instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find a positive integer that is divisible by 18 and whose cube root is a number between 8 and 8.1.
Another problem:
Find a positive integer that is a perfect square and whose square root is a number between 4 and 4.1. | nvidia |
How many of the positive integers less than 100 have a prime factor that is one less than a perfect square? | nvidia |
Let $P(x)$ be a monic polynomial of degree $n$ with complex roots. Prove that there exists a constant $M$ independent of $P$ such that $|z_i|\leq M \left|\frac{\text{Im}(a_{n-1})}{\text{Im}(a_{n-2})}\right|$ for at least one root $z_i$ of $P(x)$ whenever $\text{Im}(a_{n-1}) \neq 0$ and $\text{Im}(a_{n-2}) \neq 0$. Find... | nvidia |
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc = 1$. | nvidia |
In three-dimensional space, find the number of lattice points that have a distance of 3 from the origin.
Note: A point is a lattice point if all its coordinates are integers. | nvidia |
Compute the number of ways to arrange the letters in the word "ABRACADABRA" such that the first and last letters are both "A". | nvidia |
Determine the smallest natural number $n$ such that the number $5^{2n+1}+1$ is divisible by $12$. | nvidia |
Find $n$ such that $2^5 \cdot 3^2 \cdot n = 8!$. | nvidia |
Find the number of positive integers $b$ less than $20$ such that the congruence $bx\equiv 1\pmod{20}$ has a solution in $x$. | nvidia |
A portion of the graph of a quadratic function $f(x)$ is shown below.
Let $g(x)=-f(x)$ and $h(x)=f(-x)$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=g(x)$ intersect, and $b$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, then what is $10a+b$?
[asy]
size(150);
real ... | nvidia |
Let $g(x)$ be a polynomial of degree $6$ such that $g(0) = 0$ and $g(x) = g(-x)$ for all real numbers $x.$ Which of the following is a conclusion that can be made about the coefficients of $g(x)$?
(A) The constant coefficient is $0$.
(B) The sum of the coefficients of $g(x)$ is $0$.
(C) The coefficient of $x^5$ is $0$... | nvidia |
Find the value of $a$ so that the lines with the given equations are perpendicular. \begin{align*}
y &= 2x+13 \\
6y+ax &= 6.
\end{align*}
New Problem:
In the coordinate plane, a circle with center $(0, 0)$ intersects a line with equation $y = mx + b$ at two points. If the distance between these two points is 10 units... | nvidia |
Find the least positive integer $n$ such that $n!$ is divisible by $2^{13}$. | nvidia |
Find the smallest positive integer $n$ such that the sum of the cubes of the first $n$ positive integers is a multiple of 729. | nvidia |
Let $a$, $b$ be positive real numbers. Find the minimum value of $|a+b|+|a-b|$ given that $a$ and $b$ satisfy $a^2b^3 = 1000$ and $ab+1=2a$. | nvidia |
Find the smallest positive integer $n$ such that the cube of the smallest odd palindrome greater than $n$ is a palindrome. | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 1.$ Find all possible values of
\[\frac{x^4 + y^4 + z^4}{x^2y^2 + x^2z^2 + y^2z^2}.\] | nvidia |
A trapezoid has a fixed height of 6 inches and a perimeter of 24 inches. The lengths of its four sides are all different integers. Determine the maximum possible area of the trapezoid. | nvidia |
Let $f(n)$ denote the sum of the positive divisors of a natural number $n$. Determine the natural number $n$ satisfying $n=f(f(n))$. | nvidia |
Count the number of nonnegative integers less than $10000$ that contain exactly two $1$ digits. | nvidia |
The function $f(x)$ satisfies
\[f(xy) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(0) \neq 0.$ Find $f(10).$ | nvidia |
What is the smallest positive integer that can be expressed as the sum of two perfect squares in exactly three different ways? | nvidia |
Find the product of all the solutions of the equation $\frac{3y}{15}=\frac{2}{y}$. | nvidia |
The polynomial $$x^6-2x^5+11x^4-24x^3+16x^2+40x+25$$ has two real roots and four complex roots. Find the product of the real roots. | nvidia |
The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [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=0.3;
real who... | nvidia |
Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=3$. Determine the maximum value of $$\sqrt[3]{(a+b)(b+c)(c+a)}$$ | nvidia |
Let $ S $ be the set of all sides and diagonals of a regular pentagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | nvidia |
Find the sum of the first eight positive multiples of 17. | nvidia |
Find the sum of the distinct prime divisors of all the positive divisors of 90. | nvidia |
In an arithmetic sequence, the sum of the terms $a$, $b$, and $c$ equals the sum of the terms $d$, $e$, and $f$. The first term of the sequence is 5, and the sum of $b$ and $c$ is 23. Find the value of $d$. | nvidia |
Find the least positive integer $n$ such that $3^{2n} \equiv 1 \pmod{80}$. | nvidia |
A sphere with center $(1,-1,0)$ and radius 6 intersects the plane $z = -7$. The intersection is a circle centered at $(1,-1,-7)$. Find the distance between the center of the sphere and the point on the plane that is farthest from the center of the circle. | nvidia |
Given that $\cos 2x + \cos 3x = \sin 4x$ for some angle $x$, find the smallest positive integer $n$ such that $nx$ is an integer multiple of $2\pi$. | nvidia |
Two real numbers $x$ and $y$ satisfy $x-y=4$ and $x^3-y^3=28$. Compute $xy$. | nvidia |
The product of the positive integer divisors of a positive integer $n$ is 729. Find $n$. | nvidia |
What is the largest integer $n$ such that $n!$ is divisible by $2^{10}$? | nvidia |
A rhombus is inscribed in a unit circle so that its diagonals pass through the origin. If one of the diagonals of the rhombus lies along the positive real axis, find the product of the complex numbers representing the vertices of the rhombus. | nvidia |
Let \[x=\sqrt[3]{\sqrt{\frac{1}{256}}}\]. Find the smallest positive integer $n$ such that $x^{6n}$ is a perfect square. | nvidia |
How many positive integers $n$ less than 200 have a corresponding integer $m$ divisible by 5 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? | nvidia |
Find the smallest positive integer $x$ which is greater than $1$ and relatively prime to $120$ (recall that relatively prime means that the GCD of $x$ and $120$ is $1$) | nvidia |
Find the sum of all real solutions of the equation $|x-2| = \sqrt{3-|x-2|}$. | nvidia |
Find the sum of all real solutions to the equation:
$$\frac{x-1}{x^2+3x+2} = \frac{x-4}{x^2-7x+1}$$ | nvidia |
Let $m$ and $n$ be positive integers such that $m \leq n$. We call $m$ *representable* for $n$ if there exists an ordered pair $(a,b)$ of positive integers, with $a \leq b$, such that
\[m = (a+bi)(a-bi) = a^2 + b^2\]
and
\[n = a^2b + ab^2\]
How many integers in the set $\{100,101,102\cdots 199\}$ are representable for ... | nvidia |
Let $p(x)=x^2+ax+b$ be a polynomial with integer coefficients and positive roots $r$ and $s$. It is known that $b=r\cdot s$ and $r+s=-a$. Assuming that the values of $a$ and $b$ are known, determine the product of all possible values of $r$ and $s$. | nvidia |
Find the sum of all constants $k$ such that the quadratic $x^2 + kx - 24$ can be factored in the form $(x+c)(x+d)$, where $c$ and $d$ are integers. | nvidia |
Given that $\frac{1}{x} = \frac34$ and $\frac{1}{y} = \frac43$, find the value of $\frac12x^6y^7$.
If 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 $a$. | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=7$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | 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 |
The sum of the squares of two positive integers is 200. Find the greatest common divisor of these two integers. | nvidia |
Let $ S $ be the set of all sides and diagonals of a regular pentagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | nvidia |
The number $x^y$ is such that both $x$ and $y$ are positive integers less than or equal to 20. What is the greatest number of consecutive positive integers that can divide $x^y$? | nvidia |
In a triangle with side lengths 8, 15, and 17 inches, the three sides are cut into two equal parts each. What is the length of the longest segment that can be removed from each of the six pieces so that the remaining pieces can still form a triangle? | nvidia |
What is the least positive integer $n$ such that $4125$ is a factor of $n!$? | nvidia |
Find the number of solutions for the congruence $y - 7 \equiv 21 \pmod{15}$, given that $y$ is a positive integer less than 200. | nvidia |
Find the smallest positive value of $y$ such that $y + 2468$ results in a palindrome. | nvidia |
A positive sequence $a_n$ satisfies the property that
$\sqrt{a_1+a_2+\cdots + a_n}=n.$
Determine $\frac{a_1}{a_2}$. | nvidia |
How many different convex pentagons are there with integer sides and perimeter equal to 35, such that at least one of the internal angles is a right angle? | nvidia |
Given that
\[f(x) = \left\{
\begin{array}{cl}
|x|+a & \text{if } x < 0, \\
x-a & \text{if } x \ge 0,
\end{array}
\right.\]
and that $f(f(x)) = -x$ for all $x,$ find the value of $a.$ | nvidia |
Let $u$ be a complex number such that
$$\frac{u+1}{u^2-1}=\frac{1}{4-2i}.$$
Find $\frac{u-1}{u+1}.$ | nvidia |
What is the largest integer that is a divisor of $$1^2+2^2+3^2+\cdots+n^2$$
for all positive integers $n$? | nvidia |
Find the integer $n$, $0 \le n \le 7$, such that \[n \equiv -3737 \pmod{8}.\] | nvidia |
A point $(x,y)$ lies on the graph of the circle $x^2+y^2=15$. Consider the point $(a,b)$, where $a=|x-2|$ and $b=|y-3|$. Determine the value of $a^2+2ab+b^2$ given that $a+b=7$. | nvidia |
For real numbers $x \in (0, \frac{\pi}{2}),$ find the maximum value of
\[\frac{\sin x}{\sqrt{\cos x}}.\] | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a+b+c=1.$ Find the maximum value of
\[\frac{bc}{a+1}+\frac{ca}{b+1}+\frac{ab}{c+1}.\] | nvidia |
Find the smallest positive integer $x$ that satisfies the congruence $x^2 \equiv 1 \pmod{91}$. | nvidia |
Determine the largest positive integer that is both a divisor of 108 and a divisor of 54. | nvidia |
Find the number of positive integer divisors of 99 that are also perfect squares. | nvidia |
Let $g(z)$ be a polynomial of degree $n$ with complex coefficients, and let its roots be $w_1,$ $w_2,$ $\dots,$ $w_n.$ Suppose that there are exactly $k$ distinct values among
\[\arg(w_1), \arg(w_2), \dots, \arg(w_n).\]What is the minimum number of roots of $g(z)$ that lie on the unit circle? | nvidia |
In the equation $x^3 + bx^2 + cx + d = 0$, find the sum of all possible values of $b$ for which the equation has exactly two distinct real solutions for $x$. | nvidia |
Find the sum of all positive integers from 1 to 100 that are not divisible by 3. | nvidia |
Let x, y, z be positive real numbers such that $\frac {1}{\sqrt{x}+1} + \frac {1}{\sqrt{y}+1} + \frac {1}{\sqrt{z}+1} = 1$.
Prove that $$\frac {1}{x} + \frac {1}{y} + \frac {1}{z} \geq \frac {9}{8}.$$ | nvidia |
Find the sum of the coefficients of the polynomial \[ (x+1) \cdot (x+2) \cdot \dotsm \cdot (x+10), \] when multiplied out. | nvidia |
What is the smallest positive integer $n$ such that $n\%$ of $n$ is greater than or equal to $120$? | nvidia |
The product of two consecutive positive integers is 7020. What is the difference of the two integers? | nvidia |
Find the sum of all even numbers between 20 and 58, inclusive. | nvidia |
Let $b_1,$ $b_2,$ $\dots$ be a sequence of positive real numbers such that
\[b_n = 4b_{n - 1} - 2n + 1\]for all $n > 1.$ Find the smallest possible value of $b_1.$ | nvidia |
In triangle $ABC$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O$, and that $\frac{AO}{OA'}+\frac{BO}{OB'}+\frac{CO}{OC'}=92$, find $\frac{AO}{OA'}\cdot \frac{BO}{OB'}\cdot \frac{CO}{OC'}$.
| nvidia |
In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, the diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. Given that $AE:EC=5:7$ and $BE:ED=8:3$, compute the ratio of the area of triangle $ABE$ to the area of triangle $CDE$. | nvidia |
Find the sum of the positive divisors of 18. | nvidia |
The sum of the squares of two consecutive positive integers $x$ and $x+1$ is divided by the sum of the integers themselves. For which value of $x$ is the resulting quotient the smallest? | nvidia |
Let $x$ and $y$ be real numbers such that $x+y=2$ and $x\cdot y=-2$. Determine the value of
$$x^2\cdot y^2 \cdot x^4 \cdot y^4 \cdot x^8 \cdot y^8 \cdot x^{16} \cdot y^{16} \cdot x^{32} \cdot y^{32}.$$ | nvidia |
Find the least integer $n$ such that $\sqrt{n}+\sqrt{n+1}$ is greater than or equal to 12. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 1.$ Find the minimum value of $a + b + c.$ | nvidia |
Determine $x$ such that $\log_{10}(2x^3)=\log_{5}(x^2)$. | nvidia |
A hexagon is inscribed in a rectangle such that the vertices of the hexagon divide each side of the rectangle into four equal parts. The perimeter of the rectangle is 120 centimeters. What is the number of square centimeters in the area of the hexagon? | nvidia |
Let $f(x)=2x+1$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f(x^{-1})$. | nvidia |
Let $S$ be a set of positive integers, such that for each element $n$ in $S$, the sum of its digits is equal to $11$. Consider two subsets $A$ and $B$ of $S$, such that for each element $a$ in $A$, the number of digits in $a$ is at most $4$, and for each element $b$ in $B$, the number of digits in $b$ is at most $5$. S... | nvidia |
Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\frac mn$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
| nvidia |
Let
\[\begin{array}{ccccccccc}
\mathbf{a} & = & \left(a_{1} & , & a_{2} & , & \ldots & , & a_{n}\right)\\
\mathbf{b} & = & \left(b_{1} & , & b_{2} & , & \ldots & , & b_{n}\right)
\end{array}\]
be two geometric sequences. The first $n$ entries of their term-by-term product $\left(a_{1}b_{1},a_{2}b_{2},\ldots,a_{n}b_{n}\... | nvidia |
Let $f(x)$ be a non-constant rational function such that
\[f(x - 1) + f(x) + f(x + 1) = \frac{f(x)}{f(x^2)}\]for all real numbers $x$ except for a finite number of exceptions. Find the sum of all possible values of $f(1)$. | nvidia |
The numbers $x$ and $y$ are chosen so that $x$ is a base-6 number and $y$ is a base-8 number. When these numbers are added together, the sum is a base-10 number. If the digits of $x$ are reversed, the resulting number in base-6 is equal to $y$. Find the smallest possible value of $x$ that satisfies these conditions. | nvidia |
Let $G$ be a graph whose vertices are the numbers $\{1,2,3,\ldots,1989\}$ and where two vertices $i$ and $j$ are connected by an edge if and only if $i-j$ is divisible by either $4$ or $7$. What is the minimum number of colors needed to color all the vertices of $G$ such that no two adjacent vertices have the same colo... | nvidia |
Consider a positive real number $a$ such that the equation $x^2 + 2axy = 2ax + 2y^2$ has only one solution. Find the value of $a$. | nvidia |
Find the value of $a$ such that the equation $ax^2 - 4x + 4 = 0$ has only one solution that satisfies the inequality $x > 2$. | nvidia |
Consider the set of all complex numbers that satisfy the equation $z^3 = 1$ and have a non-zero imaginary part. Determine the number of distinct complex numbers in this set for which the real and imaginary parts form an arithmetic progression in some order. | nvidia |
Find all three-digit numbers that are divisible by 11 and whose cubes end with the digits 08. | nvidia |
Find the sum of the squares of the reciprocals of the roots of $x^3-11x^2+19x-8=0$. | nvidia |
The geometric mean of an even number of consecutive positive integers is $x$. Find the sum of the two middle integers in terms of $x$. | nvidia |
Determine the value of $x$ such that $$\sec \left(\frac{\pi x}{7}\right) = -2.$$ | nvidia |
Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{a} & \frac{1}{b} \\ \frac{1}{c} & \frac{1}{d} \end{pmatrix} \renewcommand{\arraystretch}{1}.\] | nvidia |
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