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What is the smallest integer that can be added to 11.7 so that the sum is a multiple of 0.5? | nvidia |
A line that forms a 45-degree angle with the positive x-axis intersects a circle centered at the origin. The x-coordinate of one of the intersection points is 3. What is the area of the triangle formed by the line, the x-axis, and the y-axis? | nvidia |
Let set $\mathcal{B}$ be a 75-element subset of $\{1,2,3,\ldots,120\},$ and let $T$ be the sum of the elements of $\mathcal{B}.$ Find the number of possible values of $T.$ | nvidia |
A five-digit palindrome number is a number that reads the same forwards and backwards, such as 12321. Find the smallest five-digit palindrome number that is divisible by 11 and has exactly three digits that are not zero. | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-2ax+a^2=0$$
are such that
$$x_1^3+x_2^3=54$$
Determine $a$. | nvidia |
A circle of radius 1 is surrounded by a regular polygon with $n$ sides, where each side is tangent to the circle and has a length of 2. The area of the region outside the circle and inside the polygon can be written in the form $\sqrt{a} - \pi b$, where $a$ and $b$ are positive integers. Find the value of $n$ such that... | nvidia |
Find the sum of all the positive divisors of 85. | nvidia |
A sequence is defined by $a_1 = 1$, and for each positive integer $n$, $a_{n+1}$ is the product of the first $n$ positive integers. Find the value of $a_{10}$. | nvidia |
What is the smallest positive integer $n$ such that $n^3 \equiv 2 \pmod{25}$? | nvidia |
Let $f(x)=x^3+ax^2+bx+c$ be a cubic function with real coefficients. If $f(x)$ has three distinct real roots $r_1, r_2, r_3$ such that the sum of the roots of $f^{-1}(x)$ is equal to the product of the roots of $f(x)$, find the value of $a+b+c$. | nvidia |
Let $S$ be the set of all positive integer divisors of $100$. How many different positive integers can be represented as a sum of two distinct members of $S$? | nvidia |
Let $d$ be a positive number such that when $109$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 1$. Find the value of
\[2\cos^2 0^\circ x + 2\cos^2 1^\circ x + \dots + 2\cos^2 90^\circ x.\] | nvidia |
Determine the value of $b$ such that the line $bx+4y=-8$ is parallel to the line $3x+by=6$. | nvidia |
Let $n$ be a positive integer and consider a set of $2n+1$ points chosen at random inside or on a regular $n$-gon with side length $1$. Determine the smallest possible number $a$ such that it is always possible to select a pair of points from this set such that the distance between them is equal to or less than $a$. | nvidia |
Find the number of distinct divisors of 315 that are also multiples of 5, considering only positive divisors. | nvidia |
Given that $$(a+b+c)(ab+ac+bc)=27$$and that $$a^2(b+c)+b^2(a+c)+c^2(a+b)=12$$for real numbers $a$, $b$, and $c$, what is the value of $abc$? | nvidia |
Let $f(x)=2x+1$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f(x^{-1})$. | nvidia |
Find all pairs of integers $(a,b)$ such that the roots of the equation $x^2+ax+b=0$ are of the form $n$ and $\frac{1}{n}$, where $n$ is a positive integer. | nvidia |
Consider a parabola $y^2 = 4x$ that has its focus at $F = (1, 0)$. A circle centered at the origin intersects the parabola at two points, one of which is the origin. Find the equation of the circle, knowing that the distance between the focus of the parabola and the other point of intersection is $\sqrt{5}$. | nvidia |
Find the number of non-negative integer solutions to the equation $x+y+z=11$ such that $x+y$, $y+z$, and $z+x$ are all even. | nvidia |
The sum of the squares of two consecutive even integers is 100. What is the sum of the cubes of these integers? | nvidia |
Find the number of prime positive integers that are divisors of 900. | nvidia |
Let $a$ and $b$ be positive integers such that $a^2 + b^2 = 100.$ Find the maximum value of $a + b.$ | nvidia |
Find the largest positive real number $k$ such that for all positive real numbers $a$, $b$, and $c$,
\[\sqrt[3]{abc} + k \left( \frac{a+b+c}{3} - \sqrt[3]{abc} \right) \ge \frac{ab+bc+ca}{a+b+c}.\] | nvidia |
Find the largest possible value of the expression $$\frac{(x+y)^2+(y-z)^2+(z-x)^2}{x^2},$$ where $x > y > z$ are real numbers, and $x \neq 0.$$ | nvidia |
Find the number of integers between 100 and 200 that have a digit sum equal to 10. | nvidia |
Let $a,b,$ and $c$ be positive real numbers such that
$a + b + c = 10$ and
$\ds\f{1}{a}+\ds\f{1}{b}+\ds\f{1}{c}=1$.
Determine the value of
$\ds\f{1}{ab}+\ds\f{1}{bc}+\ds\f{1}{ca}$. | nvidia |
Let $a,b,c,d,e,f,g,h,i$ be real numbers such that $a+b+c=0, d+e+f=0, g+h+i=0$. Compute the coefficient of $x^2$ in the expansion of
$$\left(ax^2+bx+c\right)\left(dx^2+ex+f\right)\left(gx^2+hx+i\right).$$ | nvidia |
Let $p$ and $q$ be two positive integers such that $p$ is a multiple of $3$ and $q$ is a multiple of $5$. The greatest common divisor of all possible numbers of the form $p^3+q^3$ is:
$\textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 15\qquad \textbf{(E)}\ 30$ | nvidia |
If the difference between the squares of two positive integers is a perfect square, then their sum has at most how many distinct prime factors? | nvidia |
Let $A$ and $B$ be two points in the plane such that the midpoint of segment $AB$ lies on the unit circle. Given that the line through $A$ and $B$ is tangent to the circle, what is the maximum value of $\frac{1}{|AB|}$? | nvidia |
If $f(x)=\frac{2^x-1}{2^x+1}$, find the value of $c$ such that the graph of the function $y=f(x)+c$ has a horizontal asymptote at $y=1$. | nvidia |
How many distinct real numbers $r$ are there such that the equation $x^{\{x\}} = r$ has exactly two solutions for $x$, where $\{x\}$ denotes the fractional part of $x$? | nvidia |
The value of
\[\frac{n}{2} + \frac{18}{n}\]is smallest for which positive integer $n$? | nvidia |
Let $f(n)$ be a function that takes an integer $n$ and returns the sum of the digits of $n$. What is the maximum possible value of $f(a) + f(b) + f(c)$ for positive integers $a$, $b$, and $c$ such that there are at least two distinct positive integers $n$ satisfying $f(na) + f(nb) = f(nc^2)$? | nvidia |
Find the maximum possible value of the real part of the complex number $a+bi$, where $a$ and $b$ are real numbers satisfying the equation $a^2+b^2=16$ and $a+b=4$. | nvidia |
Let $a$ and $b$ be positive integers such that $\text{gcd}(a,b)=15$ and $\text{lcm}(a,b)=150$. Find the smallest possible value of $\frac{a}{b}+\frac{b}{a}$. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $xyz = 1$. Find the set of all possible values of $x + y + z$. | nvidia |
Find the number of ordered pairs $(a,b)$ of real numbers such that $a - b = 3$ and $ab + 2b^2 = 5$. | nvidia |
Given that the system of equations:
\begin{align*}
3x-5y&=-11,\\
7x+2y&=-12.
\end{align*}
has solution $(x,y)=(-2,1)$, find the value of $k$ in the system of equations:
\begin{align*}
3x-5y+k&=-11,\\
7x+2y+k&=-12.
\end{align*}
such that the solution remains the same. | nvidia |
Let $n$ be a positive integer with exactly five divisors. Determine the smallest value of $m$ for which $n^m$ has more than 20 divisors. | nvidia |
Let $a,$ $b,$ and $c$ be real numbers such that
\[a+b+c=2,\] \[ab+bc+ca=-1,\]and \[abc=-6.\]Compute $(a+b+c)^3+(ab+bc+ca)^3+(abc)^3.$ | nvidia |
Find the smallest positive integer $n$ such that $n^5 - n^4$ is divisible by 11. | nvidia |
Let $\omega$ be a complex number satisfying $\omega^3 = 1$ and $\omega \neq 1.$ Let $n$ be the smallest positive integer such that there exist real numbers $r$ and $s$ satisfying
\[(r + s \omega)^n = (r + s \overline{\omega})^n.\]Determine $n.$ | nvidia |
Find the smallest positive integer k such that the repeating decimal expansion of k/7 has exactly 3 digits. | nvidia |
A square is inscribed in the circle $x^2+y^2=100$. The vertices of the square are $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, and $(x_4,y_4)$. If the coordinates of the vertices satisfy the condition $x_1y_2+x_2y_3+x_3y_4+x_4y_1=0$, find the area of the square. | nvidia |
What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer?
| nvidia |
Determine the number of positive integer divisors of $3^6 \cdot 11^4$ that are also multiples of 121. | nvidia |
Find all real numbers $x$ such that $\left\lceil \frac{x}{\left\lceil x \right\rceil}\right \rceil = 7$. | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $a$ is even, and $b$ and $c$ are odd. If $a+b+c$ is divisible by 4, then find the largest integer that always divides the expression $a^2+b^2+c^2$. | nvidia |
Find the length of the shortest curve that can be drawn inside the square with vertices (0,0), (3,0), (3,3), and (0,3) such that it divides the square into two regions, each with the same area. | nvidia |
An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free? | nvidia |
In the square $EFGH$, point $O$ is the center of the square. The side length of the square is 1. A right triangle $PQR$ is drawn in the square such that point $P$ lies on segment $EF$, point $Q$ lies on segment $FG$, and point $R$ lies on segment $GH$. The legs of the triangle are perpendicular to the sides of the squa... | nvidia |
The sum of the reciprocals of two positive integers is 1/6, and their least common multiple is 90. What is the sum of the two numbers? | nvidia |
Given that the roots of the cubic equation \(ax^3 + bx^2 + cx + d = 0\) are all integers, and knowing that two of the roots are equal, find the value of \(a\) such that the product of the roots is minimized, under the constraint that \(a\), \(b\), \(c\), and \(d\) are all positive integers. | nvidia |
How many positive integers less than 1000 can be written in the form $3^n+2^n$, where $n$ is an integer? | nvidia |
Let $x$ be a real number such that $\lfloor x \rfloor + x = 27.85$. Find $x$. | nvidia |
Determine all real numbers $t$ such that the roots of the quadratic equation $x^2-6tx+5t-1=0$ are real and nonnegative. Express your answer in interval notation. | nvidia |
Determine the number of positive integers x that satisfy the inequality \sqrt{100} \leq x \leq \sqrt{200}. | nvidia |
In an acute triangle ABC, the altitudes AD and BE intersect at H. The areas of the triangles HBD and HEC are equal. Prove that the triangle is isosceles. | nvidia |
Let $a$ and $b$ be two positive numbers with $[a+b]=7$ and $[a/b]=0$. Find the greatest possible value of $[a]+[b]$. | nvidia |
Consider a sequence $b_n$ defined as follows: $b_1=2$ and for each positive integer $n>1$, $b_n=2^nb_{n-1}+n^2$. Determine the smallest positive integer $n$ such that $b_n$ is divisible by $3$. | nvidia |
Find the product of the $x$-values that satisfy the equation $\frac{x^2-3x-4}{x-4}=\frac{x-2}{x+1}$. | nvidia |
Let ABC be a triangle. There exists a positive real number k, such that if the altitudes of triangle ABC are extended past A, B, and C, to A', B', and C', as shown, such that AA' = kBC, BB' = kAC, and CC' = kAB, then triangle A'B'C' is equilateral.
Find k.
Write another problem inspired by this one:
Given a triangl... | nvidia |
Given that $\frac{1}{x-1}+\frac{1}{y-2}+\frac{1}{z-3}=\frac{13}{6}$, find the value of $\frac{x}{x-1}+\frac{y}{y-2}+\frac{z}{z-3}$. | nvidia |
Find the sum of all divisors of the smallest perfect square that can be expressed as a sum of three consecutive odd integers. | nvidia |
Find the sum of the values of $x$ which satisfy $x^2 +1992x = 1993$. | nvidia |
Let $P(x) = x^3 + ax^2 + bx + c$ be a cubic polynomial with real coefficients, where $a, b, c$ are integers. Suppose that $P(x)$ has a real root $r$ such that $|r| > 1$. Show that there exists a positive integer $k$ such that $P(x)$ has a real root in the interval $(k, k + \frac{1}{2})$. | nvidia |
Find the number of positive 3-digit numbers that are multiples of 7 but not of 11. | nvidia |
Find the only positive real number $x$ for which $\displaystyle \frac{x-4}{9} = \frac{4}{x-9}$. | nvidia |
Consider a rectangular prism with integer edge lengths, where the length, width, and height are $a$, $b$, and $c$ respectively. The external diagonals of the prism are $\sqrt{a^2+b^2}$, $\sqrt{a^2+c^2}$, and $\sqrt{b^2+c^2}$. Suppose that these three external diagonals are the side lengths of a right triangle with hypo... | nvidia |
The values $p$ and $q$ are chosen so that the quadratic equations $x^2+px+5=0$ and $x^2+qx+6=0$ have the same discriminant. Determine the number of possible integer values for $p+q$. | nvidia |
Find the smallest positive integer value of $y$ such that $y + 1234$ results in a number that is a palindrome. | nvidia |
Determine the smallest positive integer $b$ such that there exists an integer $y$ for which $y^4 + by^2 + 1$ is not prime. | nvidia |
Find the smallest possible value of $b$ such that the polynomial $x^4 - 2bx^2 + 2025$ has four distinct real roots. | nvidia |
Let $x$ be the smallest value for which $x(x-5)(x+6)=270$. Determine the sum of all possible values of $x$. | nvidia |
The sum of three consecutive positive odd integers is twice their product. What is the sum of the reciprocals of the three integers? | nvidia |
A standard cube has a volume of $V = s^3$, where $s$ is the length of its edge. If the sum of the lengths of the edges of the cube is 240 cm, find the sum of the areas of all its faces. | nvidia |
Count the number of positive integers less than 1000 that are palindromes and have exactly three different digits. | 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 |
Let \(a,b,c\) be positive real numbers. Prove that if \(J(a,b,c)=J(b,c,a)=J(c,a,b)\), then
\[J(a,b,c)\geq 3.\] | nvidia |
In a triangle with vertices (2, 3), (7, 8), and (11, 11), find the area of the triangle formed by connecting the midpoints of the sides of the triangle. | nvidia |
How many ordered pairs of positive integers $(x,y)$ with $x$ and $y$ less than or equal to $100$ satisfy the equation $\cos(xy)=-0.8$? | nvidia |
Problem:
In a triangle with vertices at $(a,0)$, $(0,b)$, and $(c,d)$, the area is equal to half the product of the lengths of the legs of a right triangle that can be inscribed within it. Find the area of the triangle if $a = 5$, $b = 3$, and the slope of the line through $(a,0)$ and $(c,d)$ is 2. | nvidia |
Let $g(x)=5x-2$. Find the product of all $x$ that satisfy the equation $g^{-1}(x)=g((3x)^{-1})$. | nvidia |
The sum of the squares of the roots of the equation $x^2+ax+b=0$ is $4$. Find $a^2+4b$.
Let $x_1$ and $x_2$ be the roots of the equation. Then $x_1+x_2=-a$ and $x_1x_2=b$.
We have that $x_1^2+x_2^2=4$, so $(x_1+x_2)^2-2x_1x_2=4$.
Substituting the expressions for $x_1+x_2$ and $x_1x_2$, we get $a^2-2b=4$.
Therefore,... | nvidia |
Find $n$ such that $2^5 \cdot 3^2 \cdot n = 8!$. | nvidia |
The three roots of the cubic $ 30 x^3 - 50x^2 + 22x - 1$ are distinct real numbers strictly between $ 0$ and $ 1$. If the roots are $p$, $q$, and $r$, what is the sum
\[ \frac{1}{1-p} + \frac{1}{1-q} +\frac{1}{1-r} ?\] | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=7$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
How many four-digit numbers have the property that the sum of their digits is equal to 10, and the digits are in increasing order from left to right? | 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 |
Compute the number of distinct arrangements of the letters in the word "SALLOW" that can be formed using all 6 letters, without any restriction. | nvidia |
Let $x$ be a positive integer such that
$9999 \cdot x \equiv -1\pmod{10000}.$
Find the remainder when 2016 is divided by $x$. | nvidia |
Given the matrix
\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \]
where $a, b, c, d$ are positive integers such that $ad - bc = 1,$ find the remainder when the determinant of the matrix
\[ \begin{pmatrix} a^{12} & b^{12} \\ c^{12} & d^{12} \end{pmatrix} \]
is divided by the determinant of the matrix
\[ \begin{p... | nvidia |
The sum of the positive divisors of a positive integer of the form $2^i3^j$ is equal to $600$. What is $i + j$? | nvidia |
The function $$f(\mathbf{x}) = \cos(x_1)\sin(x_2) + \cos(x_2)\sin(x_3) + \dots + \cos(x_{n-1})\sin(x_n) + \cos(x_n)\sin(x_1)$$ is defined on the space $\mathbb{R}^n$, where $n$ is an arbitrary positive integer. Prove that $$\lim_{n \to \infty} \max_{\mathbf{x} \in \mathbb{R}^n} f(\mathbf{x})$$ exists. Find this limit. | nvidia |
Find the sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$. | nvidia |
Find the difference between the sum of the first 30 positive multiples of 3 and the sum of the first 25 positive multiples of 4. | nvidia |
What is the area of the largest triangle that can be formed by connecting three distinct points on the graph of the equation $x^2 + 2y^2 = 4$? | nvidia |
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