instruction_seed stringlengths 30 7.99k | source stringclasses 1
value |
|---|---|
Find the number of positive integers less than or equal to 100 that are divisible by 3 or 5. | nvidia |
Determine the remainder when the sum of the cubes of the first $1000$ positive integers is divided by $1001$. | nvidia |
What is the sum of all real numbers that are tripled when multiplied by their reciprocals? | nvidia |
Determine the number of positive integers $n$ less than or equal to $1000$ such that $\frac{n}{3500}$ is representable as a sum of distinct reciprocals of powers of $2$. | nvidia |
Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.
| nvidia |
The vertices of a regular octagon are labeled with the numbers 1 through 8. How many ways can the vertices be labeled such that the sum of the numbers on adjacent vertices is even? | nvidia |
Let $a$ and $b$ be two positive real numbers such that $a + b = 27.$ Enter the ordered pair $(a,b)$ for which $a^3 b^4$ is minimized. | nvidia |
Find $u^2+v^2$ if $u$ and $v$ are positive integers such that \[\begin{aligned} uv+u+v&=95 \\ u^2v+uv^2 &= 1323.\end{aligned}\] | nvidia |
Let $f(n)$ denote the number of distinct prime factors of a positive integer $n$, where $f(1)=0$. For example, $f(2)=1$ and $f(6)=2$. Define $T(n)$ by $T(n)=f(1)+ f(2) + \cdots + f(n).$ Determine the smallest positive integer $n$ such that $T(n)$ is a perfect square. | nvidia |
What is the greatest integer $n$ such that the sum of the cubes of the roots of the equation $x^2 - nx + 24 = 0$ is less than or equal to $0$? | nvidia |
Determine the upper bound of the sum of the cubes of the roots of the polynomial $P(x)=x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_1x+a_0$, given that the coefficients of the polynomial satisfy the condition $a_{n-1}=-a_{n-2}$ and $a_0=1$. | nvidia |
Find the area of the shaded region in the following diagram:
[asy]draw((0,0)--(10,0)--(10,10)--(0,10)--cycle);
draw((3,7)--(7,7)--(7,3)--(3,3)--cycle);
label("10",(5,0),S);
label("10",(10,5),E);
label("4",(5,3),S);
label("4",(3,5),W);
[/asy]
The area of the shaded region can be found by subtracting the area of the sm... | nvidia |
How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides $8,$ $15,$ and $17$? | nvidia |
Find the number of non-empty subsets of the set $\{1, 2, 3, \ldots, 12\}$ such that the sum of the elements in each subset does not exceed $20$. | nvidia |
A binary operation $\circ$ is defined on the set of all positive integers as follows: for all positive integers $m$ and $n$, $$m\circ n=m+n+\gcd(m,n).$$For example, $5\circ 7=5+7+\gcd(5,7)=13$, while $6\circ 12=6+12+\gcd(6,12)=20$. What is the value of $(7\circ 11)\circ(15\circ 19)$? | 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 |
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 |
The difference of squares of two consecutive natural numbers is 165. Find the sum of these numbers. | nvidia |
Find the smallest positive integer k such that the expression 3^k + 2 is divisible by a prime factor greater than 7. | nvidia |
Problem:
In a triangular region bounded by the coordinate axes and the line $2x + 3y = 6$, find the point $P$ that minimizes the sum of its distances to the vertices of the triangle formed by the intersection points of the line with the axes and the origin. Express the coordinates of point $P$ in terms of integers $m$ ... | nvidia |
What is the sum of the integers that are both greater than 3 and less than 12?
Find the sum of all integers between 1 and 100 that are not divisible by 3 or 5. | nvidia |
Let $s(n)$ be the sum of the digits of $n$ when written in base 7, and let $t(n)$ be the sum of the digits of $2s(n)$ when written in base 3. Find the smallest positive integer $n$ such that the decimal representation of $t(n)$ is a perfect square. | nvidia |
Find the number of positive integers $n$ less than 500 such that $f(n) = \sum_{k=1}^{n} \lceil k/3 \rceil$ is a multiple of $n$, where $\lceil x \rceil$ denotes the ceiling function. | nvidia |
What is the smallest positive integer n such that the sum of the elements in the nth row of Pascal's triangle is a power of 2? | nvidia |
Find the sum of the squares of the integers from 1 to 50. | nvidia |
Determine the number of positive integers less than 300 that have no common factors with 48. | nvidia |
Determine the largest value of the real number $a$ for which there exists a solution to the equation $|x-a|+|x-3|+|x+1|=8$. | nvidia |
A rectangle is inscribed in a circle of radius 5 inches, and one of its vertices coincides with the center of the circle. The angle formed by the two diagonals of the rectangle is 60 degrees. Find the area of the rectangle. | nvidia |
Let $a$ be a nonzero integer and $b$ be a positive number such that $a-b^3=\sqrt{\log_{10} b}$. What is the mean of the set $\{a, b, 1/b, 1/b^2\}$? | nvidia |
Let the roots of the equation
$$3x^2-8x+9=0$$
be $x_1$ and $x_2$, while the roots of the equation
$$2y^2-16y+5=0$$
are $y_1$ and $y_2$. The vertices of the square are the following points
$A(x_1,y_1)$,
$B(x_2,y_1)$,
$C(x_2,y_2)$,
$D(x_1,y_2)$.
Determine the perimeter of the square $ABCD$. | nvidia |
If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? | nvidia |
Given two functions, $f(x)$ and $g(x),$ defined by the graphs 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... | nvidia |
In the Cartesian plane, plot the lines $2x+5y=7$, $3x-y=4$, and $x+3y=8$. Determine the number of points where at least two of these lines intersect. | nvidia |
In a convex n-sided polygon, the degree measures of the angles form a geometric sequence with common ratio r. Given that the sum of the interior angles is equal to (n-2)*180, find the maximum value of r that is less than 1. | nvidia |
The equation $x^3 + 5x^2 + 9x - 8 = 0$ has three real roots. Find the sum of the reciprocals of the roots, given that the sum of the squares of the roots is 27. | nvidia |
Find the value of $a$ such that the equation $\log_a (2x+1) + \log_a (x-1) = 2$ has a solution. | nvidia |
Consider the set of all tetrahedra $OABC$ where $O$ is the origin and $A$, $B$, and $C$ are distinct points in 3D space with nonnegative integer coordinates $(x,y,z)$ such that $x^2 + y^2 + z^2 = 2025$. Find the number of such distinct tetrahedra whose volume is a positive integer. | nvidia |
Find a pair of positive integers whose sum is 25 and the product is divisible by 25. What are these two numbers? | nvidia |
Let $P$ be the product of the first $100$ positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k .$
| nvidia |
Let $a_n$ be the sequence defined by $a_1 = 1$ and $a_{n+1} = a_n + (n+1)^3$ for $n \geq 1$. Find the smallest positive integer $n$ such that $a_n$ is divisible by $100$. | nvidia |
The equation of the parabola that passes through the three points $(1, 3), (2, 3),$ and $(4, a)$ is given by
$y=b(x-2)^2+3$. Determine the sum of all the possible values of $a$. | nvidia |
Find the number of positive integers $n$, $1 \le n \le 1000$, for which the polynomial $x^2 + 2x - n$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
Let $f$ be a function that satisfies the functional equation $f(f(x))=x$ for all $x$ in its domain. What is the smallest possible value of the domain of $f$ if it is known that the domain of $f$ contains at least two elements? | nvidia |
If $n$ is a positive integer and $z$ is a complex number such that $\left|z^n\right|=2004$ and $\left|z^{n+1}\right|=2005$, what is the smallest possible value of $n$? | nvidia |
In the diagram, $\angle 1$ and $\angle 2$ are supplementary, and $\angle 3$ and $\angle 4$ are equal. The measure of $\angle 3$ is $x$ degrees. If $\angle 1$ is twice the measure of $\angle 4$, find the measure of $\angle 2$ in terms of $x$. | nvidia |
Suppose that we have a right triangle $ABC$ with the right angle at $B$ such that $AC = \sqrt{61}$ and $AB = 5.$ A circle is drawn with its center on $AB$ such that the circle is tangent to $AC$ and $BC.$ If $P$ is the point where the circle and side $AC$ meet, then what is $CP$? | nvidia |
A cone has a height of 5 units and a base radius of 4 units. A sphere is cut from the cone such that its volume is equal to that of the remaining cone. Find the radius of the sphere. | nvidia |
Find the value of $n$ such that the sum of the ceiling and floor functions of $x$, where $x$ is the positive root of the equation $x^2 + nx + 2 = 0$, is equal to 5. | nvidia |
If $x+yi$ is a complex number such that
$\ds\frac{1}{x+yi}+\ds\frac{1}{x-yi}=1$,
determine the value of $x^2+y^2$. | nvidia |
Find the sum of all natural numbers $n$ such that $n!$ is a product of two distinct prime numbers. | nvidia |
Let $f(x)$ be a polynomial of degree $n$ with positive integer coefficients. Suppose that $f(x)$ is equal to the sum of its first $m$ terms for some positive integer $m$ when $x = 1, 2, \dots, n$. Find the largest possible value of $m$ for which this is true. | nvidia |
How many three digit numbers are left if we exclude all three digit numbers in which there are exactly two digits the same, but these two digits are not adjacent? | nvidia |
Find the number of real solutions of the equation
\[\frac{x^2}{100} = \cos x.\] | nvidia |
In a right triangle with integer length sides, the length of the shorter leg is given as 15 units. What is the maximum possible length of the hypotenuse such that the triangle still has integer length sides? | nvidia |
What is the largest number of consecutive positive integers that can be multiplied together before the product exceeds 1000? | nvidia |
Given that $\frac{a}{b} \cdot \frac{c}{d} \cdot \frac{e}{f} = \frac{10!}{9!}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are positive integers, find the minimum value of $a+b+c+d+e+f$. | nvidia |
Find all real values of $a$ such that the equation $x^4 + ax^2 - x - 2 = 0$ has exactly one real root. | nvidia |
The sum of the digits of a two-digit number is 8, and the product of the digits is 20. Determine the number. | nvidia |
A circle is inscribed in a square. The length of a side of the square is $7$ meters. Given that the circle contains two distinct points $P$ and $Q$ such that their distance from a point on the circle to the center of the circle differs by $2\sqrt{2}$ meters. Calculate the area of the triangle whose vertices are $P$, $Q... | nvidia |
What is the sum of the squares of the real and imaginary parts of the complex number $(2+3i)(4-5i)$? | nvidia |
A root of unity is a complex number that is a solution to $z^n = 1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^2 + az + b = 0$ for some integers $a$ and $b$. | nvidia |
Find the value of $x$ such that $\sqrt[3]{x+\sqrt{x+1}}=2$. | nvidia |
Find all real solutions to $x^3+(x+1)^3+(x+2)^3=(x+3)^3$. Enter all the solutions, separated by commas.
Here is another problem inspired by this one:
Determine all real values of $a$, $b$, and $c$ that satisfy the equation
$$a^3+b^3+c^3=(a+b+c)^3.$$
Note that this problem requires a different approach, as the sum ... | nvidia |
In a circle with center $O$, two chords $\overline{AB}$ and $\overline{CD}$ intersect at a point $E$ such that $AE=CE$ and $BE=DE$. If the radius of the circle is $r$, and the lengths of $AB$ and $CD$ are $a$ and $b$ respectively, find the maximum possible value of $a+b$ in terms of $r$. | nvidia |
Let there exist 2007 identical containers arranged in a circle, such that the value of the $n^{th}$ container is $(2\sin{\frac{n\pi}{2007}})^p$, where $p$ is an integer. Define $S_p$ as the sum of the values of all containers. Determine the smallest integer value of $p$ greater than 2007 for which $S_p$ can be written ... | nvidia |
The number 3600 can be expressed as a sum of consecutive positive integers. What is the minimum number of consecutive positive integers that sum to 3600, if the sum starts with the number 12? | nvidia |
Given that the sum of the squares of the roots of a quadratic equation is 52 and the sum of the roots is -8, find the product of the roots. | nvidia |
Consider the sequence of real numbers defined by $x_1 = 1$ and $x_{n+1} = \sqrt{x_n} + \frac{1}{n}$ for all $n \geq 1$. Find the limit of the sequence $x_n$ as $n$ approaches infinity. | nvidia |
A rectangle is to be tiled with square tiles of the same size. If the length and width of the rectangle are integers and the number of tiles used is maximized, what is the minimum possible number of tiles that will be left unused? | nvidia |
A standard eight-sided fair die is rolled three times. The probability that the sum of all three numbers rolled is a multiple of 5 is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | nvidia |
The square of an integer is 182 greater than the integer itself. What is the sum of all integers for which this is true? | nvidia |
Determine the largest possible number of distinct digits that can appear in the units place of the cubes of natural numbers. | nvidia |
Find the number of times the digit 7 appears in the decimal representation of the product of all positive integers less than or equal to 10 that are not divisible by 3. | nvidia |
A five-digit number is formed by writing the number 54321 and then removing one of its digits. Find the sum of all possible values of the five-digit number that can be formed in this way. | nvidia |
Let $g(x)$ be a function such that $g(0) = 0$ and
\[g(x + y) = g(xy) + (x + y)^2\]for all real numbers $x$ and $y.$ Find $g(x).$ | nvidia |
Find the x-coordinate of the point on the x-axis that is equidistant from points C(3, 2) and D(0, 1). | nvidia |
Find the number of ordered triples $(a,b,c)$ of real numbers such that $a - b = 3$ and $ab + c^2 = 4.$ | nvidia |
Let $G(n)$ be the number of distinct values of $x$ on the interval $[0, 2\pi]$ that satisfy the equation $x^n = \cos x$. Find the largest integer $k$ such that $\sum_{n=2}^{2007} G(n) < 10^k$. | nvidia |
Determine all real values of $a$ for which the equation $|x-a| = |2x-a|$ has more than one distinct solution. | nvidia |
Compute the sum of all the coefficients of the polynomial obtained by expanding the product
$$(1+x)(1+2x)(1+3x)(1+4x).$$ | nvidia |
Find the value of $x$ such that $\sqrt[x]{8} = \log_{16} 2$. | nvidia |
Find the number of positive integers less than 100 that have exactly three positive divisors. | nvidia |
Given a regular hexagon with side length 10, find the radius of the largest possible sphere that can be inscribed inside the hexagon. | 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 |
Given that $\log_{10}2=0.3010$ and $\log_{10}3=0.4771$, compute the integer part of $\log_{10}(2^{51})+\log_{10}(3^{43})$. | nvidia |
Find the number of different prime factors in the product of all positive integers from 1 to 20 that are not divisible by 3. | nvidia |
What is the largest number of distinct regular tetrahedrons that can be inscribed within a regular dodecahedron, such that each vertex of the dodecahedron is a vertex of exactly one tetrahedron? | nvidia |
Find the greatest integer value of $x$ such that $\frac{x^2 - 3x + 7}{x+2}$ is an integer. | nvidia |
A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?
Here's another problem:
A right triangle with integer leg leng... | nvidia |
Given two points $(x_1,y_1)$ and $(x_2,y_2)$, find the equation of the line that passes through these points. Now, given a third point $(x_3,y_3)$, determine the condition under which this point lies on the line.
Given points $(1, 1)$ and $(2, -1)$, find the equation of the line and determine $k$ such that the point ... | nvidia |
Determine the number of prime numbers that can be expressed as the sum of two perfect cubes between 100 and 200. | nvidia |
Find the smallest positive integer that is both a perfect square and has exactly three distinct digits. | nvidia |
Three standard six-sided dice are rolled simultaneously.
Determine the probability that the sum of the three numbers rolled is at least 4 and at most 11. | nvidia |
The number 240 can be expressed as the product of two positive integers in several ways. How many of these ways involve at least one perfect square factor? | nvidia |
Consider the function $g(x) = |3\{x\} - 2|$, where $\{x\}$ denotes the fractional part of $x$. Find the smallest positive integer $m$ such that the equation $mg(xg(x)) = x$ has at least $2025$ real solutions. | nvidia |
Find the sum of all real values of $s$ for which the equation $\frac{1}{\sin x}=\frac{s-\cos x}{4}$ has exactly two real solutions in the interval $[0,2\pi)$. | nvidia |
Find the largest positive integer $n$ such that $n!$ does not contain the digit 7. | nvidia |
The digits of a three-digit number $ABC$ are rearranged to form two other three-digit numbers, and the greatest of the three numbers is subtracted from the sum of the other two. What prime number must be a factor of the difference if $A$, $B$, and $C$ are all distinct? | nvidia |
Let $\mathfrak{f}(n)$ be the sum of the squares of the positive integer divisors of $n$. Prove that there are no values of $n$, where $1 \le n \le 100$, for which $\mathfrak{f}(n)$ is prime. | nvidia |
Given a quadratic equation $ax^2+bx+c=0$, the sum of the squares of its solutions can be found using the formula:
$$x_1^2+x_2^2=(x_1+x_2)^2-2x_1x_2=\left(\frac{-b}{a}\right)^2-2\frac{c}{a}.$$
Using this formula, the sum of the squares of the solutions of the given equation $x^2-13x+4=0$ is:
$$x_1^2+x_2^2=(13)^2-2(4)... | nvidia |
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