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Find the number of positive integers n such that the sum of the squares of the lengths of the sides of a triangle with sides of length n+1, n+2, and n+3 is less than 500.
Note: The generated problem requires the application of the triangle inequality theorem and properties of quadratic expressions, which is different ... | nvidia |
Determine all real numbers $k$ for which the inequality
\[2x^2 + 5x + k \ge 0\]
holds for all $x$ in the interval $[-1,2].$ | nvidia |
Find the value of $y$ such that the equation $\displaystyle\sqrt[4]{y - \frac{x^2}{4}} = 3$ has exactly one solution. | nvidia |
Find all real numbers $a$ such that
$$\sqrt{a+\sqrt{a^2+3}} - \sqrt{a-\sqrt{a^2+3}} = \sqrt{3}.$$ | nvidia |
Given the function $f(x) = \lfloor x \rfloor + \{2x\}$, find the smallest positive integer $k$ such that $f(x) = f(x+k)$ for all real numbers $x$. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $xyz=1$. Find the minimum value of $x^2+y^2+z^2+xy+yz+zx$. | nvidia |
When a positive integer is divided by 12, the remainder is 4. Find the sum of the squares of the remainders when the same integer is divided by 6 and by 8. | nvidia |
Determine the smallest positive integer with exactly seven positive divisors. | nvidia |
Find the positive value of $m$ such that the equation $4x^2 + mx + 9 = 0$ has exactly one solution in $x$. | nvidia |
Compute the number of ways to arrange 5 identical red balls and 5 identical blue balls in a row, with the restriction that no two balls of the same color can be adjacent. | nvidia |
Find the largest positive real number $k$ such that for all nonnegative real numbers $x$, $y$, and $z$,
\[(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \ge 9 + k \left(\frac{x-y}{x+y}+\frac{y-z}{y+z}+\frac{z-x}{z+x}\right).\] | nvidia |
Find the value of $x$ that satisfies the equation $\cos^2 x + 2\sin^2 x = 1$, where $x$ is an acute angle. | nvidia |
If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? | nvidia |
Let $g(n)$ be a function that, given a positive integer $n$, returns the smallest positive integer $k$ such that $k^n$ is a multiple of $n!$. What is the smallest $n$ for which $g(n) > n$? | nvidia |
Find the range of values of $a$ for which the equation $\cos(\arcsin(x)) + \sin(\arccos(a-x)) = 1$ has at least one real solution. | nvidia |
In a garden, two paths intersect at right angles, forming four right triangles. The hypotenuse of each triangle is 5 meters long, and the lengths of the legs of the triangles are integers. Find the ratio of the areas of the two triangles with the largest and smallest areas. | nvidia |
Let's consider the sequence of positive integers such that each term is equal to the sum of its proper divisors (excluding the number itself). Find the smallest positive integer that appears twice in this sequence. | nvidia |
Find the largest integer value of $n$ such that $n^2-9n+18$ is negative. | nvidia |
Let's say that you have a square of a binomial
$$(x+b)^2=x^2+2bx+b^2.$$
Determine $b$ such that the product of the coefficients in the binomial expansion of this expression is equal to 36.
Note that the product of the coefficients in the binomial expansion of the given expression is $1\times 2b\times b^2=2b^3.$ Then... | nvidia |
Find the number of distinct prime factors of the number $N$, where $N$ is the product of all positive integers less than or equal to 10. | nvidia |
Let $a$ be a real number such that $a>0$ and
\[\sum_{n=1}^{\infty} n \cdot \frac{a^n}{2^n} = 6.\]
Find $a$. | nvidia |
Given that points $M,$ $N,$ and $P$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC}$ of triangle $ABC$ such that $\overline{MP}$ and $\overline{NP}$ divide triangle $ABC$ into four triangles of equal area, find the vector equation for point $P$ in terms of $\overrightarrow{A},$ $\overrightarrow{B},$ a... | nvidia |
Let $P(x)$ be a polynomial such that
$$P(P(x)) = (x + 2)^2.$$
Find all values of $x$ such that $P(x) = x + 2.$ | nvidia |
Let $n$ be a positive integer. Find the sum of all numbers of the form $\frac{1}{k}$ where $k$ ranges from $-n$ to $n$, excluding $k = 0$. | nvidia |
Find the area of the triangle with vertices $(3,-5),$ $(-2,0),$ and $(1,-6).$ | nvidia |
In a quadrilateral $ABCD$, the measures of two adjacent angles $A$ and $B$ are in the ratio $3:4$. The measures of the other two angles $C$ and $D$ are in the ratio $2:3$. If the sum of the measures of angles $A$ and $C$ is $150^\circ$, find the measure of the largest angle. | nvidia |
In a triangle $PQR,$ the side lengths $p,$ $q,$ and $r$ satisfy the following condition: when the altitudes of the triangle are drawn, the lengths of the three segments into which each altitude divides its opposite side are all integers. Furthermore, the length of each altitude, multiplied by the length of the segment... | nvidia |
Two integers $a$ and $b$ are such that $a+b=10$ and $a^2+b^2=52$. Compute $a^3+b^3$. | nvidia |
A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is t... | nvidia |
The real function $f$ has the property that, whenever $a,$ $b,$ $n$ are positive integers such that $a + b = 2^n,$ the equation
\[f(a) + f(b) = n^2\]holds. What is $f(2002)$? | nvidia |
Let $g : \mathbb{C} \to \mathbb{C} $ be defined by $ g(z) = z^2 - 2z - 3 $. How many complex numbers $z $ are there such that $ |z| < 2 $ and both the real and the imaginary parts of $g(z)$ are integers with absolute value at most $ 5 $? | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $xy = 48\sqrt[3]{3}$, $xz = 72\sqrt[3]{3}$, and $yz = 32\sqrt[3]{3}$. Determine the value of $xyz$. | nvidia |
Determine the smallest integer value of $y$ such that the solution set of the inequality $3-2y<11$ contains at least one integer. | nvidia |
What is the largest four-digit number that can be formed using the digits 1, 2, 3, and 4, with each digit used exactly once, and such that the product of its digits is maximized? | nvidia |
Find the number of positive integers less than 100 that are divisible by neither 3 nor 7. | nvidia |
Determine the value of $x$ in the equation $2^{x/3} + 7 = 11$, where $x$ is a positive integer. | nvidia |
Count the number of positive integers $n$ that can be expressed in the form $n = a^2b^3$, where $a$ and $b$ are positive integers and $a$ and $b$ are both less than or equal to 12. | nvidia |
What is the smallest positive integer $n$ such that the $n^{\text{th}}$ triangular number is a perfect square? | nvidia |
The graph of the function $g(x) = 2x + c$ intersects with its inverse function $g^{-1}(x)$ at the point $(2, d)$. Given that $c$ and $d$ are integers, find the value of $d$. | nvidia |
What is the number of positive integers less than or equal to 48 that have an odd number of positive factors? | nvidia |
What is the sum of all positive integers $a$ for which $\mathop{\text{lcm}}[a,24]=72$? | nvidia |
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x$, $y$, and $r$ with $|x|>|y|$,
\[(x+y)^r=x^r+rx^{r-1}y^1+\frac{r(r-1)}2x^{r-2}y^2+\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\cdots\]What are the first three digits to the right of the decimal point in the decimal representation... | 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 |
Find the largest value of $y$ that satisfies the equation $|2y-11|=22$. | nvidia |
The function $g(x)$ maps positive real numbers to real numbers such that
\[x^2g(y) - yg(x) = g(xy)\]for all positive real numbers $x$ and $y.$ Find all possible values of $g(25).$ | nvidia |
Three standard dice are tossed, and the numbers $a,b,c$ are obtained. Find the probability that $a,b,c$ form an increasing sequence. | nvidia |
Let $a$ and $b$ be positive integers such that $a^3+b^3$ is divisible by $a+b$. What is the largest possible value of $\frac{a^3+b^3}{a+b}$? | nvidia |
Given a rectangle with integer side lengths, if the area of the rectangle is 120 square units and the length of its diagonal is less than 21 units, what are the possible dimensions of the rectangle? | nvidia |
Find the sum of the sequence $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots+\frac{1}{1024}$. | nvidia |
Find the real number $a$ such that the equation
$$\sin^2(x) + \cos^2(x) + \sin(x) \cos(x) \tan(x) = a$$
holds for all real numbers $x$. | nvidia |
Determine the largest positive integer $m$ such that the sum of the digits of $2^m$ is congruent to $m^2$ modulo $7$. | nvidia |
A quadratic equation $ax^2 + bx + c = 0$ has two solutions that are reciprocals of each other. If the sum of the solutions is $5$, determine the value of $a/c$. | nvidia |
Find the sum of the first five prime numbers that have a units digit of 3. | nvidia |
Palindromes are numbers that read the same backwards and forwards, like 5665. What is the least possible positive four-digit palindrome that is divisible by 3? | nvidia |
Find the sum of all integers $m$ such that $\binom{17}{3} + \binom{17}{4} = \binom{18}{m}$. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a + b + c = 1$. Find the minimum value of
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c}.\] | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 4.$ Find the minimum value of
\[x + \frac{1}{y}.\] | nvidia |
The smallest positive integer that has exactly eight distinct positive factors is 24.
Here is another problem:
What is the largest positive integer that has exactly four distinct positive factors? | nvidia |
Determine all real numbers $x$ such that $\lfloor x\rfloor^2-\lceil x\rceil^2=-7$. | nvidia |
Find all possible positive integral values of $b$ such that the only integer solution to the equation $|x-2|+|x+1|=b$ is $x=1$. | nvidia |
A planar graph $G$ has $18$ vertices, $40$ edges, and $22$ faces, where each face is a triangle or a pentagon. A chord of $G$ is a line segment connecting two non-adjacent vertices that lie on the same face. Determine the maximum number of chords that $G$ can have. | 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 |
A bookshelf has two types of books: fiction and non-fiction. The number of fiction books, in boxes, is at least 8 boxes more than one-third the number of non-fiction books, and is no more than three times the number of non-fiction books. Find the least number of boxes of non-fiction books that can be placed on the book... | nvidia |
Determine the value of $x$ that satisfies $\sqrt[5]{x\sqrt{x^3}}=3$. | nvidia |
What is the smallest positive integer $n$ such that the sum of the digits of $n^{2018}$ is divisible by $5$? | nvidia |
Find the sum of the positive whole number divisors of 945 that are not multiples of 3. | nvidia |
The set of all possible scores that can be obtained by throwing a certain number of darts is the set of all integers from $0$ to $2007$. Each score can be obtained by throwing a specific number of darts in such a way that each dart scores a certain number of points, and no dart scores the same number of points as anoth... | nvidia |
Find the sum of all positive integers less than 100 that are cubes of numbers whose squares are also perfect squares. | nvidia |
What is the least positive integer $n$ such that $n^2$ is divisible by 7979? | nvidia |
All the angles of the quadrilateral are equal. Find the sum of the squares of the lengths of the sides of the quadrilateral if it is known that if we write the length of the sides of the quadrilateral in increasing order we will obtain a geometric progression. | nvidia |
Let $f(x)$ be a polynomial of degree $n$ with rational coefficients, and let $a = \frac{\pi}{2015}$. Find the smallest positive integer $n$ such that
\[f(0) + 2\sum_{k=1}^n f(k)\cos(ka)\sin(ka)\]
is an integer. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x + y = 1$. Find the maximum value of
\[\frac{x^2y^2}{x^2 + y^2}.\] | nvidia |
Find the largest value of m such that 7x^2+mx+90 can be factored as the product of two linear factors with integer coefficients. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x + y = 2$. Find the set of all possible values of $\frac{1}{x} + \frac{1}{y}$. | nvidia |
Determine the largest positive integer that can be added to 987,670 such that the resulting number is still divisible by 128. | nvidia |
A triangle with vertices $A$, $B$, and $C$ is inscribed in a circle with radius $r$. The area of the triangle is equal to half the area of the circle. The centroids of the triangle's vertices form another triangle with area $K$. Determine the value of $K$ in terms of $r$. | nvidia |
Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.
| nvidia |
Find the smallest positive integer $n$ such that the equation $\lfloor x \rfloor = n + \frac{1}{\{x\}}$ has a solution, where $\{x\} = x - \lfloor x \rfloor.$ | nvidia |
The equation $x^3 + 2x^2 + 3x + 1 = 0$ has a real root. Determine the sum of the squares of all roots. | nvidia |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 1000^\circ.$ | nvidia |
Find the sum of all the terms in the third row of Pascal's triangle that are divisible by 3. | nvidia |
Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{2y} = \frac{1}{7}$, what is the least possible value of $xy$? | nvidia |
What is the sum of all positive integers less than 1000 that are cubes of perfect cubes? | 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 |
Determine the smallest positive integer $n$ such that $2^n \equiv n^2 \pmod{5}$. | nvidia |
Find the number of positive integers $n \le 1000$ such that $15n$ is a perfect cube. | nvidia |
Let $x,y,z$ be positive integers with $y>1$, such that $z-y\leq y$ and
$$\frac{x+z}{y}=\frac{x}{10y}+10+z.$$Find the greatest possible value of $y$. | nvidia |
For how many ordered pairs of positive integers $(x,y),$ with $x \le 50$ and $y < x,$ are both $\frac{x}{y}$ and $\frac{x+2}{y+2}$ integers? | nvidia |
In a cube, the sum of the lengths of all edges is numerically equal to the surface area. Determine the length of one edge. | nvidia |
How many four-digit positive integers have at least one digit that is a 2 or a 3? | nvidia |
Compute the number of ways to arrange the letters in the word " Mississippi" such that the two "s"s are next to each other. | nvidia |
Let $m$ and $n$ be two positive integers such that the greatest common divisor of $m$ and $n$ is 15. If $n$ is divisible by 45, what is the least possible value of $m$? | nvidia |
A positive integer $n$ is divisible by 99 if and only if the sum of the digits of $n$ in blocks of two digits from right to left is divisible by 99. For example, 514623 is divisible by 99 because $63 + 42 + 51 = 156$ is divisible by 99. What is the largest integer less than 400 that is divisible by 99? | nvidia |
Count the number of ways to arrange the letters in the word “eleven” so that no two e’s are consecutive. | nvidia |
Find the sum of all positive real numbers x that satisfy the following system of equations:
\begin{align*}
2^x + 3^x &= 2 \cdot 5^x, \\
2^{2x} + 3^{2x} &= 2 \cdot 5^{2x}.
\end{align*} | nvidia |
The Fibonacci sequence $(F_n)$ is defined recursively by $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 2$. What is the smallest positive integer $k$ such that the sum $F_1+F_2+\cdots+F_k$ is a perfect square? | nvidia |
In a set of positive integers, each number is of the form $2^a \cdot 3^b \cdot 5^c$, where $0 \leq a, b, c \leq 5$. If the set contains exactly 20 numbers that are multiples of 6, how many numbers in the set are multiples of 15? | nvidia |
Determine all real numbers $a$ for which the equation
\[\sin (x^3 + a^3)^\circ = 0\]
has exactly two distinct solutions in the interval $[0, 2\pi]$.
One approach to solve this problem is to first notice that $x^3+a^3$ must be an integer multiple of 180, since the sine of an angle is 0 only when the angle is an integer... | nvidia |
The sum of the squares of the first 10 positive integers is equal to the sum of the squares of the first n negative integers. Determine n. | nvidia |
Let $x$ and $y$ be positive real numbers such that
\[\frac{1}{x + 2} + \frac{1}{y + 2} = \frac{1}{3}.\]Find the minimum value of $x + 2y.$ | nvidia |
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