instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find the number of integers that are both greater than 10 and less than 50, and are also multiples of 3 or 5. | nvidia |
Let $p$ and $q$ be real numbers such that $\frac{p}{x+3}+q(x-4)=\frac{2x^2-15x-12}{x+3}$. What is $p-q$? | nvidia |
Given that f(x)=log_(x+1)(2x), find all real numbers x such that f(x)>f(1) | nvidia |
Determine the value of $b$ such that the equation $(x^2+bx-1)^2+(x^2+bx-1)-12=0$ has exactly two distinct real solutions. | nvidia |
Find the sum of all possible values of $x^2+y^2+z^2$ given that $x$, $y$, and $z$ are real numbers such that
\[x + y + z = xy + yz + zx = xyz = 0.\] | nvidia |
Let $x$ be a real number. Consider the following five statements:
$0 < x^2 < 1$
$x^2 > 1$
$-1 < x < 0$
$0 < x < 1$
$0 < x - x^2 < 1$
What is the maximum number of these statements that can be true for any value of $x$? | nvidia |
What is the least positive integer with exactly five distinct positive factors? | nvidia |
Find the 200th term of the increasing sequence of positive integers formed by omitting all perfect cubes. | nvidia |
Given that $\mathbf{A}$ is a square matrix and $\mathbf{B}$ and $\mathbf{C}$ are two square matrices of the same order that satisfy the equation $\mathbf{A}\mathbf{B} = \mathbf{C}\mathbf{A},$ find the relationship between the traces of $\mathbf{A},$ $\mathbf{B},$ and $\mathbf{C}.$ | nvidia |
A regular hexagon is inscribed in a circle centered at the origin. The equation of the circle is x^2+y^2=100, and one vertex of the hexagon is (10,0). Find the length of the diagonal of the hexagon passing through the center of the circle. | nvidia |
Find the sum of $a$ and $b$ in the expression $\frac{\sqrt{a}}{b}$, which represents the positive difference between the roots of the quadratic equation $2x^2 + 5x - 6 = 0$, where $a$ is an integer not divisible by the square of any prime number and $b$ is an integer. | nvidia |
Find all perfect cubes between 100 and 200 that can be expressed as the sum of two perfect squares. | nvidia |
Let $x$ and $y$ be two real numbers such that $x+y\neq 0$. Find the set of all possible values of the expression
$\ds\frac{|x|+|y|}{|x+y|}$.
Is the number 2/3 in this set? | nvidia |
In a rectangle $ABCD$, two circles are drawn inside the angle $\angle BAD$ such that they are tangent to rays $\overline{AB}$, $\overline{AD}$, and segment $\overline{BC}$. The centers of these two circles are $O_1$ and $O_2$, and it is known that $O_1O_2=6$. If the area of the rectangle is $48$, find the perimeter of ... | nvidia |
A circle is inscribed inside a square. The radius of the circle is 5 units. A triangle is formed by connecting the center of the circle to two adjacent vertices of the square. Find the area of this triangle. | nvidia |
In a triangle, the lengths of the three sides are consecutive integers, and the largest angle is three times the smallest angle. Find the length of the side opposite the largest angle. | nvidia |
The pattern of Pascal's triangle is illustrated in the diagram shown. What is the fourth element in Row 15 of Pascal's triangle? $$
\begin{array}{ccccccccccccc}\vspace{0.1in}
\textrm{Row 0}: & \qquad & & & & & 1 & & & & & & \\ \vspace{0.1in}
\textrm{Row 1}: & \qquad & & & & 1 & & 1 & & & & &\\ \vspace{0.1in}
\textrm{R... | nvidia |
What is the largest composite number that appears before a sequence of seven consecutive odd positive integers, all of which are prime? | nvidia |
Find a value of $a$ such that the equation
$$ax^2+2\sqrt{2}x+1=0$$
has exactly one root. | nvidia |
Find the value of the product $\prod_{k=1}^{100} \sin(5k+3)$, where all angles are measured in radians. | nvidia |
Find the greatest $b$ such that $\frac{5\sqrt{(3b)^2+(2)^2}-2b^2-4}{\sqrt{4+9b^2}+2}=1$. | nvidia |
A certain positive integer $n$ can be expressed as the sum of three distinct positive integers $a$, $b$, and $c$, such that $n = a + b + c$, where $a$, $b$, and $c$ are all prime numbers. If exactly one of the statements about the primality of $a$, $b$, and $c$ is incorrect, and that incorrect statement was made about ... | nvidia |
Find the smallest value of the positive integer $n$ for which there exists a vector $\mathbf{v}$ satisfying $\|\mathbf{v}\|=n$ and $\mathbf{v}\cdot \mathbf{w}=7$, where $\mathbf{w}$ is a unit vector perpendicular to $\mathbf{v}$. | nvidia |
Let $g(x)$ be a function defined for all positive integers satisfying the conditions $g(x) > 1$ for all $x \geq 1$ and
\[g(\text{lcm}(x,y)) = g(x) + g(y)\]for all positive integers $x$ and $y$ with $\gcd(x,y)=1$. Determine $g(2010)$. | nvidia |
Find the number of positive integers that can be expressed as the sum of two perfect squares. | nvidia |
What is the largest three-digit positive integer that is a perfect square and has exactly three distinct prime factors? | nvidia |
A triangle is formed with one vertex at the vertex of the parabola $y=x^2-1$ and the other two vertices at the intersections of the line $y=r$ and the parabola. If the area of the triangle is between $8$ and $64$ inclusive, find all possible values of $r$. Express your answer in interval notation. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a + b + c = 1$. Find the minimum possible value of $\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}$. | nvidia |
Let $a$, $b$, $c$, and $d$ be positive integers such that $ab+c=bc+a$, $bc+d=cd+b$, $cd+a=da+c$, and $da+b=ab+d$. Determine $a+b+c+d$. | nvidia |
Let $x$ and $y$ be real numbers greater than 1 such that the circles of radii $x$ and $y$ are externally tangent and are circumscribed by a third circle, as shown in the figure. Determine the value of the expression $\ds\frac{(x+y)^2}{x^2+y^2-xy}$. | nvidia |
Let $a$, $b$ be positive integers and $x$ be a real number, such that
$a\cdot \lceil x\rceil+b\cdot \lfloor x\rfloor =20.$
Find the largest possible value of the expression $a\cdot b$. | nvidia |
Find the maximum possible value of $k$ such that the quadratic function $f(x) = x^2 + 4x + k$ has no real roots. | nvidia |
Determine the number of integer values of $n$ such that $\sqrt{n} + \sqrt{n+1}$ is an integer. | nvidia |
Find all values of $x$ that satisfy the equation $|x-3|=2x+4$. Express your answers in simplest fractional form. | nvidia |
Let $g(x)$ be a function that satisfies the following equation for all real numbers $x$ and $y$:
$$g(x + y) = 3^xg(y) + 5^y g(x).$$
Given that $g(2) = 1$ and $g(3) = 5$, find the value of $g(5)$. | nvidia |
The lengths of the sides of a triangle with positive area are $\log_{10}12$, $\log_{10}75$, and $\log_{10}n$, where $n$ is a positive integer. Find the number of possible values for $n$. | nvidia |
When two fair 8-sided dice are tossed, the numbers $m$ and $n$ are obtained. What is the probability that the two digit number $mn$ (where $m$ and $n$ are digits) and $m$ and $n$ are all divisible by 2? | nvidia |
Given that $n^3+100$ is divisible by $n+10$, we want to find the largest positive integer $n$.
Here is another problem:
Find the largest positive integer $m$ such that $2^{2m}+1$ is divisible by $2^m+1$.
This problem requires a different approach, focusing on properties of powers of 2 and divisibility, rather than c... | nvidia |
What is the product of all positive integer divisors of a number that can be expressed as the sum of two perfect squares in exactly two different ways? | nvidia |
The graph of a function $f(x)$ is given below.
[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 wholetickdown=tickdown;
voi... | nvidia |
Prove that the number $0.\overline{123}=0.123123\ldots$ cannot be represented as a finite decimal. | nvidia |
The equation $x^2+px+q=0$ has two distinct real roots. If $x_1$, $x_2$, $x_3$, and $x_4$ are four distinct real numbers such that $x_1$ and $x_2$ are the roots of the equation, find the value of the following expression:
$$\frac{(x_3-x_1)(x_4-x_2)}{(x_3-x_2)(x_4-x_1)}$$ | nvidia |
Find the largest positive integer $m$ such that for any positive real numbers $c$ and $d$, the equation
\[(c + di)^m = (c - di)^m\]
implies that $c = d$. | nvidia |
Let $a$, $b$, and $c$ be real numbers. Find the number of real solutions $x$ to the equation
$$|x-a|+|x-b|=|x-c|.$$ | nvidia |
Find the number of different possible positions of the circumcenter of triangle PQR, where P, Q, and R are chosen to be any three of the 40 equally spaced points around the perimeter of a square that are not collinear. | nvidia |
The sum of the reciprocals of two consecutive positive integers is $\frac{17}{72}$. What is the smaller integer? | nvidia |
Find the number of four-digit positive integers such that the product of the digits is equal to 12. | nvidia |
Consider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| nvidia |
Let A represent the set of positive integers less than 500 divisible by 17, B represent the set of positive integers less than 500 divisible by 19, and C represent the set of positive integers less than 500 divisible by both 17 and 19.
Find the number of elements in the set \( A \cup B \), i.e., the set containing the... | nvidia |
Let $w$ be a complex number such that $w^3=1$ and $w\ne 1.$ Find the real part of $\frac{1}{1-w}.$ | nvidia |
Let $a$ and $b$ be positive integers. A number $n$ is said to be $ab$-balanced if $n$ can be expressed as the sum of two distinct divisors of $ab$, each of them being greater than $1$. Find the number of distinct positive integers that are $12$-balanced. | nvidia |
Let $a$ and $b$ be real numbers such that
\[a^3 - 15a^2 + 20a - 50 = 0 \quad \text{and} \quad 8b^3 - 60b^2 - 290b + 2575 = 0.\]Compute $a + b.$
A function $f$ is defined on the set of real numbers and satisfies
$f(xy) + x = xf(y) + f(x)$
for all real numbers $x$ and $y.$
Assume that $f$ is not identically zero and tha... | nvidia |
A list of five positive integers has all of the following properties:
$\bullet$ The only integer in the list that occurs more than once is $8,$
$\bullet$ its median is $9,$ and
$\bullet$ its average (mean) is $10.$
What is the largest possible integer that could appear in the list? | nvidia |
Let $S$ be the set of all positive integers that can be formed by rearranging the digits of $123456789$. If $y\%$ of the numbers in $S$ have at least one digit that appears only once, then what is $y$? Express your answer as a decimal to the nearest tenth. | nvidia |
The positive integer divisors of 175, 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 7? | nvidia |
Let $x$ and $y$ be positive integers such that $xy - 9x + 4y = 2007$. Find the minimal value of $|x - y|$ such that $\frac{x}{y}$ is a rational number. | nvidia |
Find the number of distinct points at which the graph of y=(x-5)/(x^2+5x+6) is undefined. | nvidia |
The function
$$t(z)=\sqrt[3]{\frac{1}{(z-4)^2}}$$
is defined on a subset of the real numbers. Determine the set of all $z$ such that $t(z)$ is defined, and express the answer in interval notation. | nvidia |
Find the sum of the integer divisors of the number 48, excluding 1 and 48 themselves. | nvidia |
In the complex plane, two complex numbers $a+bi$ and $c+di$ are reflected across the line $y=x$. The reflected complex numbers are $b+ai$ and $d+ci$. Given that $a+b=3$, $c+d=5$, and $a+c=7$, find the value of $b+d$. | nvidia |
Find the smallest positive integer $n$ such that $2^n$ is a perfect square, $3^n$ is a perfect cube, and $5^n$ is a perfect fifth power. | nvidia |
The product of two positive integers plus their sum is 95. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | nvidia |
Given that the polynomial $x^2 - kx + 16$ has only positive integer roots, find the average of all distinct possibilities for $k$.
New problem:
A rectangle with integer side lengths has a perimeter of $2p$ units and an area of $q$ square units. If $p$ and $q$ are positive integers and the sum of the distinct side len... | nvidia |
A convex polygon has 12 sides, and the length of each side is a positive integer. The sum of the lengths of the sides is 100. How many different combinations of side lengths are possible? | nvidia |
Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $xyz=1$ and $x^2+y^2+z^2=50$. Find the smallest possible value of $xy+yz+zx$. | nvidia |
Let p be a prime number with exactly 5 positive factors. Find the number of positive factors of p^4 that are multiples of 5. | nvidia |
A square pyramid has a base edge of 10 inches and slant height of 13 inches.
Find the length of the segment joining one of the vertices of the base with the midpoint of the opposite edge of the base that passes through the vertex of the pyramid. | nvidia |
When three numbers are added two at a time the sums are 29, 46, and 53. What is the sum of all three numbers? | nvidia |
If $a,b,c$ are nonnegative real numbers such that $a^2+b^2+c^2=18$ and $a+b+c=6$, find the maximum value of $ab+bc+ca$. | nvidia |
Given a positive integer n, find all possible values of x and y that satisfy the equation:
\[n(x+y) = (x)(y)\]
Note: The goal is to find all possible combinations of x and y for a given value of n. | nvidia |
Let $S$ be the sum of the areas of all triangles whose vertices are the points of intersection of the diagonals of a regular octagon with side length 1. If $S=\frac{a}{\tan b}+\frac{c}{\tan d}$, where $a,b,c,d$ are positive integers, what is the greatest integer that does not exceed $\frac{100a}{b}+\frac{100c}{d}$? | nvidia |
Find the area of triangle $PQR$ given that the radii of the circles inscribed in triangles $PQR$ and $PQS$ are $6$ and $9$, respectively. | nvidia |
What is the probability that the sum of the numbers shown on the two dice is 7, and at least one of the dice shows a prime number? | nvidia |
Let $T$ be the set of all positive real numbers. The function $g : T \to T$ satisfies the following two properties:
(i) First,
\[g(x + y) = g(x)g(y)\]for all $x, y \in T.$
(ii) Second,
\[g \left( \frac{x + y}{2} \right) = \sqrt{g(x)g(y)}\]for all $x, y \in T.$
Let $m$ be the number of possible values of $g(1),$ and... | nvidia |
Compute the product of the roots of the equation
\[2x^3+5x^2-6x^2-5x-5=0,\]
given that all of the roots are real and the constant term is negative. | nvidia |
Three positive integers are the only solutions of the equation
$$5x^3-12x^2+7x-2=0$$
What is the sum of the three integers? | nvidia |
Find the sum of all positive integral values of $m$ such that $m^3-2m^2-15m+36= n^2$ for some perfect square $n^2$. Note that there is at least one such $m$. | nvidia |
Find the sum of all real solutions to the equation \[\frac{x-2}{x^2+4x+1} = \frac{x-5}{x^2-10x}.\]
A new problem:
A right triangle has a hypotenuse of length $h$ and legs of length $a$ and $b$, where $a < b$. The altitude from the vertex of the right angle to the hypotenuse has a length of $4$ units. If the area of t... | nvidia |
Determine all positive real numbers $b$ for which the roots of the equation $x^3-bx^2+2bx+5b$ are real, and find the greatest value of $b$ that satisfies this condition. | nvidia |
Consider the triangular array of numbers with 0, 1, 2, 3, $\dots$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.
\[
\begin{array}{ccccccccccc}
& & & & & 0 & & & & & \\
& & & & 1 & & 1 & & & & \\
& & & 2 & & 2 & & 2 & & & \\
& & 3 & & 4... | nvidia |
Let $a$ and $b$ be real numbers such that the roots of the cubic equation $x^3+ax+b=0$ are all real and positive. Determine the minimum value of $a+b$.
Best of luck! | nvidia |
Find the sum of the smallest positive three-digit number that is a multiple of 5 and the smallest positive four-digit number that is a multiple of 7. | nvidia |
A certain natural number N can be written as the product of an odd number of distinct prime numbers. Find the smallest N such that the sum of its prime factors is a perfect square. | nvidia |
How many positive three-digit integers less than 500 have at least two digits that are the same? | nvidia |
Consider a function $g(x)$ such that
$$\frac{g(x+y)}{x+y}=\frac{g(x)}{x}+\frac{g(y)}{y}$$
for all real numbers $x$ and $y$ not equal to zero.
If $g(1) = 1$ and $g(-1) = 2$, compute $g(10)$. | nvidia |
A sequence of points $A_1$, $A_2$, $A_3$, $\ldots$ is drawn on a circle $C$, such that the lengths of the line segments $\overline{A_1A_2}$, $\overline{A_2A_3}$, $\ldots$ form a geometric sequence with first term $\frac{1}{2}$ and common ratio $\frac{1}{3}$. What is the smallest $n$ such that the sum of the lengths of ... | nvidia |
There exists a positive real number x such that $x=\sin(x)+\tan(x)$. Find the value of x. | nvidia |
What is the smallest number of boxes that can be filled with 24 identical balls, with each box containing a number of balls that is a multiple of 6, and then rearranged to fill a different number of boxes, with each box containing a number of balls that is a multiple of 9? | nvidia |
In a 2 by 2 grid, the diagonals are drawn. Let a point $P$ be on one of the diagonals such that the circles inscribed in the two triangles formed by $P$ and the adjacent sides have the same radius. If the sides of the grid have length 5, find the length of the diagonal that doesn't contain point $P$. | nvidia |
A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$.
| nvidia |
Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\cos x - \sin x = \sqrt{2}.$ Enter all the solutions, separated by commas. | nvidia |
The sum of two positive integers is 24 and their product is 180. How many different values could be the sum of the two integers? | nvidia |
How many four-digit positive integers have at least one digit that is a 2 or a 3? | nvidia |
Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{11}{12}.\] | nvidia |
Find the smallest possible six-digit number in the form $ABC,CBA$ that is divisible by 11, where $A$, $B$, and $C$ are distinct digits. | nvidia |
Find the sum of the odd, positive integers less than 89. | nvidia |
In a certain quadrilateral, the sides are related by the equation $ab+cd=2020$, where $a$, $b$, $c$, and $d$ are positive integers. Determine the number of quadrilaterals whose area is a perfect square. | nvidia |
Let $f(x) = \left\{\begin{array}{cl}
1 & \text{if } x > 0 \\
-1 & \text{if } x \le 0
\end{array}\right.$. Find all real $a$ such that the function $g(x) = \int_{-1}^{x} f(t) \;dt$ satisfies $g(x) + g^{-1}(x) = 0$ for all $x \in (-1, 1)$. | nvidia |
Find the sum of all real values of $x$ which do not satisfy the equation
$$\frac{(x^2+ax+b)(x^2+cx+d)}{(x^2+ex+f)(x^2+gx+h)}=3$$
where $a,b,c,d,e,f,g,h$ are constants such that the given equation is an identity. | nvidia |
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