instruction_seed stringlengths 30 7.99k | source stringclasses 1
value |
|---|---|
Find the smallest even number that can be expressed as the product of three distinct perfect squares. | nvidia |
A hexagon is drawn in the complex plane such that its vertices are the geometric images of the sixth roots of unity. If the vertex corresponding to 1 is shifted by the vector 1 + i, compute the numerical value of the sum of the squares of the coordinates of all the vertices. | nvidia |
Let $a$, $b$, and $c$ be the solutions of the cubic equation $4x^3-13x^2+9x-3=0$. Find the sum of the squares of the pairwise differences $(a-b)^2+(a-c)^2+(b-c)^2$. | nvidia |
Find the greatest common divisor of the sum of the first 10 positive integers and the sum of the first 15 positive integers. | nvidia |
Find the maximum value of
\[|z^3+i|\]
where $z$ is a complex number that satisfies $|z|=1$. | nvidia |
Let $f(x)$ be a function such that $f(2x) = f(x) + 1$ for all positive integers $x$. If $f(1) = 3$, find the value of $f(32)$. | nvidia |
What is the smallest positive integer $n$ such that $17n \equiv 1234 \pmod{7}$ and $n$ is a multiple of 3? | nvidia |
Find all values of $a$ for which the equation
$\arccos x + \arcsin x = a$
has exactly one solution. | nvidia |
In trapezoid $ABCD$, the bases $AB$ and $CD$ are parallel, and points $E$ and $F$ are on the extensions of $AB$ and $CD$, respectively, such that the distance between $E$ and $F$ is equal to the height of the trapezoid. The areas of triangles $AED$ and $BFC$ are equal to 12 and 15, respectively. Find the area of trapez... | nvidia |
Find the sum of all 3-digit positive integers whose digits are in arithmetic progression. | nvidia |
Let $g$ be a function taking the positive integers to the positive integers, such that
(i) $g$ is a periodic function with period $5$ (i.e. $g(n + 5) = g(n)$ for all positive integers $n$)
(ii) $g(m + n) = g(m) + g(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m + n$ is divisible by $5,$ then... | nvidia |
In the equation $ax^2+bx+c=0$, the sum of the roots is $-\frac{b}{a}$ and the product of the roots is $\frac{c}{a}$. Given that the sum of the roots is $-\frac{11}{3}$ and the product of the roots is $\frac{1}{3}$, determine the value of $c$ if $a=2$. | nvidia |
Determine the number of positive integers that can be expressed as the sum of three distinct perfect squares in exactly two different ways, considering all possible orderings of the squares. | nvidia |
Find the smallest positive integer $m$ such that $m^2 \equiv 34399 \pmod{15}$. | nvidia |
Let $n$ be the smallest positive integer such that $2^n \equiv 1 \pmod{125}$.
Find the residue of $3^{n+2} \pmod{125}$. Your answer should be an integer in the range $0,1,2,\ldots,124$. | 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 |
In the triangle $\triangle$, the median to the hypotenuse $c$ has a length of 12 units. Determine $c$ such that the difference of the squares of the two legs is 64 square units. | nvidia |
Find all real values of x that satisfy the equation |x-3| + |x+1| = 4. | nvidia |
Let $x,y,z$ be positive real numbers such that $x+y+z=20$ and $xy+yz+zx=100$. Let $n=\max\{xy,yz,zx\}$. Find the smallest possible value of $n$. | nvidia |
Let $n$ be a positive integer and $a$ be an integer such that $a$ is its own inverse modulo $n$. Let $p(x)$ be a polynomial with integer coefficients such that $p(a) \equiv 0 \pmod{n}$. Prove that $p(a^2) \equiv 0 \pmod{n}$. | nvidia |
Find the number of positive integers less than 1000 that can be expressed as $7k+3$, where $k$ is a positive integer. | nvidia |
Find all integral values of $a$ for which the equation $y=x^2+ax+a$ has exactly one integer root. | nvidia |
Find the sum of all positive integer values of $m$ such that $\frac{m+24}{m}$ is an integer. | nvidia |
Let $a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z$ be positive integers such that
$a+b+c+d+e+f+g+h+i+j+k+l+m+n+o+p+q+r+s+t+u+v+w+x+y+z=2015$.
Let $\#(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z)$ be defined as $abc+d$. Find the minimum value of $\#(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z)$. | nvidia |
Find the smallest positive integer $m$ such that
\[\sin^m x + \cos^m x \le \frac{1}{m^2}\]for all real numbers $x$ in the interval $\left[0, \frac{\pi}{4}\right]$. | nvidia |
Given that $a$ and $b$ are integers, if the function $f(x) = ax^2 + bx + 9$ satisfies the condition that $\frac{f(3x)}{f(x)}$ is a constant for all real numbers $x$, find the product of $a$ and $b$. | 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 $\frac{b}{a}.$ | 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*} | nvidia |
What is the smallest positive integer $n$ such that the product of $n$ and a 5-digit whole number can have exactly 7 digits? | nvidia |
Let $a$, $b$, and $c$ be positive integers, each consisting of two digits, such that $a^2+b^2=c^2$ and the sum of the digits of $a$ is equal to $3$. Find the remainder when the sum of the digits of $b$ is divided by $3$. | nvidia |
Find the value(s) of $z$ such that the system of equations
$$\begin{cases} x+y=2 \\ xy+z^2=0 \end{cases}$$ has infinitely many solutions. | nvidia |
Determine the sum of all integer divisors of $180$ that also divide the sum of its digits. | nvidia |
Find the smallest positive integer $n$ such that $n^2$ is divisible by 12, 18, and 30. | nvidia |
Compute
\[\sum_{1 \le a_1 < a_2 < \dots < a_k} \frac{1}{a_1^{2}a_2^{3} \dots a_k^{k+1}}.\]
(The sum is taken over all sequences $(a_1, a_2, \dots, a_k)$ of positive integers such that $1 \le a_1 < a_2 < \dots < a_k$.) | nvidia |
Let $m$ be a positive integer such that $2^m\equiv 1\pmod{7}$. Determine $2^{m+1}+2^{m-1}\pmod{7}$. | nvidia |
Find all complex numbers $z$ such that $|z-2|=|z+2|$. | nvidia |
The two vectors $\mathbf{u}$ and $\mathbf{v}$ are represented by the ordered pairs $(3, -2)$ and $(3k, -2k)$. If $k$ is an integer such that $0\le k<10$, determine $k$ so that the vector $23\mathbf{v}-7\mathbf{u}$ has the same direction as the vector $5\mathbf{i}+6\mathbf{j}$. | nvidia |
Compute the number of pairs $(x,y)$ of positive integers such that
$$x^2+y^2\leq 100$$
and
$$x^3+y^3\geq 1000.$$ | nvidia |
Given a parabola defined by the equation y = ax^2 + bx + c, where a, b, and c are constants, find the coordinates of the vertex of the parabola if the sum of the slopes of the tangent lines at the two x-intercepts is equal to the slope of the tangent line at the vertex. | 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 |
Given that $\triangle ABC$ is an isosceles right triangle with legs of length $x$ units, and the hypotenuse has a length of $2\sqrt2$ units. Determine the length of the hypotenuse's projection on one of the legs. | nvidia |
Given that $x,$ $y,$ and $z$ are positive integers such that $x+y+z=n$, let $p(x,y,z)$ be the product of $x$, $y$, and $z$. Among all triples $(x,y,z)$ satisfying the constraint, what is the minimum value of $|p(x,y,z)-xyz|$? | nvidia |
Find the number of natural numbers $n$ such that $n^3+2n^2-3n+4$ and $3n^2+6n-6$ are relatively prime. | nvidia |
A cube has six faces, and on each face, a different positive integer is written. The sum of the numbers on any two opposite faces is always the same. The numbers on three adjacent faces form an arithmetic sequence. Find the sum of the numbers on all six faces. | nvidia |
Find the product of all constants $t$ such that the quadratic $x^2 + tx - 10$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. | nvidia |
Find the maximum value of the expression
$\sqrt{xy} + \sqrt{x} + \sqrt{y}$
where x and y are positive numbers such that
$x + y = 100$. | nvidia |
Find the sum of the digits of all the divisors of 72. | nvidia |
Given that $z$ is a complex number such that $\frac{z^2+z+1}{z^2-z+1}=3$, find $\frac{z^{2000}+z^{1999}+z^{1998}}{z^{2000}-z^{1999}+z^{1998}}$. | nvidia |
Let $P(n)$ be the product of the first $n$ fractions, where each fraction is of the form $\frac{k^3-1}{k^3+1}$, and $k$ ranges from $2$ to $n+1$. Find the value of $n$ such that $P(n) < \frac{1}{2}$. | nvidia |
A point is chosen at random on the number line between 0 and 2, and the point is colored blue. Then, another point is chosen at random on the number line between 0 and 2, and this point is colored red. What is the probability that the number of the red point is greater than half the number of the blue point, but less t... | nvidia |
Let $x$ be the smallest positive integer that is a multiple of both $5$ and $7$. Find the value of $x$ plus the smallest positive two-digit number that is a perfect square. | nvidia |
Find the sum of all positive integers less than 10000 that can be expressed as a sum of distinct powers of 2, and determine the remainder when this sum is divided by 1000. | nvidia |
Find all the possible values of $a$ for which there exist two different natural numbers $m$ and $n$, $m,n>2$, such that
$\dbinom{2m}{m}+\dbinom{2n}{n}=\dbinom{m+n}{n}$ | nvidia |
The geometric mean of five numbers is 24. Three of the numbers are 16, 32, and 48, and the other two are equal. What is the sum of the reciprocals of the two equal numbers? | nvidia |
Find the integer $n$, $1 \le n \le 6$, such that \[n \equiv 4327 \pmod{3}.\] | nvidia |
Two positive real numbers have arithmetic mean 6 and median 5. Find the two numbers, separated by commas. | nvidia |
In a regular polygon, the number of diagonals is equal to four times the number of sides. Determine the number of sides of the polygon. | nvidia |
The midpoints of the sides of a triangle form the vertices of a smaller triangle. Given that the area of the smaller triangle is 40, what is the area of the larger triangle? | nvidia |
What is the least positive integer that can be expressed as a sum of exactly $10$ consecutive positive integers? | nvidia |
Find the number of distinct positive integers that will leave a remainder of 3 when divided into 63. | nvidia |
Find the number of ways to arrange the letters in the word PENTAGON, such that no two vowels (E and A) appear together. | nvidia |
The positive integer divisors of 300, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 10? | nvidia |
Let $a$ and $b$ be positive real numbers such that $a^2 + b^2 = 1$. Find the minimum value of $a^2b^2 + a^2 + b^2 + 1$. | nvidia |
If $A$, $B$, and $C$ are three complex numbers such that $A-B+C=2-i$, $B-C+A=3+2i$, and $C-A+B=-1-i$, find $A+B+C$. | nvidia |
Let $T$ be the sum of all positive integers $m$ such that $m^2+16m-2087$ is a perfect square. Find the remainder when $T$ is divided by $1000.$ | 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 |
Find the smallest positive integer $y$ which is greater than $2$ and relatively prime to $315$ (recall that relatively prime means that the GCD of $y$ and $315$ is $1$) | nvidia |
Let $n$ be the number of odd integers, and let $x$ be the middle integer.
Then the sequence of integers is
$x-(n-1),x-(n-3),\dots,x-(2),x,x+2,\dots,x+(n-3),x+(n-1)$
and the arithmetic mean is $y=\frac{nx}{n}=x$.
The smallest integer in the sequence is $x-(n-1)$ and the largest integer is $x+(n-1)$, and their sum is... | nvidia |
Let $a$ and $b$ be the roots of the equation $2x^2-7x+1=0$. Find the number of rational numbers $p/q$ with $q\leq 1000$ such that $a<p/q<b$. | nvidia |
Let $m$ be the smallest integer whose fifth root is of the form $n + r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$. | nvidia |
The repeating decimal $0.\overline{125}$ can be represented as a fraction $\frac{a}{b}$ where $a$ and $b$ are positive integers. Find the smallest possible value of $b$ if it is known that $a$ is divisible by 5. | nvidia |
Count the number of ways to arrange the digits of the number 12345 such that the sum of the digits in the even positions is equal to the sum of the digits in the odd positions. | nvidia |
Find the positive real value of $t$ that satisfies $|t+2i\sqrt{3}| |6-4i| = 26$. | nvidia |
In a triangular array of nine distinct positive integers, each row and each column contains exactly three integers. Find the smallest possible value of the product of the three integers in the diagonal of the array. | nvidia |
Consider the congruence $7x \equiv 3 \pmod{25}$. Show that for every positive integer $k$, there exists a unique solution modulo $25^k$. Find all solutions modulo $125$. | nvidia |
A cryptographer is trying to decipher a coded message that consists of a series of integers. Each integer represents the number of letters in a word, and the sum of all these integers is congruent to 250 modulo 307. However, it is also known that the number of words in the message is 17, and the length of each word is ... | nvidia |
Determine the number of positive integers less than or equal to 1000 that are not divisible by any of the prime numbers 2, 3, or 5. | nvidia |
A sphere with center $(0,0,k)$ is tangent to the plane $z=0$ and the cone $z^2=x^2+y^2$. Given that the point $(6,0,0)$ lies on the sphere, determine the radius of the sphere. | nvidia |
On the edges of the cube are drawn two circles with radius $\frac{1}{2}$ of the side length of the cube. Then on each of three mutually adjacent faces of the cube a rhombus is drawn, so that its vertices are at the centers of the circles. Describe the intersection curve of the rhombi. What area do they enclose if the e... | nvidia |
Find the smallest positive integer that has exactly four distinct positive factors and can be expressed as the sum of two perfect squares. | nvidia |
Let $a$ and $b$ be two positive numbers such that $a^2 + b^2 = 1$. Find $\left(\frac{a}{b}\right)^3$. | nvidia |
Find all real values of $y$ that satisfy $\frac{y(y-2)}{(y+3)^2} \le 8.$ (Give your answer in interval notation.) | nvidia |
Let $a$ and $b$ be the roots of the quadratic equation
$$x^2-5x+c=0$$
so that $a+b=5$ and $ab=c$.
Given that
$$\frac{1}{a^2}+\frac{1}{b^2}=15$$
compute $c$. | nvidia |
Let p ( n ) be a polynomial of degree 4 such that p (1) = 10 , p (2) = 27 , p (3) = 52 , and p (4) = 85 . Find p (5) . | nvidia |
A circle with center at the origin intersects the line $y=3x$ at points A and B, such that $OA=15$ and $OB=20$. Find the length of the segment AB. | nvidia |
How many four-digit numbers greater than 2000 have at least two digits that are the same? | nvidia |
Suppose that $a$, $b$, and $c$ are positive real numbers such that $a^3 - 2b^2c = 5$ and $a^2 + bc = 3$. Then find the maximum possible value of $a^3 + 2b^2c$. | nvidia |
Let $A$ be a nonempty subset of $\{0,1,2,\ldots,1997\}$. Let $m$ be the number of sets $\{x,y,z\}$ of three distinct elements of $A$ which satisfy $x+y+z\equiv 3 \pmod{3}$ and let $n$ denote the number of sets $\{x,y,z,w\}$ of four distinct elements of $A$ which satisfy $x+y+z+w\equiv 4 \pmod{4}$. Find all positive int... | nvidia |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | nvidia |
Let $x$ and $y$ be positive real numbers such that
\[\frac{1}{x^2 + 2} + \frac{1}{y^2 + 2} = \frac{1}{3}.\]Find the minimum value of $x + y.$ | nvidia |
What is the sum of the squares of the real and imaginary parts of the complex number $(2+3i)(4-5i)$? | nvidia |
On square $ABCD$, point $E$ lies on side $AD$ and point $F$ lies on side $BC$, so that $BE=EF=FD=30$. Find the area of the square $ABCD$.
| nvidia |
In a sequence of consecutive multiples of 4, the sum of the first 5 terms is 200. Determine the number of terms in the sequence that are less than 60. | nvidia |
Find the product of all integers $n$ such that $\dfrac{n^3+1}{n+1}$ is a perfect cube. | nvidia |
Find the value of $n$ such that $n! = n \cdot (n-1)! + (n-2)!$. | nvidia |
Let $S$ be the set of all positive integers that are multiples of 5 and are less than 20. Let $T$ be the set of all positive integers that are multiples of 4 and are less than 20. What is the largest possible number of elements in a subset of $S \cup T$ such that no two elements in the subset add up to 20? | nvidia |
Let $S$ be the set of all positive integer divisors of $1000$. For how many elements of $S$ is it true that the decimal representation of the reciprocal of the number terminates? | nvidia |
Determine the length of the side of the square inscribed in the circle $(x-3)^2+(y-4)^2=25$. | nvidia |
The sum of the digits of a three-digit number is equal to 12, and the product of its digits is equal to 24. If the hundreds digit is 2, find the largest possible value of the number. | nvidia |
In a 4-digit number $\overline{abcd}$, where each letter represents a digit, the sum of the squares of the first and last digits equals the sum of the cubes of the two middle digits. If the number formed by the first two digits is divisible by 5, and the number formed by the last two digits is divisible by 3, determine... | nvidia |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.