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Find the largest possible value of $k$ such that $\frac{M}{k}$ is strictly between $\frac{2}{5}$ and $\frac{3}{8}$, where $M$ is a positive integer. | nvidia |
Given the expression $(a^2)^3$, where $a$ is a real number, determine the value of $a$ such that $(a^2)^3$ equals the number of ways to arrange 6 different objects in a row. | nvidia |
The sum of two numbers is 35, and 15$\%$ of the larger number is equal to 20$\%$ of the smaller number. What are the numbers? | nvidia |
Find all values of $y$ that satisfy the equation $\sqrt{y^2+5y+6}-\sqrt{y^2-7y+8}=2$.
Please let me know if you need help with anything else. | nvidia |
Given that the center of a circle is located at the origin and the coordinates of one endpoint of a diameter are $(3,4)$, find the coordinates of the other endpoint of the diameter. | nvidia |
The graph of the function
$$f(x)=\frac{x^4-4}{x^2-1}$$
consists of two distinct parts.
Determine the integer $n$ such that
$$\left[f(x)\right]^n=16.$$
is valid for any value of $x$ except
for $-1$ and $1$. | nvidia |
The roots $x_1$, $x_2$, and $x_3$ of the cubic equation $x^3+px+q=0$ are such that $x_1x_2x_3=-8$, $x_1x_2+x_2x_3+x_3x_1=14$, and $x_1^3+x_2^3+x_3^3=64$. Determine $p$. | nvidia |
Find the smallest positive integer $x$ such that $2x \equiv 3 \pmod{13}$. | nvidia |
For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 50$.
(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? | nvidia |
Find the sum of the digits of the largest perfect square whose decimal representation contains only 3 distinct digits. | nvidia |
What is the sum of all positive integers less than 100 that are squares of perfect squares? | nvidia |
Determine the number of positive integers less than 1000 that have exactly 7 positive divisors. | nvidia |
Find the area of the triangle bounded by the $y$-axis and the lines $y-3x=-2$ and $3y+x=12$. | nvidia |
In $\triangle ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$.
$\textbf{(A)}\ 4\qquad \textbf{(B)}\ \sqrt{17} \qquad \textbf{(C)}\ 4.25\qquad \textbf{(D)}\ 2\sqrt{5} \qquad \textbf{(E)}\ 4.5$
| nvidia |
Five circles are drawn on a plane such that each circle intersects the other circles in at most two points, and each pair of circles intersects in exactly one point. The five points of intersection are $A$, $B$, $C$, $D$, and $E$. Each point has a color red, blue, or green. It is known that the total number of points w... | nvidia |
The area of the base of a cone is equal to the area of the base of a cylinder. The height of the cone is half the height of the cylinder. Find the ratio of the volume of the cone to the volume of the cylinder. | nvidia |
A curve is defined by the equation $x^3 + y^3 = 27.$ Find the equations of all possible lines passing through the point $(0, 3)$ that intersect the curve at exactly two points. | nvidia |
Find the sum of all positive integer values of $k$ such that the equation $3x^3-2x^2+kx-1=0$ has at least one real root in the interval $[-1,1]$. | nvidia |
Consider a triangular array of numbers with the top row consisting of only $1$'s, and subsequent rows created by taking the previous row and appending a $1$ at the beginning and end of the row, and multiplying each element in the row by a fixed integer $n$. For example, if $n=2$, the array would look like:
$$
\begin{ar... | nvidia |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \cos 170^\circ & -\sin 170^\circ \\ \sin 170^\circ & \cos 170^\circ \end{pmatrix}^n = \mathbf{I}.\] | nvidia |
Find the number of ordered pairs $(c,d)$ of complex numbers such that
\[c^4 d^3 = c^2 d^7 = -1.\] | nvidia |
Find the area of the triangle with vertices $(4,2),$ $(1,7),$ and $(-3,1).$ | nvidia |
$\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$. | nvidia |
Let $f(n)$ denote the sum of all the positive divisors of the integer $n$. Find the number of integers $j$ between 1 and 2025 (inclusive) for which $f(j) = 1 + j + 2\sqrt{j}$. | nvidia |
Find the number of terms in the sequence $1, 2, 4, 8, \ldots, 2^n$, where $n$ is a positive integer and the sequence is formed by multiplying the previous term by 2. | nvidia |
Find the number of distinct positive divisors of 81. | nvidia |
Let $g(x)$ be a function defined for all positive real numbers $x$ such that $g(1) = 1$ and
\[g(a + b) = \frac{g(a)g(b)}{g(ab)}\]for all positive real numbers $a$ and $b.$ Find the value of $g(2).$ | nvidia |
Determine the function $f(x),$ with real coefficients, such that
\[f(x) + f\left(\frac{1}{x}\right) = [f(x^2)]^2\]for all real numbers $x.$ | nvidia |
A polynomial $P(x)$ has the property that $P(x^2) = x^2 \cdot P(x)$ for all real numbers $x$. If $P(2) = 12$, find the value of $P(3)$. | nvidia |
What is the sum of all positive integers less than 1000 that are cubes of perfect cubes? | 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 |
The numbers $a$, $b$, $c$ and $d$ are positive integers such that
$a+1=b-1=\frac{c}{2}=\frac{d+5}{3}$.
Determine the smallest possible value of $a+b+c+d$. | nvidia |
What is the largest possible value of the sum of the digits of a product of two three-digit numbers, each with digits that are either 1 or 0, when the two numbers are multiplied together? | nvidia |
Compute the sum of the series:
$$\frac{1}{1\cdot2} + \frac{1}{2\cdot3} + \frac{1}{3\cdot4} + \dots + \frac{1}{19\cdot20}$$
Note that the series is a sum of fractions where each fraction is of the form $\frac{1}{n(n+1)}$ and $n$ takes values from $1$ to $19$. | nvidia |
In the Cartesian plane, consider the set of all triangles ABC where A, B, and C are distinct points with coordinates (x, y) such that x and y are both integers and 0 ≤ x, y ≤ 100. Let S be the set of all such triangles whose perimeter is a prime number. Find the number of triangles in S whose area is also an integer. | nvidia |
In a sequence of seven consecutive integers, the sum of the squares of the three largest integers is greater than the sum of the squares of the four smallest integers. What is the smallest possible value for the largest integer in the sequence? | nvidia |
Find the sum of all positive integer values of $b$ such that the quadratic equation $4x^2 - 7x + b = 0$ has two real roots. | nvidia |
Find the smallest positive integer n such that n^3 has exactly 3 digits. | nvidia |
Let $x$ and $y$ be real numbers such that $\sqrt {x}+\sqrt {y}=1$ and $\sqrt {x}-\sqrt {y}=-\sqrt {2}$.
Find $x-y$. | nvidia |
A square is drawn such that one of its sides coincides with the line $y = 5$, and so that the endpoints of this side lie on the parabola $y = x^2 + 3x + 2$. What is the area of the square? | nvidia |
Let $P$ be a point inside a regular hexagon $ABCDEF$ such that the distance from $P$ to any of the vertices of the hexagon is the same.
Determine the ratio of the area of the hexagon to the area of the triangle formed by connecting the point $P$ to any three consecutive vertices of the hexagon. | nvidia |
Find the sum of the values of x that are excluded from the domain of the equation:
$$y = \frac{2x}{3x^2 + 5x - 2}$$ | nvidia |
A cube with edge length 6 is divided into 216 unit cubes. A plane passes through the centers of three unit cubes that share a vertex of the larger cube. The plane's intersection with the cube has an area that can be expressed as $a\sqrt{b}$. Find $a+b$. | nvidia |
Let $p(x) = x^3 + ax^2 + bx + c$ be a cubic polynomial with real coefficients such that $p(1) > 0$, $p(2) < 0$, $p(3) > 0$, and $p(4) < 0$. Determine the number of real roots of $p(x)$. | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $f(x)=ax^2+bx+c$. If $f(3x)=x^2+9x+4$ for all real numbers $x$, find $f(7)$. | nvidia |
Let $S(n)$ denote the sum of the divisors of a positive integer $n$, and let $d(n)$ denote the number of divisors of $n$. For how many positive integers $n$ can we find a positive integer $m$ such that $d(m)=S(n)$ and $S(m)=d(n)$? | nvidia |
Find the largest number of consecutive positive integers whose product is less than 1000. | nvidia |
Given a regular hexagon $ABCDEF$, a circle is circumscribed around it. The area of the triangle formed by connecting the center of the circle and points $A$ and $B$ is $4\sqrt{3}$, and the area of the triangle formed by connecting the center of the circle and points $A$ and $C$ is $2\sqrt{3}$. Determine the area of the... | nvidia |
Let $F,G$ be functions such that $F(x) = 3x-2$ and $G(x) = \frac{x+1}{2}$. Find the value of $x$ such that $F(G(F(G(x)))) = x$. | nvidia |
Count the number of positive integers less than 1000 that have the property that the sum of their digits is 7. | nvidia |
Find the side length of the largest equilateral triangle that can be inscribed in a circle of radius 5. | nvidia |
A nine-digit number is formed by arranging the digits 1 through 9 in a specific order. The number has the property that the sum of any three consecutive digits is always a multiple of 3. How many such numbers can be formed? | nvidia |
The product of two consecutive integers is 11 less than the square of the smaller integer. Find the smallest of the two integers. | nvidia |
Find the number of ordered triples of positive integers $(x,y,z)$ with $x<y<z$ for which the geometric mean of the three numbers is equal to the logarithmic mean of the two extreme numbers, and this common value is $10^9$. | nvidia |
How many four-digit positive integers exist, all of whose digits are 1's, 3's, 7's, and/or 9's, such that no two consecutive digits are the same? | nvidia |
Given a Fibonacci sequence that starts with 0 and 1, find the number of terms between the 100th and 500th term (inclusive) that are divisible by 11. | nvidia |
Find the number of ordered pairs of positive integers $(a, b)$ such that the least common multiple of $a$ and $b$ is 96, and the greatest common divisor of $a$ and $b$ is 12. | nvidia |
Find the largest value of $k$ such that the equation $x^2 + kx + k = 0$ has two distinct real roots. | nvidia |
Find a positive integer that is divisible by 21 and has a cube root between 16 and 16.5. | nvidia |
A circle is divided into six equal sectors. A triangle is formed by connecting the center of the circle to the endpoints of each sector. If all the triangles are congruent, what is the minimum number of lines that can be removed so that no triangle remains? | nvidia |
Let $g(x)$ be a function that satisfies the equation $g(g(x))=2x$ for all $x>0$. Given that $g(3)=7$, find $g(8)$. | nvidia |
Find the greatest possible value of $n$ such that $8^{n}$ divides $100!$ ($100!=100\times99\times98\times\cdots\times 3\times 2\times 1$). | nvidia |
Let $T$ be the set of complex numbers of the form $x + yi,$ where $x$ and $y$ are real numbers, such that
\[|x| + |y| \le 1.\]
Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a complex number $z \in T$ such that $z^n = -1.$ | nvidia |
Find the sum of all the positive integers less than 1000 that have exactly two digits when written in hexadecimal notation. | nvidia |
Find the product of all integer divisors of $105$ that also divide $14$. (Recall that the divisors of an integer may be positive or negative.) | nvidia |
Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$ Find the maximum value of $xy.$ | 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 |
Let's consider the set $A$ of all real numbers $x$ such that $11\le x\le12$.
Find the number of elements of the set $B=\{\lfloor \arctan x \rfloor: x\in A\}$. | nvidia |
We call a number a mountain number if its middle digit is larger than any other digit. For example, 284 is a mountain number. How many 3-digit mountain numbers are there?
Determine the number of 4-digit mountain numbers that have two identical digits, one of which is the middle digit. | nvidia |
Find the positive value of $m$ such that the equation $4x^2 - mx + 1 = 0$ has exactly one solution in $x$. | nvidia |
For how many positive integers $n$ less than or equal to 1000 is $$(\cos t+i\sin t)^n=\cos nt+i\sin nt$$true for all real $t$? | nvidia |
What is the largest positive integer $n$ such that $\frac{1}{\binom{8}{3}} + \frac{1}{\binom{8}{4}} = \frac{1}{\binom{9}{n}}$? | nvidia |
Let $a$ and $b$ be positive real numbers. Consider the sum of the areas of all the squares whose side lengths are $a, a + d, a + 2d, \dots, a + nd$, where $d = \frac{b-a}{n}$ and $n$ is a positive integer. Determine the limit of this sum as $n$ tends to infinity. | nvidia |
Determine the sum of all positive integer values of $a$ such that the quadratic equation $6x^2 - 7x + a = 0$ has two distinct real roots. | nvidia |
Let $x$ be a real number such that
$\frac{\sin x}{x} = \frac{5}{6}.$
Find the smallest positive value of $\lfloor x \rfloor.$ | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x^3 - y^3) = g(x)^3 + x^2g(y) - 2xy\]for all real numbers $x$ and $y.$
Find the number of possible values of $g(1),$ and show that there is a unique function $g$ satisfying the given conditions. | nvidia |
Find the smallest positive integer $a$ such that $a^3$ is divisible by 243 and $a^2$ is divisible by 160. | nvidia |
Let $ABCD$ be a parallelogram where the diagonals $AC$ and $BD$ intersect at $O$. The line segments $AO$ and $BO$ are extended to meet a circle centered at $O$ at $P$ and $Q$, respectively. Given that the area of $\triangle PCD$ is $35$ square units and the area of $\triangle QCD$ is $45$ square units, find the area of... | nvidia |
The product of two positive integers is 18. The positive difference of these two integers is 3. What is the sum of the two integers? | nvidia |
The product of two positive integers is equal to the number obtained by concatenating their decimal representations (assuming no leading zeros). Determine the smaller integer if the larger integer has exactly twice as many digits as the smaller integer. | nvidia |
Let $\star (x)$ be the sum of the digits of a positive integer $x$. $\mathcal{S}$ is the set of positive integers such that for all elements $n$ in $\mathcal{S}$, we have that $\star (n)=12$ and $0\le n< 10^{7}$. If $m$ is the number of elements in $\mathcal{S}$, compute $\star(m)$.
| nvidia |
Given positive integers $m$, $n$, and $p$, find all values of $k$ such that
$$m^{k}-n^{k}$$
is divisible by $p$, where $p$ is a prime number.
Find all values of $k$ for $m=11$, $n=17$, and $p=23$. | nvidia |
Let $c$ be the number of positive integers that are less than $100$, and can be written as the product of two distinct prime numbers. Let $d$ be the number of positive integers that are less than $100$, and have exactly three distinct factors. Compute $(c - d)^2$. | nvidia |
Find all positive integers n such that n is divisible by 11 and the sum of the proper divisors of n is equal to twice the number of divisors of n. | nvidia |
Let $g(n)$ be the product of the proper positive integer divisors of $n$. (Recall that a proper divisor of $n$ is a divisor other than $n$.) For how many values of $n$ does $n$ not divide $g(n)$, given that $2 \le n \le 50$? | nvidia |
Consider the sequence of positive integers whose digits are all 1, starting with 1, 11, 111, and so on. Let $T_n$ be the sum of the reciprocals of the digits of the integers from 1 to $10^n$ inclusive, excluding the integers in the sequence of all 1s. Find the smallest positive integer $n$ for which $T_n$ is an integer... | nvidia |
The difference between two numbers is 9, and the sum of the squares of each number is 153. What is the value of the product of the two numbers? | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2+y^2=1$ and $\frac{x}{y}+\frac{y}{x}$ is an integer. Find the maximum possible value of $x+y$. | nvidia |
What is the largest integer that can be subtracted from the product $77\times 78\times 79\times 80$ such that the resulting number is divisible by 17? | nvidia |
Any five points are taken inside or on a square with side length $1$. Let a be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than $a$. Then $a$ is:
$\textbf{(A)}\ \sqrt{3}/3\qquad \textbf{(B)... | nvidia |
Find the greatest value of $c$ such that $-c^2+5c-6 \ge 0$. | nvidia |
Let $M$ be a set of $n$ elements and let $f:M\to M$ be a function such that
$f(x \circ y) = f(x)\circ f(y)$
holds for all $x,y\in M$, where $\circ$ is an associative binary operation.
Find the smallest $n$ such that there exists a function $f$ that is not a constant function and is not equal to the identity function. | nvidia |
There are four complex numbers $z$ such that
\[z^3 + \overline{z}^3 = 210,\]and both the real and imaginary parts of $z$ are integers. These four complex numbers are plotted in the complex plane. Find the perimeter of the quadrilateral formed by the four complex numbers as vertices. | nvidia |
On a rectangular chessboard with 20 rows and 25 columns, a pawn is placed randomly and uniformly among the cells. Let $m$ be the probability that the pawn is placed on a square with an even number on the row and a prime number on the column (assuming that 1 is not a prime number). Let $n$ be the probability that the pa... | nvidia |
Problem:
Determine the value of $b$ for which the equation $x^2+y^2-4x+2y+b=0$ defines a circle that passes through the point $(5,3)$.
(And don't forget that $b$ can be negative). | 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 dimensions of the rectangle of maximum area that can be inscribed in a circle of radius 4 cm. | nvidia |
Let $f(x)$ be a monic polynomial with integer coefficients such that $f(x) = g(x) \cdot h(x),$ where $g(x)$ and $h(x)$ are monic polynomials with integer coefficients. Suppose that both $g(x)$ and $h(x)$ have exactly two distinct roots. If the roots of $g(x)$ are integers and the roots of $h(x)$ are $\sqrt{2}$ and $-\s... | nvidia |
Let $a$ and $b$ be real numbers such that the roots of the equation $x^2 + ax + b = 0$ are complex numbers. If the absolute value of the difference between these roots is equal to $2 \sqrt{a}$, determine $a$ such that the product of the roots is minimum.
Determine $a$. | nvidia |
Find the sum of all the integer solutions of the equation $x^2+7x+12=0$ that also satisfy the inequality $2x+5>0$. | nvidia |
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