be a positive real number. Find the maximum value of $x^3 + \sin \left( \frac{\pi}{x^3} \right)$. | nvidia |
Determine the smallest positive integer $m$ such that when the decimal representation of $m$ is reversed, the resulting number is a multiple of 24. Compute the ratio of $m$ to its reverse. | nvidia |
Find the sum of all positive integers n such that 109 is a divisor of 53^n + 1. | nvidia |
Determine $x$ such that $\sqrt[4]{(\sqrt{x})^3}=18$. | nvidia |
Let $a,$ $b,$ $c$ be real numbers such that $a + b + c = 0.$ Find the set of all possible values of $ab + ac + bc.$ | nvidia |
Let \( y \) be a positive real number. Find the largest possible value of \( n \) such that for any positive integer \( k \leq n \), the following inequality holds:
\[ k \cdot \lceil y \rceil - \lceil k \cdot y \rceil \geq 2.\] | nvidia |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.