instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find the sum of all integers between 100 and 1000 that are divisible by 7. | nvidia |
Let $P(x)$ be a polynomial such that $P(2) = 15$, $P(3) = 7$, and $P(4) = 19$. Determine the largest integer $n$ such that $\frac{P(x)-5}{x-2}$ is an integer for all integers $x$ with $2 \le x \le n$. | nvidia |
A number $x$ is equal to $7\cdot24\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? | nvidia |
There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number? | nvidia |
Prove that for positive integers n, k, l the expression
$$\sum_{i=1}^{n}\frac{i!}{(i+k)!(i+l)!}$$
is always integer and find all $(n,k,l)$ for which the expression is equal to $1$. | nvidia |
Let $T$ be the set of all positive real numbers. Let $g : T \to T$ be a function such that
\[g(x) + g(y) = g(x^y + y^x)\]for all $x,$ $y \in T$. Find the largest possible value of $g(3)$. | nvidia |
Compute the number of non-decreasing sequences of length 12, where each element is chosen from the set {1, 2, 3, 4, 5}, given that the number of non-decreasing sequences of length 10 using the same set is 3003, and the number of non-decreasing sequences of length 8 is 1287. | nvidia |
Find the smallest positive integer $k$ such that $19^k \equiv 1 \pmod{24}$. | nvidia |
How many positive three-digit integers have a sum of digits that is divisible by 5? | nvidia |
Find the sum of all positive divisors of 18 that are also divisors of 24. | nvidia |
The digits of a five-digit number are such that the sum of the digits in the even positions is equal to the sum of the digits in the odd positions, and the product of the digits in the even positions is equal to the product of the digits in the odd positions. Determine the smallest such number. | nvidia |
There are 2 complex numbers $x+yi$ and $z+wi$. If $w=4$, $x=-z$, and the sum of the numbers is $5i$, find $y$. | nvidia |
Find the smallest positive integer whose digits form a decreasing sequence and the sum of its digits is a prime number greater than 17. | nvidia |
Find the least positive integer $m$ such that $m \equiv 27 \pmod{36}$ and $m$ is a multiple of 5. | nvidia |
A bag contains ten balls, some of which are red and the rest of which are yellow. When two balls are drawn at random at the same time, the probability that both balls are red is $\frac{1}{15}$. How many balls in the bag are red? | nvidia |
Find all values of $a$ such that $\lfloor a \rfloor + a = 23.7$. | nvidia |
Consider the sequence of numbers $a_1, a_2, a_3, \ldots, a_n, \ldots$ where $a_n = \frac{1}{\sin n}$ for $n \geq 1$. Find the sum of the squares of the reciprocals of the positive integers $n$ for which $a_n$ is an integer. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a + b = 1.$ Find set of all possible values of $\frac{1}{a} + \frac{1}{b}.$ | nvidia |
Let $n$ be a positive integer such that there exist nonzero integers $a$ and $b$ so that $\sqrt[n]{a} + \sqrt[n]{b}$ is a perfect square. Find the smallest possible value of $n$. | nvidia |
What is the smallest positive integer $m$ such that the difference between $m^2$ and $m^3$ is divisible by at least two distinct prime numbers when $m$ is expressed as the sum of $r$ consecutive integers, where $r$ ranges from $1$ to $m$? | nvidia |
Find all real numbers \(a\) such that the equation
\[\left| \frac{x^2 - 1}{x + 1} \right| = |a|\]
has exactly three distinct solutions. | nvidia |
Let $p$ be an odd prime number. There are $p-1$ positive integers less than $p$ that are relatively prime to $p$. Let $a_1, a_2, \ldots, a_{p-1}$ be these numbers. Prove that the sum $S = a_1^{p-2} + a_2^{p-2} + \ldots + a_{p-1}^{p-2}$ is congruent to $0$ modulo $p$. | nvidia |
Given that
$\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}$
find the greatest integer that is less than $\frac N{100}$.
| nvidia |
Given that five distinct natural numbers are chosen from between 10 and 25, inclusive, what is the probability that the sum of their squares is even? | nvidia |
Find the largest number of positive, consecutive odd integers whose sum is equal to the sum of their squares. | nvidia |
Let $a$ and $b$ be positive real numbers such that each of the equations $x^2 + ax + 2b = 0$ and $x^2 + 2bx + a = 0$ has real roots. Find the smallest possible value of $a + b.$ | nvidia |
Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.Find the sum of all integers $n$ such that $\dfrac{12}{n}$ is also an integer.
| nvidia |
What is the largest positive integer that is both a power of 3 and the sum of two perfect squares? | nvidia |
Suppose that $a$ and $b$ are positive integers such that $(a-bi)^2 = 8-6i$. What is $a-bi$? | nvidia |
Find the sum of all the products of each pair of adjacent numbers in the sequence $2, 5, 7, 11, 13, 17$, assuming that the first and last numbers are also adjacent. | nvidia |
Let $p(x,y,z)$ be a cubic polynomial such that $p(x,y,z)=p(y,z,x)=p(z,x,y)$ for all $x,y,z$. Suppose that $p(0,0,0)=p(1,0,0)=p(0,1,0)=p(0,0,1)=p(1,1,1)=0$. Find the point $(r,s,t)$ such that $p(r,s,t)=0$ for all such polynomials, where $r$, $s$, and $t$ are not integers. | nvidia |
Find the maximum value of $x$ such that $\lfloor x \rfloor + \dfrac{1}{x} = \dfrac{5}{2}$. | nvidia |
Find the largest prime factor of the expression $a^3+b^3+c^3$, where $a,b,c$ are positive integers and $a+b+c=0$. | nvidia |
A standard cube has a volume of $V = s^3$, where $s$ is the length of its edge. If the sum of the lengths of the edges of the cube is 240 cm, find the sum of the areas of all its faces. | nvidia |
If $a$ and $b$ are positive real numbers such that $a^3b=108$ and $ab^3=48$, what is the value of $a^2b^2$? | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $a+b+c=240$, $\text{lcm}(a,b,c)=480$, and $c=2\cdot\text{gcd}(a,b)$.
Find the smallest possible value of $c$. | nvidia |
Find the sum of the digits of the number $4! \cdot 5!$. | nvidia |
There are 2025 positive numbers with both their sum and the sum of their reciprocals equal to 2026. Let $y$ be one of these numbers. Find the minimum value of $y + \frac{1}{y}$. | nvidia |
What is the number of positive integers less than or equal to 48 that have an odd number of positive factors? | nvidia |
What fraction of the two-digit numbers that are divisible by 5 are also divisible by 3? | nvidia |
What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions? | nvidia |
Let $a, b, c$, and $d$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}a^2+b^2&=&c^2+d^2&=&2008,\\ ac&=&bd&=&1000.\end{array}\]
If $S=a+b+c+d$, compute the value of $\lfloor S\rfloor$.
A similar problem could be:
Let $a, b, c$, and $d$ be positive r... | nvidia |
A number $x$ is equal to $7\cdot24\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? | nvidia |
Find the number of even integers between 40 and 60 that are divisible by 3. | nvidia |
Find the sum of all $x$ such that the expression $\frac{x^2+4x+4}{x^2+5x+6}$ is undefined. | nvidia |
Let $a$ and $b$ be the roots of the equation $kx^2+(1-k)x+1=0$. Find $k$ such that $a^3+b^3=1$. | nvidia |
Determine the sum of all the positive integer divisors of 100. | nvidia |
Count the number of five-digit integers that have exactly three distinct digits such that the digits can be arranged in a way that forms a palindrome. | nvidia |
Find the sum of the divisors of the least positive integer whose positive integral divisors have a sum of 24. | nvidia |
Let $a$ and $b$ be any two distinct positive integers. Determine the largest integer that divides all possible numbers of the form $ab(a^2-b^2)$. | nvidia |
There are two concentric circles with radii of $15$ and $18$ centimeters.
The circumference of an arc of the smaller circle that lies outside a regular hexagon inscribed in the smaller circle is $32\pi$ cm.
Find the area of the hexagon. | nvidia |
Let $g(x)$ be a function such that
\[g(x) + g(y) = g(xy)\]
for all positive real numbers $x$ and $y$. If $g(5) = 4$, what is $g(25)$? | nvidia |
The sum of the squares of the three positive integers is equal to four times the product of the smallest two integers, and the largest integer is twice the sum of the other two. Find the sum of all possible values of the product of the three integers. | nvidia |
Let $f(x)$ be a quadratic function that satisfies $f(f(x))=x$ for all real numbers $x$. If $f(1)=3$ and $f(2)=4$, find $f(3)$. | nvidia |
In a certain sequence of moves, stones are removed from a stack of $n$ stones, where $n$ is a positive integer. The moves are made in alternating order by two players, Ann and Ben, with Ann making the first move. In each move, Ann must remove one stone from the stack such that the remaining number of stones is a multip... | nvidia |
Determine the number of distinct prime divisors of $N^2$, where $N$ is the least positive integer divisible by all positive integers less than or equal to 8. | nvidia |
Find all values of $b$ such that $b+\lceil b\rceil=17.8$. | nvidia |
The sequence $a_n=\frac{3^n+1}{4}$ is defined for all positive integers $n$. Calculate $\left\lfloor\sum_{n=1}^\infty\frac{1}{a_n}\right\rfloor$. | nvidia |
Determine the sum of all single-digit replacements for $z$ such that the number ${24{,}z38}$ is divisible by 6. | nvidia |
Find the number of positive integers that are divisors of both 72 and 108. | nvidia |
Of the numbers 1, 2, 3, ..., 15, which number has the greatest number of divisors (the dots mean that we are including all the integers between 1 and 15)? | nvidia |
Find the least positive integer such that there exists a subset of it such that the elements of the subset have a sum that is divisible by any positive integer less than or equal to 13. | nvidia |
Given a square with side length $s$, find the largest $n$ such that the square can be tiled with L-shaped pieces of width $\frac{s}{n}$ made of two squares of size $\frac{s}{n}$ by $\frac{s}{n}$, leaving no empty space between the tiles. | nvidia |
Let a and b be two positive integers such that (a+1) and (b+1) are prime numbers.
If ab has exactly two positive divisors, find the value of a+b. | nvidia |
Find the smallest positive integer $n$ such that the quadratic equation $2x^2 + 3x + n \equiv 0 \pmod {15}$ has a solution. | nvidia |
Count the number of four-digit positive integers that have only even digits. | nvidia |
Find the range of the expression $\sqrt{x^2+6x+9}+\sqrt{x^2+2x+1}+\sqrt{x^2-2x+1}$ over the real numbers $x$. | nvidia |
Find the maximum value of $xy$ over all real numbers $x$ and $y$ such that $2x^2 + 5y^2 = 14$. | nvidia |
Find the sum of all solutions to the equation $(x-6)^2=25$. | nvidia |
How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\frac14$? | nvidia |
The area of a square equals the square of a length of the side of the square. The perimeter of a square equals the sum of the lengths of all four sides. The sum of the areas of two squares is 65, while the difference in their areas is 33. Find the sum of their perimeters. | nvidia |
Let $f(x)$ and $g(x)$ be two functions that satisfy the following properties:
* $f(x) > 0$ for all real numbers $x$
* $g(x) = f(x - 2)$ for all real numbers $x$
* $f(x) + g(x) = h(x)$ for all real numbers $x,$ where $h(x)$ is a function that satisfies $h(x) > 0$ for all real numbers $x$
Find the largest possibl... | nvidia |
Given a convex quadrilateral $ABCD$ with fixed perimeter of length 100, determine the minimal area that can be enclosed by the kite formed by connecting the midpoints of its diagonals. | nvidia |
Find the least positive integer n such that n^2 + n is a multiple of some but not all prime numbers less than or equal to n. | nvidia |
A bookshelf has five shelves, each of which can hold one book of a different color: red, blue, green, yellow, and purple. The books must be arranged in such a way that no two books of the same color are on adjacent shelves. What is the smallest number of arrangements that can be made such that the books on the top and ... | nvidia |
Find all three-digit numbers that have exactly three different digits and have a units digit of 3, such that when the hundreds and tens digits are swapped, the resulting number is also a three-digit number with exactly three different digits. | nvidia |
The product of the digits of 3214 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 12? | nvidia |
Find the number of real solutions to the equation $\sqrt[x^2]{x+1} = \sqrt[x]{x+1}$, where $x$ is a positive real number. | nvidia |
Find the value of $x$ such that $\arctan x = \frac{\pi}{4}$. | nvidia |
What is the greatest whole number that must be a divisor of the sum of any three consecutive positive integers? | nvidia |
Find the equation of the plane that passes through the points $(1,2,3)$, $(4,5,6)$, and $(7,8,9)$, given that the plane is perpendicular to the line passing through the points $(10,11,12)$ and $(13,14,15)$. | nvidia |
Find the values of $x$ for which the function $f(x) = \frac{2x^2+5x-3}{x^2-4x-5}$ is defined, where $x$ is a real number.
Note: I've created a problem that requires finding the domain of a rational function with a quadratic expression in the denominator, which is a slightly different approach from the original problem... | nvidia |
Determine how many perfect squares exist between 1 and 30, where a perfect square is a positive integer that can be expressed as the square of another integer. | nvidia |
A circle passes through the points $(0,0)$, $(2,2)$, and $(4,k)$. Find the value of $k$ such that the area of the triangle formed by these three points is a maximum. | nvidia |
Find the largest possible value of $x$ such that $3^x$ divides evenly into $13^{12}$. | nvidia |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 1000^\circ.$ | nvidia |
A box contains $n$ marbles, each labeled with a distinct integer from 1 to $n$. Two marbles are drawn at random without replacement. The probability that the two marbles have consecutive labels is $\frac{1}{6}$. Determine $n$. | 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 $|a^2b+b^2c+c^2a|$. | nvidia |
A $100\times 100$ square grid is drawn on a piece of paper, and then $50\times 50$ squares are randomly selected and colored red. What is the minimum number of straight lines that can be drawn on the grid such that every red square is intersected by at least one line? | nvidia |
Find the sum of the real roots of the polynomial $f(x) = x^{2007} + \sum_{k=1}^{2006} kx^{2006-k}$. | nvidia |
The vertices of a triangle are located at $(0,0)$, $(2,5)$, and $(-1,-3)$. Find the sum of the coordinates of the centroid of the triangle. | nvidia |
Let $a_0$ be a positive integer. A sequence of positive integers is constructed in the following way: for every positive integer $k$, let $a_k$ be the largest integer such that $k$ is divisible by $a_k$ and $k$ is also divisible by $a_{k-1}$.
Find the minimum possible value of $a_0$ such that the sequence $a_1,a_2,\ld... | nvidia |
Find the smallest prime factor of the sum of the squares of the first $100$ positive integers. | nvidia |
Let $a$, $b$ and $c$ be positive integers. Consider two arithmetic progressions $(ak+c)$ and $(bk-c)$. Determine the smallest possible value of $k$ for which there exists a positive integer $n$ such that $an+c$ and $bn-c$ share a common factor greater than 1. | nvidia |
Problem:
Prove that for every positive integer $n$, there exist positive integers $a$ and $b$ such that $\frac{a}{b} = \sum_{k=1}^n \frac{(-1)^k}{2^k k}$.
Solution:
The new problem requires a different approach, using a constructive proof involving telescoping series. The original problem involves a telescoping s... | nvidia |
Let $X$ be the number of 5-digit numbers that are palindromes, and $Y$ be the number of 6-digit numbers that are multiples of 3. Find $X+Y$. | nvidia |
Two real numbers $x$ and $y$ satisfy $x-y=4$ and $x^3-y^3=28$. Compute $xy$. | 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 |
The sum of the squares of three numbers is 90. One of the numbers is 7, and the other two numbers differ by 2. What is the sum of the two unknown numbers? | nvidia |
Find the sum of the infinite series $\sum_{n=1}^{\infty}n\sin\left(\frac{\pi}{3n}\right)$. | nvidia |
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