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The sum of the squares of two consecutive positive integers is 100. Find the value of their product. | nvidia |
The rectangle with the largest area that can be inscribed in a triangle with base length $b$ and height $h$ is a square, and its area is equal to $bh/2$. What is the ratio of the side length of this square to the side length of a square with the same area that is inscribed in the sector of a circle with angle $90^\circ... | nvidia |
Let $m$ be a real number such that $m > 1$ and
\[\sum_{n=1}^{\infty} \frac{3n+2}{m^n} = \frac{11}{3}.\]
Find $m$. | nvidia |
Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than $30$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$. | nvidia |
Given a regular octagon with side length 1 unit, the vertices are connected by line segments in such a way that they form triangles, with the additional condition that no more than two of these line segments can intersect at the same interior point. What is the maximum number of equilateral triangles formed within the ... | nvidia |
Let $ABCD$ be a square of side length 1, and let $E$ and $F$ be two points chosen randomly inside the square. What is the probability that the area of $\triangle EDF$ is less than the area of $\triangle EBF$? | nvidia |
The remainders of the sequence $a, a^2, a^3, \ldots, a^{100}$ when divided by 5 are written in a row. If $a$ is a positive integer such that the resulting sequence of remainders contains exactly 20 zeros, what is the number of possible values of $a$ modulo 25? | nvidia |
Let $\mathcal{T}_{n}$ be the set of sequences of positive integers with length $n$ such that the sum of any 3 consecutive numbers is at most 10. For example, the sequence 1, 2, 3, 4, 1 works, but the sequence 1, 3, 4, 2, 1 does not. Find the number of elements in $\mathcal{T}_{8}$. | nvidia |
How many integers $M$ less than $2000$ can be expressed as the sum of $k$ consecutive even positive integers from exactly 3 values of $k\ge 1$? | nvidia |
Let $x$ be a real number. Define
$$\binom{x}{0}=1, \qquad \binom{x}{n}=\frac{x(x-1)\dots (x-n+1)}{n!}, \qquad n=1,2,\dots.$$
Prove that for any positive integer $n,$
$$\sum_{k=0}^{n} (-1)^{n-k}\binom{n}{k}\binom{x+k}{n}=\begin{cases} 0& \text{if} & x=0,1,2,\dots,n-1\\ 1&\text{if}& x=n. \end{cases}$$ | nvidia |
Consider the congruence $7x \equiv 3 \pmod{25}$. Show that for every positive integer $k$, there exists a unique solution modulo $25^k$. Find all solutions modulo $125$. | nvidia |
Find the sum of all possible values of x for which the inequality $$|x-1| + |x+2| > 5$$ is satisfied. | nvidia |
Determine the value of $x$ such that $2\log_{3}{x}-\log_{3}{9}=2$. | nvidia |
Determine the number of positive integers $n \geq 2$ for which the expression $\sum_{i=1}^n i \cdot (i-1)$ results in a prime number. | nvidia |
The sum of the reciprocals of two consecutive positive integers is 13/84. What is their product? | nvidia |
Find the coordinates of the point that is equidistant from the vertices of a triangle with vertices at (1,2), (3,4), and (5,6), and lies on the line passing through the points (2,3) and (4,5). | nvidia |
Find the number of pairs $(m,n)$ of positive integers satisfying $m+n=2015$ and $m\mid n$. | nvidia |
A number is divisible by $9$ if the sum of its digits is divisible by $9.$ For example, the number $19\,836$ is divisible by $9$ but $19\,825$ is not.
If $D\,767\,E89$ is divisible by $9,$ where $D$ and $E$ each represent a single digit, what is the sum of all possible values of the sum $D+E?$ | nvidia |
What is the smallest positive integer $n$ such that there are exactly 25 positive integers congruent to 1 (mod $n$) between 1 and 500? | nvidia |
Find the number of real solutions $(a,b,c)$ to the system of equations:
$$a^3+b^3+c^3=3abc$$
$$a^2+b^2+c^2=1$$
$$ab+bc+ca=0$$
Note that all three equations are symmetric in $a,b,c$. Can you use this symmetry to simplify the problem? | nvidia |
Compute the number of ordered pairs of complex numbers $(z, w)$ with $|z|, |w| \leq 2$ such that $z^3 + w^4$ is a real number. | nvidia |
Find a nonzero $p$ such that $px^2-12x+4=0$ has only one solution. | nvidia |
Find a positive integer that is divisible by 14 and has a square root between 25 and 25.3.
Here's another problem:
Find a positive integer that is divisible by 11 and has a cube root between 7 and 7.2. | nvidia |
The sum of the roots of the equation $x^4 - 2x^2 - 1 = 0$ is 0, and the product of the roots is 1. Find the value of the sum of the squares of the roots of the equation. | nvidia |
Find the sum of the coefficients of the polynomial obtained by multiplying the binomial $x+2$ by the trinomial $x^2+x+1$. | nvidia |
Find the largest integer value of $m$ such that $m^2-7m+12$ is negative. | nvidia |
Let $n$ be a positive integer and consider the factorization of the polynomial
$x^n + x^{n - 1} + \dots + x + 1$
into irreducible polynomials with integer coefficients.
Find the smallest value of $n$ such that the polynomial has at least three distinct factors. | nvidia |
A cube is inscribed in a sphere with radius $r$. Find the maximum possible value of the volume of the cube. | nvidia |
The product of two positive integers plus their sum is 103. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | nvidia |
Find the smallest integer $n$ such that $\sqrt{3n} > 12$. | nvidia |
Determine the sum of all single-digit replacements for $y$ such that the number ${12{,}y54}$ is divisible by 9. | nvidia |
Find the largest possible value of $a$ such that the equation $ax^2+2x+3=x(2x+a)$ has no real solutions. | nvidia |
Let $m$ and $n$ be positive integers such that $\gcd(m,n)=16$ and $\text{lcm}(m,n)=576$. If $m<n$, what is the largest possible value of $n-m$? | nvidia |
What is the sum of all positive integer values of $n$ such that $\frac{n+18}{n}$ is an integer? | nvidia |
Find the smallest positive integer that is both an integer power of 11 and is not a palindrome. | nvidia |
Find the sum of the first seven prime numbers that have a units digit of 7. | nvidia |
Let $S$ be the sum of the squares of the first $50$ positive even integers. Find the largest integer $m$ such that $S$ is divisible by $2^m$. | nvidia |
Compute the smallest positive integer $n$ such that
\[\sum_{k = 0}^n \log_2 \left( 1 + \frac{1}{2^{2^k}} \right) \ge 1 + \log_2 \frac{2014}{2015}.\] | nvidia |
The integer $n$ is called social if there exists a set of positive integers, containing $n$, whose product is equal to $3^n$. Find the largest social integer. | nvidia |
In the expansion of $(a+b)^n$, where $a$ and $b$ are positive integers, the sum of the digits of the coefficients is $2n$. What is the largest value of $n$ such that there exists a pair of positive integers $a$ and $b$ that satisfy this condition? | nvidia |
Find the number of distinct positive integers that can be expressed as the sum of two different elements from the set {1, 2, 3,..., 20, 21, 22}. | nvidia |
Given that
\begin{eqnarray*}&(1)& x\text{ and }y\text{ are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& y\text{ is the number formed by reversing the digits of }x\text{; and}\\ &(3)& z=|x-y|. \end{eqnarray*}
How many distinct values of $z$ are possible?
| nvidia |
Find the sum of the four positive integers that are divisors of each number in the list $$48, 80, 24, 160.$$ | nvidia |
Find the sum of the infinite series:
$$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$$
if it is known that the sum of the squares of its terms is equal to $\frac{4}{3}$. | nvidia |
Given that the function $f(x) = \sin x + \cos x$ has a period of $2\pi$, find the smallest positive real number $a$ such that the function $g(x) = \sin(ax) + \cos(ax)$ has a period of $\pi$. | nvidia |
Find all 4-digit numbers of the form $abba$ that are divisible by 9, where $a$ and $b$ are digits. What is the maximum value of $a$? | nvidia |
Find the largest number of positive, consecutive integers whose sum is 21 and whose product is also a positive integer. | nvidia |
In three-dimensional space, find the number of lattice points that have a distance of 5 from the origin. | nvidia |
Determine the largest size of an equilateral triangle, of integer side length, that can be cut from a regular hexagon with side length 15 units. | nvidia |
Find the smallest positive integer $x$ such that the sum of its binary digits is 6. | nvidia |
Find the only positive real number $x$ for which $\displaystyle \frac{x-4}{9} = \frac{4}{x-9}$. | nvidia |
Problem:
A box contains 8 identical marbles, each marked with a different number from 1 to 8. If Anna randomly selects 4 marbles from the box, what is the probability that the sum of the numbers on the selected marbles is greater than the sum of the numbers on the remaining marbles? | nvidia |
The function $g(x)$ satisfies
\[g(x+y) = g(x) + \frac{y}{x}\]for all positive real numbers $x$ and $y.$ If $g(2) = 3,$ find $g(5).$ | nvidia |
In a certain arithmetic sequence, the sum of the first three terms is equal to the sum of the last three terms, and the sum of all six terms is 90. Determine the common difference of the sequence. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) + y) = g(x) + g(g(y) + g(-x)) - x^2\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the sum of all possible values of $g(2).$ Find $m + t^2.$ | nvidia |
Find the equation of the parabola that passes through the points $(2,0)$, $(3,2)$, and $(4,6)$. | nvidia |
In a regular octahedron, the centers of the eight triangular faces are the vertices of a smaller octahedron. The ratio of the surface area of the smaller octahedron to that of the larger is $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | nvidia |
Let $\mathbf{B}$ be a $3 \times 3$ matrix, with real entries, such that $\mathbf{B}^2 = \mathbf{B}.$ Find the number of different possible values that the determinant of $\mathbf{B}$ can be. If you think the answer is infinite, then enter "infinite". | nvidia |
Given that the point $(2,5)$ is on the graph of $y=g(x)$, there is one point that must be on the graph of $3y=2g(5x)-1$. What is the product of the coordinates of that point? | nvidia |
Find the sum of all positive integers between 1 and 100 that are not divisible by 2 or 3. | nvidia |
Let $p$ be the probability of getting heads when flipping a biased coin, and let $q=1-p$ be the probability of getting tails. Consider the sequence of results when flipping the coin $n$ times, where $n$ is a fixed positive integer.
We say that the sequence has a 'head-to-head' run if there is at least one pair of hea... | nvidia |
The average of $n$ consecutive positive integers is equal to $n$. What is the value of the largest integer? | nvidia |
A four-digit number $n$ is called a multiple suffix if there exists a two-digit number $m$ such that the last two digits of $n$ are equal to $m$ and $n$ is divisible by $m$. For example, 1236 is a multiple suffix because the last two digits are equal to 36, and 1236 is divisible by 36.
Find the smallest multiple suffi... | nvidia |
Determine the sum of all single-digit replacements for $z$ such that the number ${24{,}z38}$ is divisible by 6. | nvidia |
A cube is painted so that each face is a different color: red, white, blue, green, yellow, and orange. Three cuts are made through the cube so that each cut intersects all six faces exactly once. What is the minimum number of different colors that must be present on each of the three cross-sections created by these cut... | nvidia |
The equation of a certain curve is given by
$$xy = \frac{a(x^2+y^2)}{x+y} + b,$$
where $a$ and $b$ are parameters that vary over all real numbers.
Find the point that lies on this curve for any $a$ and $b$. | nvidia |
If $a$,$b$, and $c$ are positive real numbers such that $a(b+c) = 152$, $b(c+a) = 162$, and $c(a+b) = 170$, then find $abc.$ | nvidia |
Let $a_n$ be a sequence defined by
$$a_n=\sqrt{2+a_{n-1}}$$
with $a_0=2$. Let $S_n$ be the sum of the first $n$ terms of this sequence, that is,
$$S_n=a_0+a_1+a_2+\dots+a_{n-1}.$$
Find the value of $\frac{S_{2024}}{2024}$. | nvidia |
Find the smallest possible value of $x$ such that $3x^2+30x-90=x(x+19)$. | nvidia |
Find all real values of $x$ that satisfy $\frac{1}{x^2+1} + \frac{2}{x^2+4} + \frac{3}{x^2+9} \ge 1.$ (Give your answer in interval notation.) | nvidia |
The set of all the integers $x$ such that $x,$ $x+1,$ and $x+2$ are all prime numbers has exactly $k$ elements. What is the largest possible value of $k$? | nvidia |
Find the smallest positive integer $n$ such that $2n+3$ is a multiple of 5 and $3n-2$ is a multiple of 4. | nvidia |
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 3. If $\frac{S_{3n}}{S_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term. | nvidia |
If $x$ is a positive number such that \[\sqrt{9x}\cdot\sqrt{12x}\cdot\sqrt{2x}\cdot\sqrt{18x}=24,\]find all possible values for $x$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x + y) + xf(y) - xy - x + 1\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(1),$ and let $s$ be the sum of all possible values of $f(1).$ Find $n \times s.$ | nvidia |
Let $\mathbf{a}$ and $\mathbf{b}$ be two three-dimensional unit vectors such that the angle between them is $45^\circ.$ Let $\mathbf{c}$ be a vector that forms a rhombus with $\mathbf{a}$, $\mathbf{b}$, and the origin. Find the maximum possible value of the magnitude of $\mathbf{c}$. | nvidia |
Determine the number of integers $n$ that satisfy the inequality $\sqrt{2n-1} < n < \sqrt{2n+1}$.
(Note: This problem requires a different approach, as it involves finding integers that satisfy a specific inequality involving square roots, rather than simply counting integers between two fixed square roots.) | nvidia |
The vertices of a square $ABCD$ are represented by the complex numbers $a, b, ci,$ and $d$, where $|a|=|b|=|c|=|d|=1$ and $a+b+c+d=0$. Find the maximum possible value of $|a+b+d|$. | nvidia |
Determine the least possible value of a positive integer $c$ such that $c$ has $4$ factors and there exists a positive integer $d$ with $5$ factors, where $d$ is divisible by $c$ and $d-c$ is a perfect square. | nvidia |
Find the value of $x$ such that $\sin(x) = \cos(\frac{\pi}{6})$. | nvidia |
Find the largest possible value of $\mathrm{lcm}(7p,14q)$, given that $\mathrm{lcm}(p,q) = 24$, where $p$ and $q$ are positive integers. | nvidia |
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2005 \left( \frac{1}{x} + \frac{1}{y} + 2004 \right)\]for all $x,$ $y > 0.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times... | nvidia |
The function $f(x)$ satisfies
\[f(xy) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(0) \neq 0.$ Find $f(10).$ | nvidia |
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2005 \left( \frac{1}{x} + \frac{1}{y} + 2004 \right)\]for all $x,$ $y > 0.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times... | nvidia |
For each positive integer $n$, let $f(n)$ denote the unique positive integer $m$ such that $|m-\sqrt[3]{n}|<\frac{1}{3}$. For example, $f(8)=2$ and $f(30)=3$. Find $T=\sum_{n=1}^{2015} f(n)$. | nvidia |
Let $x$ be a real number and $A$ a positive integer such that
$$x^2 \cdot x^4 \cdot x^6 \cdots x^{2A} = 2010^2.$$
Determine the sum of all possible values of $x.$ | nvidia |
What is the sum of all the positive three-digit integers that remain unchanged when their digits are rotated by one position? For example, 137 remains unchanged when its digits are rotated: 371 -> 713 -> 137.
Find the sum of all such integers. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 4f(x) y\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | nvidia |
The decimal representation of the number $\frac{1}{4!5!6!7!}$ terminates. Find the smallest positive integer $n$ such that $\frac{1}{4!5!6!\cdots n!}$ has a non-terminating decimal representation. | nvidia |
Find the largest value of $x$ such that the equation
$\cos x + \cos 2x = 1$ holds true. | nvidia |
Let $a$, $b$, and $c$ be the roots of the cubic polynomial $P(x)=x^3-x^2-5x-1$. Determine the smallest positive integer $n$ such that $P(n) \equiv 0 \pmod{a+b+c}$. | nvidia |
Let $g(x)$ be a function defined for all real numbers satisfying the conditions $g(x) \neq 0$ for all $x$ and
\[g(x + y) = \frac{g(xy)}{g(x) + g(y)}\]for all $x$ and $y.$ Determine $g(2010).$ | nvidia |
Find all positive values of $d$ such that the inequality $x^2-4x+d<0$ has real solutions for $x$. Express your answer in interval notation. | nvidia |
Let $m$ be a real number such that $m > 2$ and
\[\sum_{n=1}^{\infty} \frac{3n+2}{m^n} = \frac{11}{3}.\]Find $m.$ | nvidia |
A palindrome is a number that remains the same when its digits are reversed, such as 121 or 12321. In a certain base b, the 3-digit palindrome $aba_b$ is equal to $b^2+2$. Find the largest possible value of $a$ that can occur in such a palindrome. | nvidia |
What is the sum of all the coordinates of all the points of intersection of the two circles defined by $x^2-4x +y^2-6y+13=0$ and $x^2-10x+y^2-6y+29=0$? | nvidia |
Determine the number of pairs $(a,b)$ of real numbers such that $10, a, b, ab$ is an arithmetic progression. | nvidia |
If $a$, $b$, $c$ are three distinct positive numbers in a geometric progression such that $a+b+c=24$, find $\ln\left(\frac{b^{2}}{ac}\right)$. | nvidia |
Find the values of x that satisfy the equation:
$$3x^3-7x^2+2x+\sin x=0$$
where x is a real number. | nvidia |
What is the smallest positive integer $k$ for which there exists a positive integer $n$ such that $k^n$ can be expressed as the sum of exactly two perfect squares? | nvidia |
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