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Find the smallest positive integer $n$ such that $n^2$ is divisible by 10 but not by 15. | nvidia |
The product of two numbers is 180. The quotient of the two numbers is 3/2. What is the larger of the two numbers? | nvidia |
Find the sum of all possible values of $y$ such that $\left(\sqrt{2}\right)^{y^2-6y+8} = \left(\sqrt{2}\right)^{y-4} \cdot \left(\sqrt{2}\right)^{y-2}$. | nvidia |
The sum of the lengths of all the edges of a cube is 60 cm. Find the number of cubic centimeters in the volume of the cube. | nvidia |
The sum of the cubes of the roots of the equation $x^3 - 4x^2 + 5x - 3 = 0$ is equal to $5$ times the sum of their squares. Find the value of the sum of the fourth powers of the roots. | nvidia |
The two-digit positive integer $n$ has the property that when divided by $4$ or $7$ its remainder is always $3$. What is the smallest value of $n$ that is a multiple of $3$? | nvidia |
Determine the largest possible value of $m$ such that
$$f(x) = (x^2-4x+4)(x^2-8x+4)(x^2+4x+4)(x^2+8x+4)$$
is divisible by $x^m$, where $x$ is a multiple of $1280$. | nvidia |
Determine all values of $n$ for which there exists a solution to the system of congruences
$$nx\equiv n \pmod {1007},$$
$$nx \equiv n+1 \pmod {1009},$$
where $n$ and $x$ are integers. | nvidia |
Find the smallest positive integer $n$ such that $\frac{n}{4}$ is a perfect square, $\frac{n}{9}$ is a perfect cube, and $\frac{n}{16}$ is a perfect fourth power. How many of the divisors of $n$ are not multiples of 6? | nvidia |
The vertices of a triangle are given by the complex numbers $z_1, z_2,$ and $z_3.$ When each of these complex numbers is divided by $2+i$, the triangle formed by the resulting points has an area of $15$. Find the area of the original triangle. | nvidia |
Let $g(x)=5x-7$ and $h(x)=\frac{1}{2x+1}$. Find the sum of all $x$ that satisfy the equation $g(g^{-1}(x))=h(h^{-1}(x))$. | 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 |
Find the area of the triangle with vertices $(-1,4),$ $(7,0),$ and $(11,5).$ | nvidia |
What is the greatest integer less than or equal to\[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?\]
$\textbf{(A) }80\qquad \textbf{(B) }81 \qquad \textbf{(C) }96 \qquad \textbf{(D) }97 \qquad \textbf{(E) }625\qquad$
| nvidia |
Find the smallest positive integer that is divisible by both 14 and 21, and also has exactly three distinct prime factors. | nvidia |
Find the number of ways to arrange the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in a row such that the number of even digits is greater than the number of odd digits and the number of digits greater than 4 is greater than the number of digits less than 4. | nvidia |
For a positive real number $x$, let $a(x)=\min\{x, x^2\}, b(x)=\max\{x, x^2\}$. Define the function
$f(x)=\frac{a(x)}{b(x)}$.
Let $x_1$ be the value of $x$ such that $f(x)$ reaches its maximum over the interval $[0, 2]$. Find $f(x_1)$ | nvidia |
Let $a, b, c$ be real numbers such that
\begin{align*}
b+c & = 18, \\
c+a & = 20, \\
a+b & = 22.
\end{align*}
Find $\sqrt[3]{abc(a+b+c)}$. | nvidia |
Find the sum of all real solutions to the equation \[\frac{x-2}{x^2+4x+1} = \frac{x-5}{x^2-10x}.\] | nvidia |
A circle with a radius of 2 units is drawn with its center at the point (-2,0). A line segment is drawn from the point (0,4) to the circle such that it is tangent to the circle at a point P. Find the coordinates of point P. | nvidia |
Find the largest integer that always divides into the difference between the square of a prime number and its fifth power. | nvidia |
Find the sum of all $x$ such that the expression $\frac{x^2-3x-4}{x^2+5x-6}$ is not continuous. | 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 |
Let $x$ and $y$ be two positive numbers such that the minimum value of $x^2+y^2$ is 16. Determine the largest value of $c$ such that $0$ is in the range of $f(x,y) = x^2 + 2xy + y^2 + c$. | nvidia |
Given the function $y=f(x)$ and the point $(4,7)$ on its graph, find the equation of the graph of $y=f(2x-3)+2$ and determine the point on this new graph that has the same $x$-coordinate as the original point $(4,7)$. | nvidia |
What is the largest positive three-digit number divisible by 11 which has all distinct digits and none of the digits is equal to 9? | nvidia |
What is the greatest positive integer n such that n divides evenly into the sum of the product of each two consecutive integers from 1 to 5? | nvidia |
Find the base $b$ such that the base $b$ number $123_{b}$ is equal to the decimal number $1.5$ times the base $b$ number $110_{b}$. | nvidia |
The sum of the squares of two positive integers is 193. The product of the two integers is 84. What is the sum of the two integers? | nvidia |
Find the largest real number $c$ such that \[x_1^2 + x_2^2 + \dots + x_{101}^2 \geq cM^2\]whenever $x_1,x_2,\ldots,x_{101}$ are real numbers such that $x_1+x_2+\cdots+x_{101}=0$ and $M$ is the median of $x_1,x_2,\ldots,x_{101}.$ | nvidia |
Let $x$ be a real number such that $0 < x < 1$. Determine the value of $[\sqrt{x+1} + \sqrt{x^2+1}]$. | nvidia |
Find all complex numbers $z$ such that
\[|z^2 + 2i| = |2z - 5|.\]Enter all complex numbers, separated by commas. | nvidia |
Find the greatest possible four-digit number that is divisible by 8 and has exactly two digits repeated. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of
\[\frac{x + y}{xyz}.\] | nvidia |
Determine the number of positive integer values of $x$ such that $\frac{3x-2}{x+1}$ is an odd integer. | nvidia |
Suppose $x$ and $y$ are integers such that $xy+5x+4y=-5$. Find the greatest possible value of $y$. | nvidia |
The function
\[f(z)=az^3+bz^2+cz+d\]where $a,b,c,$ and $d$ are integers, has exactly one complex zero that is not a real number. What is the minimum number of zeros that any such function can have? | nvidia |
Find the largest number of consecutive positive integers whose sum is a proper divisor of 864. | nvidia |
Given the vertices of a triangle as (0, 0), (4, 6), and (3, k), find the value of k such that the area of the triangle is 10 square units. | nvidia |
If the point $(4,8)$ is on the graph of $y=f(x)$, and $g(x)=3f(x)$ for all $x$, then there is one point that must be on the graph of $y=g(x)$. What is the product of the coordinates of that point? | 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 positive value of $m$ such that the equation $x^2+4mx+2m^2=0$ has two solutions in $x$ that differ by exactly $3$ units. | nvidia |
Find the smallest positive integer $y$ which is greater than $2$ and relatively prime to $210$ (recall that relatively prime means that the GCD of $y$ and $210$ is $1$) | nvidia |
Find the largest value of $n$ for which there exist unit vectors $\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_n$ in a three-dimensional space such that $\|\mathbf{x}_i-\mathbf{x}_j\|$ is an integer for all $i$ and $j$. | nvidia |
Let $a,b,c,d$ be positive integers such that the polynomial $f(x)=(ax+b)(cx+d)$ has no multiple roots. Given that $\gcd(f(x),x)=1$ for every $x$ that is a multiple of $ac$, find the maximum possible value of $\gcd(a,c)$. | nvidia |
Find the smallest positive integer $a$ such that $a$ is a multiple of $15$ and $a$ is a multiple of $20$. | nvidia |
A fair coin is flipped three times, resulting in a sequence of heads and tails. A bug eats one of the coins at random, so that one of the tosses is now unknown. Given that the bug is equally likely to have eaten any of the three coins, what is the probability that an odd number of the remaining two tosses are heads? | nvidia |
Find the number of ordered pairs of integers $(m,n)$ with $1 \leq m \leq 50$ and $n \geq 1$ such that the polynomial $x^2-mx+n$ can be factored into the product of two distinct linear factors with integer coefficients. | nvidia |
Find the largest integer $n$ such that the least common multiple of all the prime factors of $159137$ is greater than $n$ but less than $2n$. | nvidia |
Given a triangle with vertices A(0, 0), B(-3, 2), and C(2, 7), find the equation of the line that passes through the centroid of the triangle and is perpendicular to side AB. | nvidia |
Find the value of $b$ that satisfies the equation $161_{b}+134_{b}=315_{b}$. | nvidia |
Find the sum of the odd integers between 1 and 100 that are not divisible by 3. | nvidia |
Determine the largest integer value of x such that the inequality $$x - \frac{5}{3} < -2$$ holds true. | nvidia |
A box contains three types of balls: red, blue, and green. The number of red balls is twice the number of blue balls, and the number of green balls is three times the number of blue balls. If the total number of balls is 120, what is the largest possible sum of the numbers of each type of ball that can be chosen such t... | nvidia |
Find the number of integers $n$ for which $\cos(n) \le \cos(2n + 1) < \cos(3n - 2)$. | nvidia |
Find the smallest positive integer n such that it has exactly 6 distinct positive factors, including 1 and itself. | 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 $P(x)$ be a quadratic polynomial with real coefficients such that
\[P(x)P(-x)=36x^4-28x^2+1.\]Find the sum of all possible values of the coefficient of $x$ in $P(x)$. | nvidia |
The product of the positive integer divisors of a positive integer $n$ is 729. Find $n$. | nvidia |
The polynomial $P(z)=z^{28}-z^8-1$ has roots of unity as solutions. Consider a new polynomial
$Q(z)=z^{42}+az^{21}+bz^7+c$. Find the values of $a, b,$ and $c$ such that all the roots of $Q(z)$ that lie on the unit circle are also roots of $P(z)$. What is the sum of the values of $a, b,$ and $c$? | nvidia |
Determine the number of three-digit integers that have exactly two distinct digits.
(For example, 121 and 343 are such numbers while 111, 123, 222 are not.)
Find the amount of such numbers. | nvidia |
Count the number of six-digit numbers that read the same forwards and backwards, where the first and last digits can be any number from 1 to 9, and the middle digits can be any number from 0 to 9. | nvidia |
Determine the largest positive integer n, such that the positive divisors of n are 1, 2, 4, 5, and 10 only. | nvidia |
Find the smallest positive integer $n$ such that the equation
$$\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}+\ldots+\sqrt{x+n}=10\sqrt{x}$$
has a solution. | nvidia |
Find the smallest positive integer whose cube has a digits sum that is a perfect square. | nvidia |
A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of 9 that is not a lucky integer? | nvidia |
Let $a$ and $b$ be positive real numbers such that $2a + 7b = 35.$ Find the maximum value of $ab.$ | nvidia |
Find the number of positive integers less than 100 whose square ends with the digits 76. | nvidia |
A convex hexagon $P_1P_2P_3P_4P_5P_6$ is said to be equi-areal if there exists a point $Q$ in its interior such that the area of each triangle $QP_iP_{i+1}$, $i=1,2,3,4,5$, is the same, and the area of triangle $QP_6P_1$ is $2$ times the area of each triangle $QP_iP_{i+1}$, $i=1,2,3,4,5$. Determine the maximum number o... | nvidia |
A circle is inscribed in a triangle with sides of lengths 5, 12, and 13. The area of the triangle can be expressed as $A\sqrt{B}$. Find the value of $A+B$, where $A$ and $B$ are positive integers. | nvidia |
Find all values of $a$ for which the graph of $|z-1|=|z-3|$ intersects the graph of $|z-2|=a$ in exactly two points. | nvidia |
Let $f(x)=2x+1$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f(x^{-1})$.
Another problem:
Let $f(x) = \frac{x^2 - 1}{x^2 + 1}$. Find the product of all $x$ that satisfy the equation $f(f(x)) = f(x^{-1})$. | nvidia |
How many positive integers less than 100 can be expressed as the sum of a perfect square and a perfect cube? | nvidia |
What is the smallest positive integer that must divide the sum of the first twenty terms of any arithmetic sequence whose terms are positive integers? | nvidia |
Let $x$ and $y$ be the two roots of the equation $x^2-kx+2=0$. Find $k$ such that $\frac{x}{y}+\frac{y}{x}=3$. | nvidia |
Find the sum of the positive factors of 72. | nvidia |
In a right triangle with legs parallel to the coordinate axes, the vertices are $(0,0)$, $(0,4)$, and $(a,0)$ for some positive integer $a$. The medians of the triangle intersect at a point with coordinates that are integers divided by $5$. Find the value of $a$. | nvidia |
Consider a geometric sequence with positive integer terms, where the common ratio is also an integer. What is the smallest positive integer that can be a common factor of the product of the first five terms of any such sequence? | nvidia |
Three fair coins are flipped simultaneously. The sum of the numbers on the top of the coins is calculated, where the numbers on the coins are 1, 2, and 3. What is the probability that the sum is a prime number? | nvidia |
Let $f(z)$ be a polynomial such that $f(z)=f(z^{-1})$. If there are exactly four distinct roots that lie on the unit circle, what is the maximum number of real roots that $f(z)$ can have? | nvidia |
Find the maximum number of consecutive positive odd integers that can be multiplied together before the product exceeds 1000. | nvidia |
Consider a rectangle with its lower-left corner at (0,0) and its upper-right corner at (x,y), where x and y are positive integers. Inside this rectangle, there are circular "bubbles" of radius 1/3 centered at each integer coordinate point. Determine the minimum number of bubbles that must be popped in order to draw a s... | nvidia |
Let $f(x)$ be a polynomial function such that
$$f(x^2+1)=x^4+4x^2$$
for all real values $x$. Let $n$ be the number of points where the graph of
$y=f(x)$ intersects the $x$-axis. Find $n(n-1)$. | nvidia |
Consider the following equation, $az^2+bz+c=0$, where $a$, $b$, and $c$ are integers. Determine the number of roots of this equation that also satisfy the condition $|z|=\sqrt{2}$, for some positive integer $n$.
Note: $|z|$ denotes the modulus of the complex number $z$. | nvidia |
Find the smallest positive integer $k$ such that $5^{2k} \equiv 1 \pmod{11}$ | nvidia |
What is the minimum number of consecutive positive integers that can be multiplied together such that the product exceeds 1000 and at least one of the integers is a prime number? | nvidia |
What is the sum of the digits of the smallest positive integer $n$ such that $n$ is a multiple of 3 and $n$ can be expressed as the sum of three distinct perfect squares? | nvidia |
Let $\mathbb{P}$ be the set of positive real numbers. Find the number of functions $g : \mathbb{P} \to \mathbb{P}$ such that
\[g\left(\frac{x}{g(y)}\right) = \frac{x}{y}\]for all positive real numbers $x$ and $y.$ | 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 |
Prove that the determinant of the matrix
\[\begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}\]
is equal to \((x+y+z)(x^2+y^2+z^2-xy-xz-yz)\). | nvidia |
Given a square with side length $a$ and a circle inscribed within it, the area of the square not covered by the circle is equal to the area of the circle. Determine the side length of the square.
(Finished) | nvidia |
What is the probability that when two regular six-sided dice are rolled, the sum of the numbers showing on the dice is 7 and at least one die shows a 3? | nvidia |
Consider the sequence of functions $f_n(x)$ defined recursively as follows:
\[f_1(x) = \frac{1}{x}, \quad f_{n+1}(x) = f_n\left(\frac{1}{x+1}\right),\]
for all positive integers $n$ and nonzero real numbers $x$. As $n$ approaches infinity, find the value that $\frac{1}{f_n(1)}$ approaches. | nvidia |
The numbers from 1 to 6 are written on six cards. Find the smallest possible sum of the numbers on two cards, such that the product of these numbers is divisible by 8. | nvidia |
Given positive integers $m$ and $n$ such that $\frac{1}{3m} + \frac{1}{n} = \frac{1}{9}$, what is the least possible value of $mn$? | nvidia |
Consider a sequence of positive integers $(b_n)$ such that
$$b_n=3b_{n-1}+\sqrt{b_{n-2}}$$for all $n>2$.
Determine whether there is a value of $b_2$ for which the sequence $(b_n)$ is bounded. | nvidia |
Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$. Find $g^{-1}(f(-7))$. | nvidia |
Let \(P(x) = x^4 + ax^3 + bx^2 + cx + d\) be a polynomial with real coefficients and no real roots. It is known that one of its roots is \(\sqrt{3} + i\) and that the product of the imaginary parts of all roots is \(5\). Find \(d\). | nvidia |
Find the equation of the line that passes through the point $(2,3)$ and is perpendicular to the line $3x+4y=12$. Express your answer in the form $Ax+By+C=0$, where $A,B,C$ are integers such that $A>0$ and $\gcd(|A|,|B|,|C|)=1$. | nvidia |
Find the number of distinct products that can be obtained by multiplying three or more different elements from the set {2, 3, 4, 7, 13}. | nvidia |
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