instruction_seed stringlengths 30 7.99k | source stringclasses 1
value |
|---|---|
Let $n$ be the smallest positive integer that satisfies the following conditions:
- $n$ is divisible by 12.
- $\frac{n}{12}$ is a perfect square.
- $\frac{n}{15}$ is a perfect cube.
Determine the smallest possible value of $n$ that satisfies all the given conditions and find the sum of its distinct prime factors. | nvidia |
Given that $0.\overline{ab}= \frac{x}{99}$, where $a$, $b$, and $x$ are integers and $10a+b$ is a two-digit positive integer, determine $a$ and $b$ if $x = 74$. | nvidia |
Let $a$ and $b$ be two positive integers such that $a+b$ and $a-b$ are both perfect squares.
Find the smallest possible value of $b$ given that $a+b$ is a multiple of 5. | nvidia |
Let $z$ be a complex number such that $z^2+\bar{z}^2=2$ and $|z|=3$. Find the complex number $z$. | nvidia |
Let $x$ be a positive integer such that its cube $x^3$ ends with four zeros.
What is the least possible number of zeros that the integer $x$ can have? | nvidia |
Find the smallest positive value of $y$ such that $2y + 1234$ is a perfect square. | nvidia |
Given two complex numbers $c$ and $d$ with $|c|=3$ and $|d|=5$, find the value of $k$ such that $|c+d|=k$. | nvidia |
Find the largest positive integer $m$ such that
\[\tan^m x + \cot^m x \ge \frac{1}{m}\]for all real numbers $x.$ | nvidia |
Compute the value of $\sin 45^\circ$ using the fact that the sum of the squares of the sine and cosine of any angle is equal to 1. | nvidia |
Find the largest three-digit number that is divisible by each of the first three prime numbers. | nvidia |
The equation $x^3-6x^2+11x-6=0$ has three solutions. One of the solutions is a rational number of the form $\frac{a}{b}$, where $a$ and $b$ are positive integers. What is the value of $a+b$? | nvidia |
Find the sum of the reciprocals of the divisors of $32$. | nvidia |
Let $m$ and $n$ be positive integers such that $m+n=7$ and $m^2+n^2=25$. Determine $m^4 + n^4$. | nvidia |
Count the number of non-similar rectangles with integer side lengths that have an area of 24 square units. | nvidia |
Find the sum of all possible values of $d$ such that the number ${14d62}$ is divisible by 9. | nvidia |
The diagonals of a convex hexagon $ABCDEF$ intersect at a point $G$ such that the triangles $AGB$, $BGC$, $CGD$, $DGE$, $EGF$, and $FGA$ are all isosceles, and the sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$ are all equal to 2. Find the area of the hexagon. | nvidia |
Find the largest possible value of the expression $$\frac{(a+b)(b+c)(c+a)}{abc},$$where $a > b > c > 0$ are real numbers. | nvidia |
Determine the number of positive integers less than or equal to 100 that are not divisible by any of the first three prime numbers (2, 3, 5). | nvidia |
What is the smallest positive integer $n$ such that the decimal representation of $2^n$ has exactly $5$ digits? | nvidia |
Let A be the set of all positive integers whose prime factorization has at least one prime raised to the power of 2004.
Let B be the set of all positive integers whose prime factorization has exactly 2004 distinct prime factors.
Find the number of elements in A \cup B. | nvidia |
Consider the expression \[(3^{1507}+7^{1506})^2-(3^{1507}-7^{1506})^2.\]
Find a positive integer $m$ such that the value of this expression is equal to $m\cdot14^{1506}$. | nvidia |
Let $A$ be the area of the right triangle whose legs are the roots of the equation $x^2-4x+3=0$ and $B$ be the area of the right triangle whose legs are the roots of the equation $x^2-2x+4=0$. Determine $\ds\frac{A}{B}$. | nvidia |
The following system of equations has a solution:
\begin{align*}
x+2y+3z&=6,\\
4x+5y+6z&=16,\\
7x+8y+9z&=a.
\end{align*}
Determine $a$ such that the system of equations has infinitely many solutions. | nvidia |
In a conference, the number of attendees from Europe is triple the number of attendees from Asia. There are 90 attendees from Asia, and none of the attendees from Asia are also from Europe. The total number of men attending the conference is three times the number of women. How many people are attending the conference ... | nvidia |
Let $\tau (n)$ denote the number of positive integer divisors of $n$ (including $1$ and $n$). Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$.
| nvidia |
Find the area of the triangle formed by the points $(1,2)$, $(4,5)$, and $(7,8)$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 4f(x) y\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | nvidia |
Let $a$, $b$, and $c$ be the roots of the cubic equation $x^3 + mx^2 + nx + p = 0$. Find the number of cubic equations for which the sum of the cubes of their roots is equal to the sum of the squares of their roots multiplied by a root.
That is, find the number of triples $(m,n,p)$ such that $$a^3+b^3+c^3=(a^2+b^2+c^2... | nvidia |
What is the sum of all values of $z$ such that $z^2=12z-7$? | nvidia |
Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 4.$ Find the minimum value of
\[x^2 y^2 + x^4 + y^4.\] | nvidia |
Determine the smallest positive integer $n$ such that $3n\equiv 13\pmod{34}$. | nvidia |
Consider a sequence of real numbers $(a_n)$ such that $a_i + a_j > i \cdot j$ for all positive integers $i$ and $j.$ We call this sequence "exclusive". Find the largest possible value of $a_8$ such that $(a_n)$ is exclusive and the product $a_1 \cdot a_2 \cdot \dots \cdot a_{10}$ is minimized. | nvidia |
Find the number of real solutions of the equation
\[\frac{x}{50} = \cos x.\] | nvidia |
Let $A_n$ denote the sum of the squares of the digits of the $n$-digit number $A$. For example, $1234_5 = 1^2 + 2^2 + 3^2 + 4^2 = 30$. Find all positive integers $n$ such that $n^2_n = n$.
Note: The subscript $_n$ denotes the number $n$ written in base $n$. | nvidia |
Let $x$ be a positive real number. Find the maximum value of $8\ln x-\sqrt[3]{x}-\frac{1}{x}$ | nvidia |
How many integers $m \neq 0$ satisfy the inequality $\frac{1}{|m|}\geq \frac{1}{8}$?
Determine the number of integers $n$ such that $n^2-64\leq 0$ and $\frac{n}{8}+\frac{8}{n}\geq 2$. | nvidia |
Ava is creating a list of positive integers that are divisible by 15 and have exactly three distinct prime factors. She starts with the smallest such number and continues until she reaches the smallest positive integer that is divisible by 15 and has four distinct prime factors. How many integers are in Ava's list? | nvidia |
The graph of the parabola $y=ax^2+bx+c$ passes through the points $(1,6)$, $(3,24)$ and $(5,42)$. Find the area of the triangle formed by these three points. | nvidia |
In a triangular array of numbers, each number is the sum of the two numbers directly above it. The first row contains only 1, and each subsequent row contains one more number than the previous row. Find the sum of the squares of the numbers in the 10th row. | nvidia |
Find the greatest value of $b$ such that $b^2-12b+27 \le 0$. | nvidia |
Determine $c$ such that the roots of the equation $$5x^2+11x+c=0$$ are reciprocals of each other. | nvidia |
A sequence of $n$ non-zero digits is said to be "balanced" if for any two digits in the sequence that are $k$ positions apart (where $1 \leq k \leq n/2$), the sum of the two digits is equal. For example, the sequence $1, 2, 3, 4$ is balanced because $1+4=5=2+3$. How many balanced sequences of length 6 exist, where the ... | nvidia |
The product of two positive integers is 360. If one integer is a multiple of 4, what is the largest possible value of the other integer such that their sum is a multiple of 7? | nvidia |
In triangle $ABC$, $AB=13$, $AC=15$, and $BC=14$. The point $D$ is on side $BC$ such that $AD$ is perpendicular to $BC$. Find the area of triangle $ABC$. | nvidia |
Given that the point $(9,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=\frac{f(2x)}2+2$. What is the sum of coordinates of that point? | nvidia |
Find the number of positive divisors of 2002. | nvidia |
Given the points $(-3,5)$ and $(2,-5)$, find the equation of the circle that passes through these two points and has its center on the line $y=x$. | nvidia |
Find the largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$. | nvidia |
Find the smallest positive integer n such that when its digits are reversed, the resulting integer is 1/29 of the original integer n, and n is a perfect square.
Note: This problem requires a different approach than the original one, as it involves finding a perfect square number that meets the condition, rather than j... | nvidia |
Find the sum of the $x$-intercept and $y$-intercept of the curve described by the equation $y=\ds\frac{4}{x}$. | nvidia |
Given the set of all integers from 1 to 2010, consider a function $g(n)$ that denotes the product of all positive divisors of $n$. How many distinct pairs of integers $(a, b)$ exist in this range such that $g(a) = g(b)$ and $a \neq b$, yet the number of divisors of $a$ is not equal to the number of divisors of $b$? | nvidia |
Find the smallest positive integer $n$ such that for any $n$ distinct points on the unit circle, there exist at least two points that are endpoints of a diameter. | nvidia |
The sum of three positive integers is 20. Two of the integers are prime numbers whose sum is greater than the third integer. Find the maximum value of the third integer. | nvidia |
Find the sum of all two-digit positive integers formed by the digits $1$, $2$, $3$, and $4$ without repetition. | nvidia |
Find the product of all positive integral values of $n$ such that $n^2-35n+306= p$ for some prime number $p$. Note that there is at least one such $n$.
Another problem:
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 $a$.
Note: This problem requires a different... | nvidia |
For the functions $f(x)=\frac{x^2+Ax+B}{x^2+Cx+D}$ and $g(x)=\frac{x^2+Ex+F}{x^2+Gx+H}$, it is known that $\lim_{x\rightarrow\infty}f(x)=\lim_{x\rightarrow\infty}g(x)$, the vertical asymptotes of $f$ and $g$ are $x=1$ and $x=-1$, respectively, and that $f(0)=2$ and $g(0)=3$. Given that $A+B=C+D=E+F=G+H$, determine $|B-... | nvidia |
Two fair, eight-sided dice are rolled. What is the probability that the sum of the two numbers showing is greater than 9? | nvidia |
Let the element whose symbol begins with the letter A in the periodic table of chemical elements be denoted by a number -- the sum of the positions in the alphabet of all the letters in its Latin name. For example, carbon is denoted by the number $3+1+18+2+15+14=53$. Determine the sum of the numbers denoted to argon, s... | nvidia |
In triangle $ABC$, point $P$ is chosen such that the circumcenters of triangles $ABP$, $BCP$, and $CAP$ are collinear. Find the ratio of the area of triangle $ABC$ to the area of triangle $APB$. | nvidia |
The absolute value of the sum of a number and its reciprocal is equal to 3. Find the sum of all possible values of the number. | nvidia |
Find the number of positive divisors of $n$ such that $n$ is a multiple of $6$ and has exactly $3$ distinct prime factors. | nvidia |
Find the number of complex numbers $w$ such that $\left| w \right| < 25$ and satisfy the equation
\[
e^{2w} = \frac{w^2 + 2}{w^2 - 2} \,.
\] | nvidia |
Let's call a positive integer "special" if it is divisible by exactly two different prime numbers, and both of those primes are greater than 5. What is the smallest positive integer that is the sum of two "special" integers? | nvidia |
Find all values of $m$ so that the domain of
\[f(x) = \frac{mx^2 + 4x - 3}{-3x^2 + x + m}\]
is the set of all real numbers. | nvidia |
Let $f$ be a function taking the positive integers to the positive integers, such that
(i) $f$ is increasing (i.e. $f(n + 1) > f(n)$ for all positive integers $n$)
(ii) $f(mn) = f(m) f(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^n = n^m,$ then $f(m) = n$ or $f(n) = m.$
Find the sum of al... | nvidia |
The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the the smallest possible value of $m+n.$ | nvidia |
Let $x^3-ax^2+bx+c$ be a cubic polynomial with roots $x_1$, $x_2$, and $x_3$. If $x_1$, $x_2$, and $x_3$ are positive integers such that $x_1x_2+x_1x_3+x_2x_3=30$, how many different values of $a$ are possible? | nvidia |
Find the value of $x$ such that
$$\left(\frac{2}{3} \right)^{x} \cdot \left(\frac{5}{7} \right) = \left(\frac{5}{7} \right)^{x} \cdot \left(\frac{2}{3} \right).$$ | nvidia |
Let's consider a cubic polynomial with integer coefficients of the form $p(x) = x^3 + bx^2 + cx + d$, where $b$, $c$, and $d$ are integers. We know that this polynomial has at least one integer root, let's denote it $r$. Can you find the sum of all possible integer values of $b$ for which the polynomial $p(x)$ has exac... | nvidia |
Let $m$ and $n$ be positive integers such that $m^2+n^2$ is divisible by 49. Find the remainder when $m^3+n^3$ is divided by 49. | nvidia |
The function
$$y=\frac{a}{x}+\frac{b}{\sqrt x}+cx$$
has a minimum value of 16 when $x$ is a positive integer.
Determine $x$ if $a,b,c$ are positive integers satisfying
$a:b:c=2:3:5$. | nvidia |
Find all real numbers $x$ such that \[2 \le \frac{x}{3x-2} < 5.\](Give your answer in interval notation.) | nvidia |
In the equilateral triangle $ABC$, point $P$ is located inside the triangle so that the sum of the measures of angles $\angle PAB$ and $\angle PBC$ is $60$ degrees less than the measure of $\angle PAC$.
Find the measure of $\angle PAC$. | nvidia |
Find the sum of the coordinates of the midpoint of the segment with endpoints $(4, 8)$ and $(-2, 3)$. | nvidia |
Find the least positive integer such that when multiplied by 17, the result is divisible by 7 and 11. | nvidia |
Let $m$ and $n$ be positive integers such that $m$ is the smallest number of consecutive integers that add up to $n$. Find the largest value of $m$ for which $n$ is a perfect square. | nvidia |
The real function $f(x)$ satisfies the condition $\sin(\arccos(f(x)))=x$ for all $x$ in the domain $[-1,1]$. Find the value of $\displaystyle\int_{-1}^{1} f(x) dx$. | nvidia |
Find the number of five-digit numbers that have exactly two different digits. | nvidia |
On the entire real line define the function $f(x)$ satisfying
$f(x+y)=\frac{f(x)+f(y)}{1+f(x)f(y)}$
and having the property
$\int\limits_{-\infty}^{+\infty}f(x)dx=1.$
Prove that $f(x)$ is an odd function of $x$. | nvidia |
A bookshelf contains 12 books, each written by one of two authors, Alice and Bob. The books are arranged randomly on the shelf. What is the probability that there is at least one pair of books written by the same author that are next to each other on the shelf? | nvidia |
Factor the expression $x^8+2x^6y^2+2x^2y^6+y^8$ as the product of two polynomials of degree 4, with all coefficients being integers. Determine the sum of the coefficients in this factorization. | nvidia |
Let $A$ be a $2 \times 2$ matrix with integer entries such that $A^3 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Find the smallest possible value of the sum of the absolute values of the entries of $A$. | 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 |
Find the sum of the rational roots of $g(x)=x^3-9x^2+16x-4$. | nvidia |
A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 3-primable positive integers are there that are less than 1000? | nvidia |
If the sum of two squares of natural numbers can be represented as (2^x + 2^(x-1))^2, where x>1, find the sum of these two natural numbers. | nvidia |
In a regular $n$-gon, the diagonals and sides form $t$ triangles, of which $u$ are equilateral. Find the value of $n$ such that $u/t$ is maximized. | nvidia |
Given a fourth-degree polynomial $x^4 - 12x^3 + 47x^2 - 60x + p$, find the value of $p$ such that the polynomial can be factored into the product of two quadratic polynomials with integer coefficients. | nvidia |
Let $a$ and $b$ be positive real numbers such that $(a+b)^2-ab=\frac{1}{a}+\frac{1}{b}$. Find the minimum possible value of $a+b$. | nvidia |
A portion of the graph of a cubic function $p(x)$ is shown below.
Let $q(x)=-p(x)$ and $r(x)=p(-x)$. If $a$ is the number of points where the graphs of $y=p(x)$ and $y=q(x)$ intersect, and $b$ is the number of points where the graphs of $y=p(x)$ and $y=r(x)$ intersect, then what is $20a+b$?
[asy]
size(150);
real tick... | nvidia |
The geometric mean of an even number of consecutive even integers is $y$. Find the product of the smallest and largest of the integers in terms of $y$. | nvidia |
What is the smallest positive integer $n$ such that $n^2+2n+17$ is a prime number? | nvidia |
Triangle $DEF$ has vertices $D(0, 12)$, $E(3, 0)$, $F(12, 0)$. A vertical line with equation $x=s$ intersects line segment $ \overline{DE} $ at $V$ and line segment $ \overline{DF} $ at $W$, forming $\triangle DVW$ with perimeter 27. Compute $s$. | nvidia |
A rectangular grid with 5 rows and 6 columns is given. Find the number of rectangles that can be formed using the grid points as vertices, such that at least two adjacent sides of the rectangle lie on the boundary of the grid. | nvidia |
Determine $b$ such that the product of the numbers $32_{b}$ and $4B_{b}$ is equal to $1542_{b}$, where $B_{b}=11_{10}$. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x + y) = g(x)g(y) + x^2 + y^2\]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 |
What is the smallest positive integer that can be written as the sum of two different decimal numbers with exactly two digits each? | nvidia |
Find the positive real value of $t$ that satisfies $\sqrt{(t-2)^2 + 5^2} \cdot \sqrt{3^2 + 4^2} = 65$. | nvidia |
Find the length of the shortest path that lies entirely on the surface of the cylinder $x^2+y^2=1$ and connects the points $(1,0,2)$ and $(1,0,-3)$. | nvidia |
Evaluate the product \[ (x-2) \cdot (x-1) \cdot x \cdot (x+1) \cdot (x+2), \] where $x=3$. | nvidia |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.