instruction_seed stringlengths 30 7.99k | source stringclasses 1
value |
|---|---|
The sum of the digits of the number $7^m$, where $m$ is a positive integer greater than or equal to 10, is divisible by 9.
Prove that $m$ is divisible by 3. | nvidia |
Find the sum of all the roots of the polynomial
$$x^4 - 6x^3 + 13x^2 - 12x + 4$$
that are in the form of $\frac{1}{a}$, where $a$ is an integer. | nvidia |
Let $\mathbf{A}$ be a $3 \times 3$ matrix with real entries such that $\mathbf{A}^T \mathbf{A} = \mathbf{I}$ and $\det(\mathbf{A}) = 1$. If the sum of the squares of the entries in the first row of $\mathbf{A}$ is 2, find the possible values of the sum of the squares of the entries in the second row of $\mathbf{A}$. | nvidia |
In a set of 20 distinct real numbers, what is the minimum number of elements that must be selected to guarantee that the selection contains at least one pair of numbers whose sum is a perfect square? | nvidia |
Find the smallest positive integer whose sum of divisors is a perfect square. | nvidia |
Find the sum of the interior numbers of the eighth row in Pascal's Triangle, given that the sum of the interior numbers in the fifth row is 14 and the sum of the interior numbers in the sixth row is 30. | nvidia |
Determine the greatest integer $n$ such that $3^n$ is less than 10000. | nvidia |
An integer between 1000 and 9999, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there? | nvidia |
The function $f(x) = \tan x$ has exactly three real roots on the interval $(0,4\pi)$. Find all possible real values of $c$ such that the equation $f(x) = c\tan2x$ also has exactly three real roots on the same interval. | nvidia |
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 $a$.
What is the value of $a$ in the equation $x^2+ax+a^2=0$ if the sum of the squares of its roots is equal to 1.75? | nvidia |
Two spheres, one centered at $(0,0,0)$ with radius 3, and the other centered at $(4,1,1)$ with radius 4, are given. Determine the equation of the plane containing the two points on the spheres that are closest to each other. | nvidia |
Let $g(x)$ be a polynomial of degree 2018 with real coefficients, and let its roots be $s_1,$ $s_2,$ $\dots,$ $s_{2018}.$ There are exactly 1012 distinct values among
\[|s_1|, |s_2|, \dots, |s_{2018}|.\]What is the minimum number of real roots that $g(x)$ can have? | nvidia |
The value of $\sqrt{73}$ is between two positive, consecutive integers. What is the product of these two integers?
Another problem inspired by this one:
Find the value of $n$ such that the sum of the squares of the two consecutive integers closest to $\sqrt{n}$ is equal to $n$. | nvidia |
The figure shows two circles with radii $10$ and $8$, whose centers are $14$ units apart. A line is drawn from the point of intersection, $P$, to a point $Q$ on one of the circles such that $PQ$ is perpendicular to the line joining the centers of the circles. A point $R$ is chosen on the other circle such that the tria... | nvidia |
Find all values of $x$ that satisfy the inequality $(x^2+2x+1)(x^2-4x+4)>0$. | nvidia |
The expression $\frac{4k+8}{4}$ simplifies to an expression of the form $ak+b$ where $a$ and $b$ are integers. Find $\frac{a}{b}$ . | nvidia |
Given the equation:
\[5x^2-kx+1=0\]
Find the value of $k$ such that the product of the roots is equal to the difference of the roots.
Note: I've changed the equation and the condition to create a new problem that requires a different approach to solve. The original problem was about finding the value of $k$ such tha... | nvidia |
Problem:
Find the largest real number $k$ such that for all positive real numbers $a,$ $b,$ and $c,$
\[a^2 + b^2 + c^2 + k(ab + bc + ca) \ge (a + b + c)^2.\] | nvidia |
Let $$m,n$$ be positive integers such that $$n\leq m$$, and let $$M$$ denote the set $$\{1,2,\ldots,m\}$$. For each subset $$X$$ of $$M$$, find the maximum possible value of $$\prod_{x\in X}\frac{x+n}{x}$$, and for each integer $$i$$ from $$0$$ to $$mn$$, give the number of sets in $$M$$ for which $$\prod_{x\in X}\frac... | nvidia |
An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $16$, and one of the base angles is $\arcsin(.8)$. Find the area of the trapezoid.
$\textbf{(A)}\ 72\qquad \textbf{(B)}\ 75\qquad \textbf{(C)}\ 80\qquad \textbf{(D)}\ 90\qquad \textbf{(E)}\ \text{not uniquely determined}$
| nvidia |
Find the positive value of $n$ such that the equation $9x^2+nx+1=0$ has exactly one solution in $x$. | nvidia |
The sum of the squares of two positive integers is 193. The product of the two integers is 84. What is the sum of the two integers? | nvidia |
If $n$ is the smallest positive integer for which there exist positive real numbers $a$ and $b$ such that
\[(a + bi)^n = (a - bi)^n,\]compute $\frac{a}{b}.$ | nvidia |
Problem:
Find all real values of $x$ that satisfy the equation
\[\left(3^{\frac{1}{x}} + 2^{\frac{1}{x}}\right)^{2x-5} = 1.\] | nvidia |
Let $x$ and $y$ be two positive integers such that $x^2 + y^2 = 50$. Enter the ordered pair $(x,y)$ for which $x^3 y^4$ is maximized, subject to the constraint that $x$ and $y$ are relatively prime. | nvidia |
Consider the positive integers 24 and 28. They can be expressed as sums of two different prime numbers (7 + 17 and 11 + 17). Determine the largest number less than 50 that can be expressed as the sum of two distinct primes in exactly three different ways. | nvidia |
The sum of the squares of the first $n$ positive integers is equal to $n(n+1)(2n+1)/6$. Prove that the sum of the squares of the first $n$ positive odd integers is equal to $n(2n+1)(2n-1)/3$. | nvidia |
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 |
Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c=1$. Find the minimum value of
$$\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right)\left(\frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c}\right).$$ | nvidia |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | nvidia |
Find the value of $x$ that satisfies the equation:
$$\frac{x+3}{x+5}-\frac{2}{x-2}=1$$ | nvidia |
Determine the value of $a$ such that the quadratic equation $ax^2 + 3x + 1 = 0$ has only one real solution. | nvidia |
Find the number of positive integers not exceeding 100 that have exactly one even digit. | nvidia |
Milton spilled some ink on his homework paper. He can't read the coefficient of $x$, but he knows that the equation has two distinct negative, integer solutions. What is the sum of all of the distinct possible integers that could be under the ink stain?
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)-... | nvidia |
A sequence of positive integers is called "unpredictable" if no three consecutive terms are in arithmetic progression. Find the largest possible value of $n$ such that there exists an unpredictable sequence of length $n$ consisting only of integers between 1 and 100. | nvidia |
Given that $a$, $b$, and $c$ are positive integers, find the smallest value of the expression $\frac{a(b-c)^2}{bc}$ where $\frac{a}{b}$ and $\frac{a}{c}$ are both integers. | nvidia |
The fourth degree polynomial $P(x) = 12x^4+20x^3+5x^2-7x+13$ can be written as the product of two quadratic polynomials, both with positive leading coefficients. Find the minimum value of the sum of the squares of these two leading coefficients. | nvidia |
Let $*$ be a binary operation on positive integers such that for every $n,m$, $n*m$ is also a positive integer. The operation $*$ satisfies the following properties:
(i) For every $n$, $1*n=n$.
(ii) For every $m,n$, $m*(n+1)=m*n+m$.
(iii) For every $m,n$, $(m+1)*n=m*n+n$.
Calculate $6*3$. | nvidia |
Determine $k$ such that the roots of the equation $2x^2+4x-k=0$ are exactly the base and height of a triangle whose area is equal to the area of the triangle with vertices $(1, 2)$, $(2, 5)$, and $(5, 1)$. | nvidia |
Find the sum of all positive integers $n$ such that $n^2+20n+21$ is a perfect square. | nvidia |
A positive integer is called symmetric if it remains unchanged when its digits are reversed. How many symmetric positive integers are there that are greater than 10 and less than 1000, and whose digits are all perfect squares? | nvidia |
A square and a regular hexagon are inscribed in the same circle. The side length of the square is $12$ cm. Find the area of the hexagon. | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that $xyz = 8.$ Find the minimum value of $x + 2y + 4z.$ | nvidia |
Determine all positive real values of $x$ for which $2x^2 + 15x$ is a perfect cube. | nvidia |
Let $g(x)$ be a function defined for all integers $x \ge 0$ such that $g(0) = 1$ and $g(a+b) = g(a)g(b) - g(ab)$ for all integers $a, b \ge 0.$ Find $g(2023).$ | nvidia |
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 |
Let $x$ and $y$ be real numbers such that $2(x^2 + y^2) = x + y.$ Find the maximum value of $x - y.$ | nvidia |
What is the sum of the values of $x$ that satisfy the equation $x^2-5x+5=9$? | nvidia |
Let $f(x)$ be a polynomial of degree 2006 with real coefficients, and let its roots be $r_1,$ $r_2,$ $\dots,$ $r_{2006}.$ There are exactly 1006 distinct values among
\[|r_1|, |r_2|, \dots, |r_{2006}|.\]What is the minimum number of real roots that $f(x)$ can have? | nvidia |
Find $x$ such that $\log_x 4=\log_{27} 3$. | nvidia |
Given a positive integer $n$, find the sum of all non-negative integral values of $m$ such that the number $n^m-m^n$ is a perfect square. Note that there is at least one such value of $m$. | nvidia |
Let $n$ be a positive integer and $\omega$ be a complex number such that $\omega^{2n} = 1$ and $\omega^{n} \neq 1.$ Let $A_n$ be the number of ordered pairs of complex numbers $(a,b)$ such that
\[a^n b^n = a^{2n} b = 1.\]Determine $A_1 + A_2 + A_3 + \dotsb.$ | nvidia |
Let $a$ and $b$ be two positive integers such that $a^2-b^2$ is a prime number.
Prove that $a-b=1$. | nvidia |
There are three points that are $7$ units from the line $y=11$ and $11$ units from the point $(4,11)$. What is the sum of the $x$- and $y$-coordinates of all three of these points? | nvidia |
Let $a$ and $b$ be positive integers such that $b>1$ and $b$ is not divisible by $3$. Find the largest positive integer $k$ that divides $a^{b-1} + a^{b-2} + \ldots + a^2 + a + 1$ for any choice of $a$. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 9.$ Find the maximum value of
\[ax + by + cz,\]
given that $x,$ $y,$ and $z$ are positive real numbers such that $x^2 + y^2 + z^2 = 1.$ | nvidia |
Find all values of $x$ such that $\lceil x \rceil - x = 0.5$. | nvidia |
Let $a$ and $b$ be positive integers such that the repeating decimal $0.\overline{ab}$ can be expressed as $\frac{1}{x}+\frac{1}{y}$ where $x$ and $y$ are positive integers with $x>y$ and $x+y<20$. What is the smallest possible value of $a+b$? | nvidia |
What is the smallest positive integer that can be written in the form $1503m + 18181n$, where $m$ and $n$ are integers? | nvidia |
In a particular ellipse, the product of the lengths of its semi-major and semi-minor axes is 50. Determine the area of the ellipse knowing that its perimeter is 30. | nvidia |
Find the values of $x$ for which the expression $\sqrt{x^3 + x^3 + \sqrt{3x}}$ is a real number. | nvidia |
Suppose that $x$ and $y$ are positive integers for which $x$ has $4$ factors and $y$ has $x$ factors. If $y$ is divisible by $x$, then what is the least possible value of $y$? | nvidia |
A parabola with equation y = x^2 is drawn, and a circle is drawn with its center at the vertex of the parabola. The circle is tangent to the parabola at two points, and its radius is such that it intersects the parabola at exactly two other points. Compute the radius of the circle. | nvidia |
Given that $a,$ $b,$ $c,$ and $d$ are non-zero complex numbers, find all possible values of $k$ such that the equations $ax^4+bx^3+cx^2+dx+e=0$ and $bx^4+cx^3+dx^2+ex+a=0$ have a common root $k$, assuming that the coefficients satisfy the condition $a+b+c+d+e=0$. | nvidia |
Consider a function $g$ defined on the positive integers, satisfying the equation
\[g(x + y) = g(x) \cdot g(y)\]for all positive integers $x$ and $y.$ We know that $g(3) = 13$ and $g(7) = 53.$ Find $g(17).$ | nvidia |
Find the largest integer that can be obtained by summing some of the nine smallest distinct positive integer multiples of 7. | nvidia |
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are... | nvidia |
If $m$ is the product of all positive integers less than $101$ that are multiples of $7$, and $n$ is the product of all positive integers less than $101$ that are multiples of $11$, find the greatest common divisor of $m$ and $n$. | nvidia |
A positive integer is called lucky if it can be expressed as the product of two distinct primes. Find the largest unlucky number less than 100. | nvidia |
Let $a_1,a_2,a_3,\ldots$ be a sequence of real numbers such that $a_1=0$ and for $n=2,3,4,\ldots$,
$a_n=a_{n-1}+\f{1}{(a_{n-1}+1)^2}$.
Find $\lim\limits_{n\to\infty}a_n$. | nvidia |
Find the value of $x$ such that $x^2+2$, $x^2+14$, and $x^2+26$ are consecutive terms of a geometric sequence. | nvidia |
Find the sum of all three-digit positive integers that are perfect cubes and have a digit sum equal to 18. | nvidia |
Let $a$ be the largest negative three-digit integer that is a multiple of $11$ and let $b$ be the smallest positive two-digit integer that is a multiple of $5$. Determine the difference between the product of the digits of $a$ and the sum of the digits of $b$. | nvidia |
A cube with edge length $x$ is placed inside a rectangular prism with dimensions $x$, $y$, and $z$, such that the cube touches all six faces of the prism. If the volume of the cube is 1/8 of the volume of the prism, find the positive integer $y$ such that $y \equiv 7 \pmod{9}$. | nvidia |
What is the smallest positive integer $n$ such that $\frac{n}{n+101}$ is equal to a terminating decimal? | nvidia |
Problem:
A bookshelf has $n$ shelves, and each shelf can hold exactly $k$ books. If the bookshelf is currently empty, and 480 books are to be placed on it, how many different pairs of $(n, k)$ can be chosen such that all the books can be placed on the shelves, with each shelf holding the same number of books, and each ... | nvidia |
Find the sum of all non-negative integers $a$, $b$, and $c$ such that $\binom{10}{a} \cdot \binom{20}{b} \cdot \binom{30}{c} = \binom{60}{20}$. | nvidia |
Let \(m\) and \(n\) be two distinct positive integers, each having an odd number of positive divisors. Determine the smallest possible number of positive divisors that \(m^{2}+n^{2}\) can have. | nvidia |
Let $a$ and $b$ be the roots of the equation $2x^2-6x-5=0$. Determine the number of real solutions of the inequality $|x^2+ax+b| < \frac{1}{2}$. | nvidia |
Given that $x+y = 10$ and $2x+y = 13$, evaluate $x^2-y^2$.
Let $x$ and $y$ be two numbers such that $x+y=10$ and $2x+y=13$. Find the value of $x^2-y^2$. | nvidia |
Find the least positive integer $n$ such that $n^2$ is divisible by $23^3$ and $n^3$ is divisible by $29^2$. | nvidia |
Given the equation of a parabola $y = ax^2$, determine the value of $a$ such that the area enclosed by the parabola and the line $y = 2$ is equal to $\frac{\pi}{2}$ square units. | nvidia |
Let $g$ be a function that maps real numbers to real numbers, such that $g(0) = 0$ and
\[g(x + y) - g(x - y) = \frac{g(2x) - g(2y)}{x + y}\]for all real numbers $x \neq -y$. Find all possible values of $g(2)$. | nvidia |
Find the sum of all $x$ that satisfy the equation $f(x)+f^{-1}(x)=f(x^2)$, where $f(x)=x^2+1$. | nvidia |
Find all values of the parameter $a$ for which the equation $ax^2+(a-1)x+1=0$ has exactly one solution, and the product of these values of $a$. | nvidia |
Let $p(x)$ be a polynomial of degree 2007 such that $p(x) = p(-x)$. Prove that if $|p(1)| < 10^{-2006}$, then $|p(2)| < 1$. | nvidia |
The line $y = 2$ intersects the graph of $y = 3x^2 + 2x - 4$ at the points $A$ and $B$. The distance between $A$ and $B$ can be written as $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are positive integers that do not share any factors other than one. Find the value of $m + n$. | nvidia |
Find the sum of all numbers $y$ such that $y + \frac{y^2}{y-1} = 3$. | nvidia |
Let $\triangle ABC$ be a right triangle with integer side lengths such that the area of the circle inscribed within its boundaries is 121. Given that the perimeter of $\triangle ABC$ is 840, what is its area? | nvidia |
Let $T$ be the set of all positive real numbers. The function $g : T \to T$ satisfies the following two properties:
(i) First,
\[g(x^2) = g(x) + 1\]for all $x \in T.$
(ii) Second,
\[g(x) + g(y) = g(xy) + g\left(\frac{x}{y}\right)\]for all $x \in T$ and $y \in T.$
Let $m$ be the number of possible values of $g(2),$ a... | nvidia |
Let $A$ be the set of all possible last digits of numbers of the form $3^n$, where $n$ is a positive integer. Let $P$ be the product of all the elements in $A$. Find the smallest power of 3 that has the same last digit as $P$. | nvidia |
In the multiplication problem below, a digit represented by $\triangle$ has been replaced by the letter. Find the value of $\triangle$.
$$
\begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
& & & 4 & 2 & \triangle_6 \\
&\times & & & & 3_6 \\
\cline{2-6}
& & & \triangle & 1 & 4_6\\
\end{array}
$$ | nvidia |
Let $x,$ $y,$ $z$ be a three-term geometric series where all the terms are positive, such that $x+y+z = 24.$ Find the largest possible value of $y.$ | nvidia |
Two of the roots of the equation \[ax^3+bx^2+cx+d=0\]are $3$ and $-2.$ Given that $a \neq 0,$ compute $\frac{b+c}{a}.$ | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $a>b>c$. The system of equations
$$ab+c \equiv b(a+c) \pmod{c(a+b)}$$
and
$$c(a+b)+1 \equiv ac+1 \pmod{ab}$$
has a unique solution $(a,b,c)$. Find $c^{-1} \pmod{a}$. | nvidia |
Let $S$ be the sum of the divisors of $120$. Determine the smallest positive integer $n$ such that $S$ is divisible by $n^2$. | nvidia |
In a graph, the number of vertices is equal to the number of edges. At each vertex, exactly three edges meet. Some vertices are "special" and at these vertices, all three edges are "short", while at the rest of the vertices, two of the edges are "short" and one is "long". Let S denote the number of special vertices, an... | nvidia |
Let $x$ be a positive real number such that $x + \frac{1}{x} = 98.$ Find\[\sqrt{x} + \frac{1}{\sqrt{x}}.\] | nvidia |
Let $g$ be a function taking the positive real numbers to the positive real numbers, such that
\[g(a)g(b) = g(\sqrt{a^2 + b^2})g(\sqrt{|a^2 - b^2|})\]for all positive real numbers $a$ and $b$.
Find the smallest possible value of $g(1)$ if $g(0) = 1$. | nvidia |
Find the number of distinct four-digit positive integers such that the sum of their digits equals 18 and none of the digits are zero. | nvidia |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.