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A sequence of positive integers has the property that the product of the first n terms is less than the product of the last n terms for all n, where n < 6. What is the maximum product of the six positive integers in the sequence? | nvidia |
A line through the origin intersects the graph of the parabola $y=x^2$ at two points. Determine the equation of the line through the origin such that the vectors from the origin to these two points have equal projections onto the vector $\begin{pmatrix} 1 \\ 2 \end{pmatrix}.$ | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $a<b<c$. Compute the number of possible combinations of these integers such that $a+b+c=6$. | nvidia |
Given that $z$ is a complex number satisfying the equation $z + \frac{1}{z} = r$, where $r$ is a real number and $|r| < 2$, we need to find $|z|$.
A new problem inspired by this one could be:
Let $w$ be a complex number such that $w^2 + \frac{1}{w^2} = s$, where $s$ is a real number and $|s| < 1$. Find $|w|$. | nvidia |
What is the number of 10-digit positive integers that have all digits either 3 or 4, and have exactly three 3's? | nvidia |
Find the product of the squares of the roots of $2x^4+5x^3-6x^2-8x+3=0$. | nvidia |
An $\textit{abundant number}$ is a positive integer, the sum of whose distinct proper factors is greater than the number. (The $\textit{proper factors}$ of a number are all of its factors except the number itself.) How many numbers less than 25 are abundant numbers? | nvidia |
Let $a$ be a positive real number such that
\[\sum_{k=0}^{\infty} \arctan\left(\frac{a}{2^{2^k}}\right) = \arctan 2014.\]
Find $\log_2 (a+1)$. | nvidia |
Find the value of $x$ that satisfies the equation $\sin x + \cos x = \sqrt{2}$, where $x$ is in radians. | nvidia |
Find the smallest positive integer that is a multiple of 7 and has exactly three distinct prime factors. | nvidia |
A certain sequence consists of 20 consecutive positive integers, the sum of which is 1150. The sequence is split into two sub-sequences: one consisting of the 4 smallest integers, and the other consisting of the remaining 16 integers. If the sum of the integers in the first sub-sequence is x, and the sum of the integer... | nvidia |
Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial such that the sum of its roots is 0, the sum of the products of the roots taken two at a time is $-15,$ and the sum of the cubes of the roots is $-27.$ Find the value of $a.$ | nvidia |
Find the range of the real-valued function $f(x)=\frac{x}{x-3}$ that takes $x$ from the set of all real numbers except $x=3$. | nvidia |
Find the smallest positive integer n such that the product of its proper divisors is greater than n^2. | nvidia |
If $a$ and $b$ are positive integers such that $\text{lcm}(a,b) = 18$, then what is the largest possible value of $\text{lcm}(2a,3b)$? | nvidia |
Given that the arithmetic mean of an odd number of consecutive odd integers is $y$, find the difference between the largest and smallest of the integers in terms of $y$ and the number of integers. | nvidia |
Let $g(x) = |3\{x\} - 2|$. Find the smallest positive integer $m$ such that the equation $mg(xg(x)) = x$ has at least $2024$ real solutions.
Note: the fractional part of $x$ is a real number $y=\{x\}$ such that $0\le y<1$ and $x-y$ is an integer. | nvidia |
Find the smallest positive integer greater than 50 that is coprime with 15. | nvidia |
Determine the Cartesian coordinates of the point $P$, given that its polar coordinates are $(4,\arctan 2)$. Express your answer in the form $(x,y)$, where $x$ and $y$ are real numbers. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x + y = 1$. Find the maximum value of $\frac{x^2}{1-y} + \frac{y^2}{1-x}$. | nvidia |
A right triangle with integer leg lengths is called "unique" if the number of square units in its area is equal to three times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of unique right triangles? | nvidia |
Find the smallest integer $k$ such that for any positive integer $n$, the number $\mathop{\text{lcm}}[n, k]$ is a multiple of $\mathop{\text{lcm}}[n, 21]$. | nvidia |
Find the largest possible degree of the polynomial $p(x)$ such that the graph of the rational function $\frac{p(x)}{x^3-2x^2+3x+1}$ has a slant asymptote. | nvidia |
Let $a$ and $b$ be positive real numbers. Find the minimum value of $\frac{\sqrt{a+b}}{\sqrt{a}}+\frac{\sqrt{a+b}}{\sqrt{b}}$. | nvidia |
The product of two positive integers $x$ and $y$ is 300, and the sum of their reciprocals is $\frac{1}{5}$. What is the maximal possible value of $x + y$? | nvidia |
Find the value of $a$, given that the sum of the third and fourth terms of the arithmetic sequence $1, 1+a, 1+2a, 1+3a, \ldots$ is twice the sum of the first two terms. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 1.$ Find the minimum value of
\[\frac{x}{y} + \frac{y}{x}.\] | nvidia |
Consider a function $g : \mathbb{Z}^+ \to \mathbb{Z}^+$ such that for any positive integers $m$, $n$, and $p$,
\[g(m \times n + p) = g(m) + g(p)\times g(n).\]
Given that there exists a unique positive integer $a$ such that $g(a) = a$, determine the possible values of $a$. | nvidia |
Find the number of positive integers less than or equal to 1000 that are not divisible by 2, 3, or 5. | nvidia |
A rectangle $ABCD$ is inscribed in a circle with center $O$ and radius $r$. The diagonals $AC$ and $BD$ of the rectangle intersect at point $E$, which lies on the positive $y$-axis. The length of the diagonal $AC$ is $2r$, and the length of the side $AB$ is $2r-1$. Find the perimeter of the rectangle $ABCD$. | nvidia |
Find the polynomial $p(x),$ with real coefficients, such that
\[p(x^3) - p(x^3 - 2) = [p(x)]^2 + 12\]for all real numbers $x.$ | nvidia |
Find the value of $x$ that satisfies the equation:
$$\frac{x+3}{x+5}-\frac{2}{x-2}=1$$ | nvidia |
Alice chose a positive integer and found that the sum of its digits was divisible by 3. What is the maximum number of digits the integer could have if Alice wanted the sum of its digits to be as small as possible?
Answer: 2.
Explanation: The smallest possible sum of digits divisible by 3 is 3. This can be achieved wi... | nvidia |
Let $P$ be a point inside a tetrahedron $ABCD$ such that the volume of $PBCD$ is $\frac{1}{6}$ of the volume of $ABCD$. Find the ratio of the area of the tetrahedron's cross-section at $P$ to the area of the base $BCD$. | nvidia |
Find the largest positive integer that divides the sum of any three consecutive positive integers. | nvidia |
How many four-digit numbers $n$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $n$? | nvidia |
There are three real numbers $x$ that are not in the domain of $$f(x) = \frac{1}{1+\frac{1}{1+\frac 1x}}.$$ What is the sum of those three numbers? | nvidia |
How many positive integers $n$ have the property that $\frac{1}{n}$ can be expressed as a repeating decimal with a period of exactly 4 digits and a non-zero ten-thousands digit? | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(0) = 0$ and
\[f(x^3 + y^3) = (x + y) (f(x)^2 - f(y)^2)\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $f(1),$ and let $t$ be the product of all possible values of $f(1).$ Find $m + t.$ | nvidia |
Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = 60$ and $P(x)$ is divisible by $(x-1)^2.$ Find the number of divisors of $P(1)$ that are not divisible by 3. | nvidia |
A jar contains quarters, nickels, and pennies. The value of the quarters is $10.00, the value of the nickels is $10.00, and the value of the pennies is $10.00. If the number of pennies is 100 more than the number of quarters, and the number of nickels is 10 more than the number of quarters, how many quarters are in the... | nvidia |
Given that $a$ and $b$ are positive integers, find the number of factors that the expression $a^4 \cdot b^4$ has, where $a$ is the smallest prime number greater than 10 and $b$ is the largest prime number less than 20. | nvidia |
The real and imaginary parts of a complex number z are both positive integers. If the sum of these two parts is 7, find the smallest possible value of |z|. | nvidia |
Find the largest value of $n$ such that $3x^2 +nx + 72$ can be factored as the product of two linear factors with integer coefficients.
New problem:
Consider a quadratic polynomial of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are integers. Suppose that the polynomial can be factored as the product of two line... | nvidia |
Let $g$ be a function defined on the positive integers, such that
\[g(xy) = g(x) + g(y)\]for all positive integers $x$ and $y.$ Given $g(15) = 12$ and $g(90) = 18,$ find $g(1350).$ | nvidia |
If $x$ and $y$ are two positive numbers such that $x^2+y^2=\frac{7}{12}$ and $xy=\frac{1}{12}$, find the value of $\frac{x^3+y^3}{x^2-y^2}$. | nvidia |
Find the sum of all real values of $x$ that satisfy
\[x = 1 - x + x^2 - x^3 + x^4 - x^5 + \dotsb.\]
Another problem inspired by this one:
A sequence is defined by $a_0 = 0$ and $a_n = 2a_{n-1} + n$ for $n \ge 1$. Find the sum of all real values of $x$ that satisfy $a_x = 2x$. | nvidia |
Count the number of non-decreasing sequences of four distinct digits that can be formed using the digits $1,2,3,4,5,6$ if the sequence cannot start with $1$. | nvidia |
The geometric mean of three numbers is 12. One of the numbers is 8, and the other two are in a ratio of 2:3. What is the sum of the two numbers that are in a ratio? | nvidia |
Let $A_n$ denote the sum of the first $n$ positive odd integers. Find the largest positive integer $m$ such that for all positive integers $n$,
$$\frac{A_n}{\gcd (A_{2n}, A_{3n})}$$ is an integer. | nvidia |
A sequence of real numbers $a_1,a_2,a_3,...$ has the property that for every positive integer $k$ greater than or equal to $3$, $a_k=a_{k-1}^2-a_{k-2}^2$. What is the smallest positive integer $n$ such that there exists a sequence that satisfies the given property and $a_n<a_{n-1}$? | nvidia |
Find the product of all positive integers that are divisors of both 120 and 210. | nvidia |
Circle $A$ has its center at $(0, 0)$ and a radius of $5$ units. What is the sum of the $x$-coordinates of the two points on circle $A$ that are also on the line $y = 3x + 2$? | nvidia |
Given the equation of a parabola $y = ax^2$, determine the value of $a$ such that the area enclosed by the parabola and the line $y = 2$ is equal to $\frac{\pi}{2}$ square units. | nvidia |
A rectangle is divided into four smaller rectangles by two lines that intersect at a point inside the rectangle. The area of the top-left rectangle is 12 square units, the area of the top-right rectangle is 20 square units, and the area of the bottom-left rectangle is 15 square units. What is the area of the bottom-rig... | nvidia |
Find the sum of the infinite series $1+2\left(\dfrac{1}{1998}\right)+3\left(\dfrac{1}{1998}\right)^2+4\left(\dfrac{1}{1998}\right)^3+\cdots$. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R} $ be defined by $ g(x) = x^3 + 2x^2 + 3x + 1 $. How many real numbers $x $ are there such that $ x > 0 $ and $g(x)$ can be expressed as the sum of two perfect squares? | nvidia |
If $a$ is a positive real number, then express the following expression in terms of $a$:
$$\frac{\left(\sqrt{a}+3\right)^2}{\sqrt{a}-3}\cdot\frac{\sqrt{a}-3}{\sqrt{a}+3}\cdot\frac{1}{\sqrt{a}+3}$$ | nvidia |
Determine all possible values of the parameter $a$ so that the equation
$$ax^3+2x^2-5x-1=0$$
has at least one rational root. | nvidia |
Find the smallest positive four-digit number that is divisible by each of the four smallest prime numbers. | nvidia |
Let $\mathbf{P}$ be a $3 \times 3$ matrix, with real entries, such that $\mathbf{P}^2 = \mathbf{I}.$ Find the number of different possible values that the trace of $\mathbf{P}$ can have. | nvidia |
A circle is inscribed in a square with side length 6. Find the radius of the circle such that the sum of the areas of the four regions outside the circle is a minimum. | nvidia |
A real number $b$ is chosen randomly and uniformly from the interval $[-15, 20]$. Find the probability that the roots of the polynomial
\[x^3 - 3bx^2 + (3b + 1)x - 1\]are all real. | nvidia |
The vertices of a triangle are $(2,4)$, $(7,6)$, and $(5,1)$. Reflect this triangle across the line $y=x$. Find the sum of the coordinates of the point which is the image of the vertex $(2,4)$, after this reflection. | nvidia |
Given an equilateral triangle with side length 20, find the sum of the squares of the distances from the vertices to the orthocenter of the triangle. | nvidia |
A circular table has 15 seats, numbered from 1 to 15. A certain number of people are seated around the table, such that the sum of the seat numbers of every pair of people sitting next to each other is always a perfect square. What is the maximum number of people that can be seated around the table? | nvidia |
Let $x,y,z$ be positive real numbers such that $x+y+z=6$ and $xy+yz+zx=8$. Let $n=\max\{xy,yz,zx\}$. Find the smallest possible value of $n$. | nvidia |
What is the largest possible value of $n$ such that the decimal expansion of $1/n$ has a repeating block of length at most 4? | nvidia |
What is the sum of the digits of the sum of all natural numbers from 1 to 99, inclusive? | nvidia |
Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a+b+c=32.$ Find the largest possible value of $b.$ | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $abc = 1.$ Find the minimum value of
\[a^2 + b^2 + c^2.\] | nvidia |
In a three-dimensional space, a line segment with endpoints $A(2,0,-3)$ and $B(-3,2,5)$ intersects a paraboloid defined by the equation $z=x^2+y^2$ at two points. Find the area of the triangle formed by these two intersection points and the origin. | nvidia |
Find the largest positive real number $a$ such that
\[|x|^a + |y|^a \ge \frac{|x+y|^a}{a}\]for all real numbers $x$ and $y.$ | nvidia |
Find all $x$ for which the expression $\arcsin \left( \frac{1}{\sqrt{3}} \cdot \cos x \right) + \arcsin \left( \frac{1}{\sqrt{3}} \cdot \sin x \right)$ is defined. | nvidia |
The area of the smallest circle that will contain a square of side length 4 is what? | nvidia |
Find the only real number that can be expressed in the form \[(a + bi)^3 - 107i,\]where $i^2 = -1,$ and $a$ and $b$ are positive integers. | nvidia |
Determine the complex number $z$ satisfying the equation $2z-3\bar{z}=-2-30i$. Note that $\bar{z}$ denotes the conjugate of $z$. | nvidia |
Find the number of solutions for the congruence $y - 7 \equiv 21 \pmod{15}$, given that $y$ is a positive integer less than 200. | nvidia |
Find all values of x such that ⌈x⌉ - x = 0.8. | nvidia |
The area of the smallest square that will contain a circle of radius 4 is what? | nvidia |
Let $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $? | nvidia |
Find the range of all possible values of $k$ such that the equation $x^2 + kx - 3 = 0$ has two real roots $x_1$ and $x_2$ satisfying the condition $|x_1| + |x_2| < 5$. | nvidia |
Find the sum of three consecutive odd integers if the sum of the first and third integers is 210. | nvidia |
Find the largest positive integer $m$ such that $m^3$ can be expressed as the difference of two squares and $5m + 2$ is a perfect cube. | nvidia |
Find a positive integer $m$ such that $m!$ is divisible by $2^{10} \cdot 5^3$. | nvidia |
When three fair coins are flipped, the numbers $x,y,z$ are obtained. Find the probability that $x+y+z = 5$. | nvidia |
Determine the maximum number of consecutive zeros that can occur at the end of a perfect square. | nvidia |
In a right triangle with integer length sides, the hypotenuse has length 39 units. How many units is the length of the shorter leg?
Consider a right triangle with integer length sides, where the hypotenuse has length 39 units. Now, instead of finding the length of the shorter leg, find the largest possible integer val... | nvidia |
Let $Q$ be the product of the first 150 positive even integers. Find the largest integer $m$ such that $Q$ is divisible by $2^m$. | nvidia |
Let $f(n)$ be the sum of the proper positive integer divisors of $n$. For how many values of $n$ is $f(n)$ greater than $n$, given that $3 \le n \le 60$? | nvidia |
What is the smallest positive integer that can be expressed as the sum of exactly 5 distinct prime numbers? | nvidia |
Consider a regular $n$-sided polygon with $n$ even, and let $k$ be a positive integer less than $n$ that is relatively prime to $n$. A $k$-diagonal of the polygon is a line segment connecting two vertices that are $k$ vertices apart. How many non-overlapping $k$-diagonals can be drawn in the polygon such that no two di... | nvidia |
Problem:
A coin is flipped repeatedly. The probability that the first $k$ flips are all heads is $P_k$. Find the smallest positive integer $n$ such that the probability that the first $n$ flips are all heads and the $(n+1)$th flip is tails is greater than $\frac{P_n}{2}$. | nvidia |
Find the number of functions $f(n),$ taking the integers to the integers, such that
\[f(a + b) + f(ab) = f(a) f(b) + 1\]for all integers $a$ and $b.$ | nvidia |
A triangle is formed by three points in the plane, each of which has a positive integer coordinate. If the sum of the absolute values of the differences of the $x$-coordinates is equal to the sum of the absolute values of the differences of the $y$-coordinates, find the smallest possible area of such a triangle. | nvidia |
Find the area of a regular hexagon inscribed in a circle of radius $5$ cm, knowing that the distance from the center of the circle to the midpoint of a side of the hexagon is $4$ cm. | nvidia |
Let $x$ be a positive integer such that the three-digit number $4x3$ is divisible by 9. What is the smallest possible value of $x$? | nvidia |
Find the smallest positive integer $b$ such that $b$ is a multiple of $3$ and the sum of the digits of $b$ is a multiple of $5$. | 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 |
What is the sum of the squares of the positive integers from 1 to 100? | nvidia |
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