instruction_seed stringlengths 30 7.99k | source stringclasses 1
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The least value of x such that x^2 + (4 + i)^2 is a real number. | nvidia |
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of the possible values of $a?$
$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 18$
| nvidia |
Find the number of integers between $100$ and $200$ that are divisible by $3$ but not divisible by $4$. | nvidia |
Palindromes are numbers that read the same backwards and forwards, like 5665. What is the least possible positive four-digit palindrome that is divisible by 3? | nvidia |
What is the smallest positive integer $x$ such that $5^x > 3^4 \cdot 2^5$? | nvidia |
Let $x$, $y$, and $z$ be distinct real numbers that sum to $0$. Compute \[ \dfrac {x^2y^2+y^2z^2+z^2x^2}{x^4+y^4+z^4}. \] | nvidia |
Find the smallest positive integer $n$ such that any $n$-digit integer, when its digits are arranged in reverse order, forms a number that is a multiple of the original number. | nvidia |
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 3. If $\frac{S_{3n}}{S_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2+y^2=1$. Find the minimum possible value of $xy+\frac{1}{xy}$. | nvidia |
Count the number of positive integers $m$ such that there exists an integer $n$, for which $m$ is the smallest positive integer satisfying $2^n + m = 3^k$, where $k$ is a positive integer. | nvidia |
Square $ABCD$ has center $O,\ AB=900,\ E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F, m\angle EOF =45^\circ,$ and $EF=400.$ Given that $BF=p+q\sqrt{r},$ where $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$
| nvidia |
In a triangle $ABC$, the side lengths are positive integers. The length of side $AB$ is given by the sum of the lengths of sides $AC$ and $BC$. If the perimeter of the triangle is 42, find the maximum possible value of the length of side $AB$. | nvidia |
The equation $x^2-7x-18=0$ has two solutions. The solution that is not an integer has the form $\frac{a+\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$. What is $a+b+c$? | nvidia |
Find all values of $a$ for which the product of the solutions of
\[x^2 + ax + 1 = 0\]equals the sum of the cubes of these solutions. Enter all possible values of $a,$ separated by commas. | nvidia |
Let $f(n)$ be the sum of all positive integers $k$ such that $k$ and $n$ are relatively prime, where $1 \leq k \leq n$.
Find the value of
\[\frac{\sum_{d|n} \frac{f(d)}{d}}{\sum_{d|n} f(d)},\]
where the sums are taken over all positive divisors $d$ of $n$. | nvidia |
Given that $f(x) = x^2 + bx + c$ is a quadratic function where $c$ can vary, determine the largest possible value of $c$ for which the graph of $f(x)$ still touches the x-axis, given that the vertex of the parabola is at $(-b/2, -b^2/4)$. | nvidia |
Find all real solutions to $x^3+(x+1)^3+(x+2)^3=(x+3)^3$.
Problem:
Find all real solutions to $\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}=\sqrt{x+3}$. | nvidia |
The Fibonacci sequence is defined recursively by $F(1)=1$, $F(2)=1$, and $F(n)=F(n-1)+F(n-2)$ for $n>2$. Prove that for any positive integer $k$, there exists a positive integer $n$ such that $F(n)$ is divisible by $k$. | nvidia |
Let $F$ be a function from the set of all positive integers to the set of all positive integers. Suppose that for all positive integers $a$ and $b$, $F(a^2 + b^2) = F(a)^2 + F(b)^2$, and $F(1) = 1$. Find the number of positive integers $n$ such that $F(n) = n$. | nvidia |
Let $m$ be a positive integer such that $m^2+2m+3$ is divisible by $m+5$. Find all possible values of $m$. | nvidia |
Find the sum of the squares of two positive integers $x$ and $y$ that satisfy the equation $(x+1)(y+1) = 91$.
(Note: I tried to create a problem that requires a different approach to solve, using a similar type of equation but with a different structure, and asking for a different type of quantity, in this case, the s... | nvidia |
The circle centered at $(3,2)$ is tangent to the line $x-y=a$. Find the value of $a$ such that the circle also intersects the line $y=x$ at a single point. | nvidia |
Find the smallest positive integer that is both an integer power of 11 and is not a palindrome. | nvidia |
An ellipse with semi-axes of lengths $a$ and $b$ is inscribed inside a square with side length $4$. The ellipse is tangent to the square at the midpoints of the sides, and the center of the ellipse is at the center of the square. Find the radius of the circle that passes through the four vertices of the square and is t... | nvidia |
Two positive numbers have the property that the sum of their squares is 200 and the difference of their fourth powers is 2800. What is the value of the larger number? | nvidia |
Given a positive integer $n$, let $T$ be the set of complex numbers of the form $a + bi,$ where $a$ and $b$ are rational numbers, such that
\[\frac{2}{3} \le a \le \frac{3}{4}.\]Find the largest positive integer $m$ such that for all positive integers $n \le m,$ there exists a complex number $z \in T$ such that $z^n \n... | nvidia |
Given that $x$ and $y$ are positive integers, how many different values of $y$ satisfy the equation $x + y = 10$ for all possible values of $x$? | nvidia |
Find the number of positive integers $n$ such that $\sqrt{n} + \sqrt{n+1}$ is also an integer. | nvidia |
Find all positive integers n such that $n^2+ 1$ has exactly 5 positive factors. | nvidia |
Let $f(x)=x^3+x^2+ax+b$ be a cubic polynomial with leading coefficient $1$ such that its roots are the reciprocals of the roots of the polynomial $x^3-6x^2+11x-6$. Find the ordered pair $(a,b)$. | nvidia |
Let $S$ be the set of all subsets of the set $\{1,2,\ldots,10\}$ such that each subset in $S$ has either $4$ or $5$ elements. Find the number of subsets in $S$ that contain exactly $3$ even numbers. | nvidia |
A sequence of consecutive numbers starting from $n$ is given, and the product of any three consecutive terms is a constant value $k$. Find the value of $k$ such that the sum of the first four terms of the sequence is equal to 24. | nvidia |
A circle with radius $r$ is inscribed in a triangle $\triangle ABC$ such that the area of the sectors of the circle enclosed by the sides of the triangle are $3$, $6$, and $12$, respectively. Find the area of $\triangle ABC$. | nvidia |
Find the absolute value of the difference between the sum of the squares of the first 10 odd numbers and the sum of the squares of the first 10 even numbers. | nvidia |
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x$, $y$, and $r$ with $|x|>|y|$,
\[(x+y)^r=x^r+rx^{r-1}y^1+\frac{r(r-1)}2x^{r-2}y^2+\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\cdots\]What are the first two digits to the right of the decimal point in the decimal representation o... | nvidia |
Consider the function
$$F(x)=\frac{x^2+ax+1}{x^2+bx+1},$$
where $a$ and $b$ are constants. Determine the values of $a$ and $b$ such that $F(x)$ is even, odd, or neither. | nvidia |
Let
\[\begin{array}{lcl}
P(x) & = & a_n x^n+a_{n-1}x^{n-1}+\cdots+a_0, \\
Q(x) & = & x^n +x^{n-1}+\cdots+x+1
\end{array}\]be polynomials with integer coefficients,
where $a_n$ is positive and
$|a_i|\leq 198$, for $i=0,1,\ldots,n-1$.
Prove that if
\[P(x)-Q(x)=R(x)\big(P(x)+Q(x)\big),\]for some non-zero polynomial $R(x)... | nvidia |
Count the number of integers between 100 and 999 that have exactly two digits equal to 3, and all other digits are prime. | nvidia |
Let g(x)=3cos(2πx)+1. Determine the number of values of x on the interval −2≤x≤2 that satisfy the equation g(g(x))=−g(x). | nvidia |
Find the number of integers from 1 to 1000 that are divisible by 3, but not divisible by 7. | nvidia |
Determine the smallest positive integer $m$ such that all the roots of $z^6 + z^3 + 1 = 0$ are $m^{\text{th}}$ roots of unity. | nvidia |
If $a$, $b$, and $c$ are integers such that $a<b<c$ and if the average of the three numbers $5a$, $5b$, and $5c$ is 15, determine the value of $b$. | nvidia |
The sum of the lengths of all the edges of a cube is 60 cm. Find the number of cubic centimeters in the volume of the cube. | nvidia |
Find the smallest positive integer $n$ such that the sum of the squares of its digits is greater than the number itself. | nvidia |
Find the largest positive integer $N$ that has the property that the digits of its base-3 representation, read in reverse order, form a base-4 number that is three times $N$. | nvidia |
Let $f(x)$ be a function such that $f(0) = 1$ and
\[f(xy) = f \left( \frac{x^2 + y^2}{2} \right) + (x - y)^2\]for all real numbers $x$ and $y.$ Find $f(x).$ | nvidia |
A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order. | nvidia |
If $x$ and $y$ are positive real numbers such that $5x^2 + 10xy = x^3 + 2x^2 y,$ what is the value of $x$? | nvidia |
Let $S$ be the set of complex numbers of the form $x + yi,$ where $x$ and $y$ are real numbers, such that
\[\frac{\sqrt{2}}{2} \le x \le \frac{\sqrt{3}}{2}.\]Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a complex number $z \in S$ such that $z^n = 1.$ | nvidia |
Let $a$ and $b$ be two distinct positive integers, and let $n$ be a positive integer such that
$$(a-b)^n = a^n - a^{n-1}b + a^{n-2}b^2 - \cdots + ab^{n-1} - b^n.$$Determine the value of $a+b$. | nvidia |
Given the complex numbers $a$, $b$, and $c$ such that $|a|=|b|=|c|=1$ and
$$a+b+c = -1,$$
find the value of
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$ | nvidia |
Let x and y be real numbers such that x^3+y^3=2x^2y+5x^2=10y^2.
Solve the equation x^3*y^3*(x+y)=16. | 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 $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(0) = 0$ and
\[g(x^2 + y^2) = xg(x) + yg(y)\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(3),$ and let $t$ be the product of all possible values of $g(3).$ Find $m \times t.$ | nvidia |
What is the largest possible value of the product of four positive integers whose sum is 24? | nvidia |
Find the area of the triangle bounded by the $y$-axis and the lines $y-3x=-2$ and $3y+x=12$. | nvidia |
In the triangle ABC, the cevian AD divides the triangle into two triangles of areas $60 \text{ and } 30$. The cevian BE divides the triangle into two triangles of areas $45 \text{ and } 45$, while the cevian CF divides the triangle into two triangles of areas $36 \text{ and } 54$. If these cevians are concurrent, find ... | nvidia |
The lengths of the sides of a triangle are consecutive positive integers.
Given that one of the angles between the consecutive sides has a measure of $\arccos \frac{18}{35}$,
determine the length of the side opposite to that angle. | nvidia |
Determine all natural numbers $n$ such that $5^n-2$ is a prime number. | nvidia |
Determine the smallest positive integer $k$ such that $1000k$ is divisible by $1763$, and calculate the whole number remainder when $1763$ is divided by $k$. | nvidia |
Let $x,$ $y,$ and $z$ be three positive integers whose sum is 24. If each of these numbers is a power of 2 and no one of these numbers is more than four times any other, then find the maximum value of the product $xyz.$ | nvidia |
Find the largest value of $c$ such that $1$ is in the range of $f(x)=x^2-5x+c$. | nvidia |
Find the last three digits of the sum of the cubes of the first 200 positive integers. | nvidia |
Find all $x$ such that $\frac{x^2+5x-6}{x^2-4}<0$. Express your answer in interval notation. | nvidia |
Find the product of the roots of $x^3+2x^2-7x-12=0$ that lie in the interval $[-2, 3]$. | nvidia |
Find the number of non-negative integers $n$ such that $\lfloor 3.5n-2\rfloor<11$. | nvidia |
We have a function $f(x)$ defined as follows:
$$
f(x) = \begin{cases}
x^2 + 1 & \text{if } x \geq 0 \\
x^2 - 1 & \text{if } x < 0
\end{cases}
$$
Find the set of values of $x$ such that $\lim_{x\to a} f(x) = f(a)$ for any real number $a$. | nvidia |
Consider the operation $a\circ b=\sqrt{ab}$ for all positive real numbers $a$ and $b$. Find $2\circ (3\circ 4)$. | nvidia |
Problem:
The graphs $y = 3(x-h)^2 + j$ and $y = 2(x-h)^2 + k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$.
Write another problem inspired by this one:
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... | nvidia |
Determine the smallest positive integer $m$ such that
\[\begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}^m = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.\] | nvidia |
What is the greatest integer less than or equal to\[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?\]
$\textbf{(A) }80\qquad \textbf{(B) }81 \qquad \textbf{(C) }96 \qquad \textbf{(D) }97 \qquad \textbf{(E) }625\qquad$
| nvidia |
Let $x$ and $y$ be positive real numbers such that $2\lfloor x\rfloor=5\lfloor y\rfloor$ and $\lfloor x\rfloor+\lfloor y\rfloor=7$. Determine the smallest possible value of $\lfloor x+y\rfloor$. | nvidia |
A gardener plants six flowers in a row, each of which is either red or yellow. The gardener then randomly selects two flowers and swaps them. The probability that the gardener will end up with exactly two yellow flowers on the ends and exactly two red flowers in the middle is $\frac{m}{n}$, where $m$ and $n$ are relati... | nvidia |
If $a$ and $b$ are positive integers such that $\dfrac{2^a}{3^b}$ is simplified to a form where it's equal to $2^x3^y$,
what's the value of $xy$ if $\dfrac{a}{b}$ is expressed as $0.\overline{6}$? | nvidia |
Determine all values of the positive integer $n$ for which $n^3 + n^2 + n + 1$ is a multiple of $n + 1$. | nvidia |
Let $x$ and $y$ be two positive numbers such that $xy=288$ and $\frac{x}{y}+\frac{y}{x}=5$. Find the value of $x+y$. | nvidia |
The function $f(x)$ satisfies
\[f(x + y) = f(x) + f(y)\]for all real numbers $x$ and $y,$ and $f(1) = 1.$ Find all possible values of $f(2).$ | nvidia |
Let $f(x)$ be a function such that $f(0) = 1$ and
\[f(xy) = f \left( \frac{x^2 + y^2}{2} \right) + (x - y)^2\]for all real numbers $x$ and $y.$ Find $f(x).$ | nvidia |
Find the area of the triangle with vertices $(3,-5),$ $(-2,0),$ and $(1,-6).$ | nvidia |
Let $x,$ $y,$ and $z$ be three positive real numbers whose sum is 1. If no one of these numbers is more than twice any other, then find the minimum value of the product $xyz.$ | nvidia |
In the geometric sequence $a_1, a_1r, a_1r^2, \ldots$, the sum of the first $n$ terms is denoted by $S_n$ and the sum of the reciprocals of the first $n$ terms is denoted by $R_n$. If $\frac{S_{2n}}{R_n}$ is a constant that does not depend on $n$, for all positive integers $n$, then find the common ratio $r$. | nvidia |
Determine the smallest whole number greater than 50 that has exactly 5 positive factors. | nvidia |
Consider the rational function
\[f(x)=\frac{x^2+ax+b}{x^2+cx+d}.\]
Find $k$ such that the function $f(x)+k$ has no horizontal asymptote. | nvidia |
Find the sum of all real numbers $x$ such that the cube of $x$ is 64 more than the square of $x$. | nvidia |
What is the probability that the sum of the numbers on two dice rolled simultaneously is less than 7? | nvidia |
Determine the sum of the digits of the smallest repeating block in the decimal expansion of $\frac{13}{23}$. | nvidia |
Find the number of integers that satisfy the inequality $\frac{x-3}{2} \ge \frac{x+1}{3}$ and also belong to the set of all odd integers between -5 and 7. | nvidia |
Find the smallest positive integer $m$ such that the expression $\frac{m^3-m^2-1}{m-1}$ is a prime number for some but not all values of $m$ when $m$ is a positive integer greater than 1. | nvidia |
If the point $(a,b)$ is on the graph of $y=f(x)$, and $g(x)=\frac{1}{f(x)}$ for all $x\neq a$, then there is one point that must be on the graph of $y=g(x)$. What is the product of the coordinates of that point? | nvidia |
A regular octahedron has its vertices at the midpoints of the edges of a regular tetrahedron. The sum of the x-coordinates of the vertices of the octahedron is 12. Find the sum of the x-coordinates of the vertices of the tetrahedron. | nvidia |
Given two equations:
$$ab=50$$
$$b+a^2=40$$
where $a$ and $b$ are positive real numbers, find the value of $a+b$. | nvidia |
The product of the first $n$ positive integers is equal to the product of the reciprocals of the first $n$ positive integers plus a constant. Find the constant. | 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 |
Given that $y$ is real and $y^4+\frac{1}{y^4}=80$, find $y+\frac{1}{y}$. | nvidia |
Determine the value of $c$ for which the polynomial $x^4-2x^3+cx^2-2x+1$ has exactly three distinct real roots. | nvidia |
Let $f(n)$ denote the largest prime factor of $n$. Prove that for every positive integer $n$, there exist positive integers $a$ and $b$ such that $f(a+b) > \max\{f(a), f(b)\}$. | nvidia |
Let $a$, $b$, $c$, $d$ be positive real numbers such that $a^2+b^2+c^2+d^2=1$. Prove that
$$a^2b^2(1-c^2-d^2)+b^2c^2(1-a^2-d^2)+c^2d^2(1-a^2-b^2)+a^2d^2(1-b^2-c^2)\leq 0.$$ | nvidia |
A box contains only dimes and quarters. The value of the quarters is $2\frac{3}{4}$ times the value of the dimes. If the total value of the coins is $57, determine the maximum number of dimes that the box could contain. | nvidia |
What is the smallest positive integer whose square can be expressed as the sum of the cubes of its digits? | nvidia |
Let $f(x)$ be a function such that $f(0) = 10,000$ and $f(n) = 1.06f(n - 1)$ for all positive integers $n.$ Then find the least positive integer $n$ such that $f(n) > 20,000.$ | nvidia |
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