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Find the positive integer $N$ such that the sum of the reciprocals of its digits is equal to the sum of the reciprocals of the squares of its digits. | nvidia |
Let $g : \mathbb{C} \to \mathbb{C} $ be defined by $ g(w) = w^3 + 2iw - 3 $. How many complex numbers $w $ are there such that $ \text{Re}(w) < 0 $ and both the real and the imaginary parts of $g(w)$ are integers with absolute value at most $ 15 $? | nvidia |
The Fibonacci numbers $F_n$ are given by the recursion $F_0 = 0, F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$. Find the largest integer $n$ such that $F_n$ is a perfect square. | nvidia |
What is the sum of the digits of the sum of all the even positive integers between 20 and 200? | nvidia |
The vertices of a triangle are $(2,5), (7,12),$ and $(x,y)$. The midpoint of the side joining the first two vertices lies on the line $y=x+2$. Find the sum of the coordinates of the third vertex. | nvidia |
Let $a$ and $b$ be positive integers. Given that $a$ and $b$ are the products of the first $n$ primes and the next $n$ primes, respectively. Determine $n$ if $\ds\f{a}{b}=\ds\f{7}{11}$. | nvidia |
Given a rational number $\dfrac{p}{q}$ where $p$ and $q$ are relatively prime, find the smallest number of decimal digits that repeat in the decimal expansion of $\dfrac{p}{q}$. | nvidia |
Find a nonzero $q$ such that the system of equations
$$y=2x^2-12x+1$$
$$y=qx^2-12x+q$$
has exactly two solutions. | nvidia |
Find the sum of the 2010 fractions of the form $\frac{1}{n(n+3)}$ if the values of $n$ are the positive integers from 1 through 2010. Express your answer as a decimal to the nearest thousandth. | nvidia |
What is the smallest positive integer that has exactly six distinct positive factors? | nvidia |
The sum of three different numbers is 67. The two larger numbers differ by 7 and the two smaller numbers differ by 3. What is the value of the largest number? | nvidia |
There is a unique two-digit positive integer $t$ for which the last two digits of $11\cdot t$ are $36$.
What is $t$? | nvidia |
In a rectangle, the length of the diagonal is $\frac{5}{4}$ times the length of one of its sides. Find the length of the other side, given that the perimeter of the rectangle is 80 cm. | nvidia |
Find the smallest positive integer $n$ such that $2^n$ is a perfect square, $3^n$ is a perfect cube, and $5^n$ is a perfect fifth power. | nvidia |
A box contains 100 balls, each labeled with a number from 1 to 100. If 30 of the balls are red, and the sum of the numbers on the red balls is 1650, what is the largest possible sum of the numbers on the remaining 70 balls? | nvidia |
Let $a$, $b$, $c$ be complex numbers such that $|a|=|b|=|c|=1$ and $|a+b+c|=0$. Find the largest possible value of $|a^3+b^3+c^3|$. | nvidia |
Find the value of $x$ that maximizes the area of a rectangle with a fixed perimeter of 24 units, where the length and width of the rectangle are $x$ and $y$ respectively, and $y=x-2$. | nvidia |
A positive integer $n$ is called superabundant if the product of its divisors exceeds $n^2$. Find the smallest superabundant number. | nvidia |
Let $f(x)$ be a polynomial function such that for all real values of $x$, if $f(x)$ is a perfect cube, then $x$ is an integer. What is the least possible degree of such a function? | nvidia |
A circle is inscribed in a triangle. If the radius of the circle is 4 cm and the sum of the lengths of the sides of the triangle is 30 cm, find the area of the triangle. | nvidia |
Find the value of $x$ such that $\sin (2x) = \sin (3x) = \sin (5x)$. | nvidia |
Find the smallest positive integer $n$ such that the product of the absolute values of the complex numbers $(2 + 3i)$, $(4 + 5i)$, and $(6 + 7i)$ is equal to the absolute value of the complex number $(1 + i)^n$. | nvidia |
What is the smallest positive integer $n$ such that $2^n-1$ is a prime number greater than 1000? | nvidia |
The sides of a triangle are equal to 36, 132, and x, where x is an integer. Determine the least possible value of x that allows the triangle to exist. | nvidia |
Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is greater than 6? | nvidia |
Consider a triangle with the altitude $H$ and the side length $L$.
There is a regular hexagon inscribed inside the triangle such that the two upper vertices of the hexagon are located on the sides of the triangle, two lower vertices are located on the base of the triangle and the other two vertices are located on the a... | nvidia |
Find the value of $x$ that satisfies the equation $\displaystyle\lim_{n\to\infty}\left(\frac{\lfloor x^n\rfloor}{x^n}\right)^n=\frac{1}{e}$, where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. | nvidia |
Find the smallest possible degree of a nonzero polynomial with integer coefficients that has all of the complex numbers
$$\sqrt[3]{1}, \; \sqrt[3]{2}, \; \sqrt[3]{3}, \; \dots, \; \sqrt[3]{1000}$$
as roots. | nvidia |
Two congruent regular tetrahedra are placed inside a larger cube such that their vertices coincide with the cube's corners. The edge length of each tetrahedron is equal to half the edge length of the cube.
Determine the number of points with integer coordinates lying inside the region that is common to both tetrahedra... | nvidia |
Find the sum of all values of $x$ for which the expression $\frac{x^2+1}{x^2-2x\cos(\theta)+1}$ is undefined, where $\theta$ is a constant angle. | nvidia |
The sum of the numbers from $1$ to $20$ is $210$. A certain number is eliminated from this sequence, and the sum of the remaining numbers is $196$. What number was eliminated? | nvidia |
What is the smallest positive integer with exactly 12 positive divisors that are also multiples of 3? | 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 all real numbers $x$ such that
$$\sqrt{x^2+1}-\sqrt{x^2-2x+2}=x-3.$$ | nvidia |
In the equation
$$5x^2-kx+1=0$$
determine $k$ such that the difference of the roots be equal to unity.
Write another problem inspired by this one:
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$.
The positive four-digit integers that use each of the four dig... | nvidia |
Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$. | nvidia |
The function $g$ is defined on the set of all real numbers and has the property that $g(g(x))=x$ for all $x$. Given that $g(1)=2$, $g(2)=-1$, $g(3)=1$, and $g(4)=-3$, determine the value of $g(g(g(g(1))))+g(g(g(3)))+g(g(g(4)))$. | nvidia |
Find the greatest integer value of $x$ that satisfies the inequality $\frac{x}{7}+\frac{5}{14}>1$. | nvidia |
Find the least positive integer with exactly four distinct positive divisors. | nvidia |
Find the sum of the first seven prime numbers that have a units digit of 7. | nvidia |
Let $p(x)$ be a monic polynomial with integer coefficients and $r$ be a rational number such that $p(r) = r.$ Find all possible values of $p(10) - r.$ | nvidia |
Find all 4-digit multiples of 33 of the form $3a6b$ where $a$ and $b$ are digits and the sum of the digits of the number is divisible by 3. What is the minimum value of $a$? | nvidia |
The function $f(n)$ is defined as follows: $f(1) = 3$, and for $n > 1$, $f(n)$ is the sum of all positive integers $k$ such that $f(n-1)$ is divisible by $k$. What is the smallest value of $n$ such that $f(n) > 10$? | nvidia |
Find the number of positive real numbers $r$ such that $\lfloor r \rfloor + \lfloor 2r \rfloor + \lfloor 3r \rfloor = \lceil 6r \rceil - 1$. | nvidia |
A function $g$ satisfies the relation $g(x+y)=g(x)+g(y)$ for all real numbers $x$ and $y$. If $g(3) = 1$ and $g(4) = 2$, find the value of $g(7)$. | nvidia |
Find the smallest positive integer $y$ that satisfies $y+7832 \equiv 4211 \pmod{15}$. | nvidia |
Find the value of $b$ such that the equation $\log_b (x+1) = \log_b (x) + 1$ has exactly one solution. | nvidia |
Let $\mathcal{S}$ be a set of positive integers such that for every element $x$ in $\mathcal{S}$, $x$ can be written as a linear combination of the elements of $\mathcal{S}\setminus\{x\}$, with integer coefficients. Find the maximum possible size of $\mathcal{S}$. | nvidia |
What is the sum of all positive integers less than 1000 that are perfect cubes of integers? | nvidia |
Determine $k$ such that the equation
$$x^2-24x+k=0$$
has two roots the sum of whose squares is equal to 650. | nvidia |
What is the maximum number of distinct digits that can be used in a 3-digit number if the sum of its digits is 14 and none of the digits are greater than 6? | nvidia |
How many positive four-digit integers less than 1000 have at least two digits that are the same? | nvidia |
Find the smallest positive integer $n$ such that $n$ is not divisible by the product of its distinct prime divisors, but $n+1$ is. | nvidia |
If the point $(2,5)$ is on the graph of $y=f(x)$, and $k(x)=f(x^2)$ for all $x$, then there is one point that must be on the graph of $y=k(x)$. What is the product of the coordinates of that point? | nvidia |
How many positive integers less than 100 have exactly three positive divisors? | 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 |
Find the sum of all positive divisors of 56 that are also divisors of 24 and are not multiples of 3. | nvidia |
Find the value of $\left|\left(\cos \theta + i \sin \theta \right)^n \right|$, where $n$ is a positive integer and $\theta$ is an acute angle. | nvidia |
Find the number of three-digit numbers that can be formed by choosing three different digits from the set {1, 2, 3, 4, 5}, where the hundreds digit is greater than the tens digit, and the units digit is less than the tens digit. | nvidia |
Two numbers are independently selected from the set of positive integers less than or equal to 5. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | nvidia |
Find the sum of the absolute values of the roots of the equation $x^3-3x^2-2x+6=0$. | nvidia |
For the functions $f(x)=\frac{x^2+Ax+B}{x^2+Cx+D}$ and $g(x)=\frac{x^2+Ex+F}{x^2+Gx+H}$, it is known that $\lim_{x\rightarrow\infty}f(x)=\lim_{x\rightarrow\infty}g(x)$, the vertical asymptotes of $f$ and $g$ are $x=1$ and $x=-1$, respectively, and that $f(0)=2$ and $g(0)=3$. Given that $A+B=C+D=E+F=G+H$, determine $|B-... | nvidia |
A circle is inscribed in a square in such a way that the square's vertices are all on the circle. A smaller square is drawn on the diagonal of the original square such that the diagonal of the original square is a side of the smaller square. If the area of the smaller square is 25, what is the perimeter of the original... | nvidia |
Find $x$ such that $\ln (3x^2+1) = 1$. | nvidia |
Find the largest of four consecutive three-digit positive integers that are less than 500 and are multiples of 4. | nvidia |
Find the sum of the first five prime numbers that have a units digit of 3. | nvidia |
If $x$ is a positive integer such that
$$2019^{2x}\cdot12^{8x}+2019^{-2x}\cdot12^{-8x}=2,$$
what is the smallest value of $x$? | nvidia |
What is the least positive integer $n$ such that the least common multiple of $n$ and $2n$ is 36? | nvidia |
Find the number of pairs of positive integers $(x, y)$ such that $x + y = 30$ and $\gcd(x, y) = 5$. | nvidia |
Find all real numbers $b$ such that the equation $\frac{x}{b} + \frac{b}{x} = \frac{5}{2}$ has exactly one solution for $x$. | nvidia |
Find the number of ways to arrange 7 different books on a shelf such that the largest book is always on one of the ends, and the smallest book is always on the opposite end. | nvidia |
Given a function $g(x)$ that satisfies the equation $g(x)g(y) = g(x+y)$ for all real numbers $x$ and $y$, if $g(2) = 3$, find $g(-4)$. | nvidia |
Let $T$ be the set of complex numbers of the form $a + bi,$ where $a$ and $b$ are real numbers, such that
\[\frac{1}{\sqrt{2}} \le a \le \frac{\sqrt{2}}{2}.\]Find the smallest positive integer $k$ such that for all positive integers $n \ge k,$ there exists a complex number $w \in T$ such that $w^n = -1.$ | nvidia |
Find the sum of all positive integers $n$ such that $1.2n-4.4<5.2$. | nvidia |
How many real numbers $x$ satisfy the inequality $2^x(x-3)(x-5)>0$? | nvidia |
Given a square with a side length of 5 cm, inscribe a circle within it such that the distance between the center of the square and the center of the circle is minimized. What is the diameter of the inscribed circle? | nvidia |
The sum of the squares of the solutions to the equation $(2x+3)^2(x-1)=14$ is equal to $51$. Find the value of the sum of the solutions. | nvidia |
Determine the number of different real numbers $a$ such that the equation $z^3-az^2+2=0$ has at least one complex root $z$ satisfying $|z|=2$. | nvidia |
Find the largest value of $m$ such that $3x^2+mx+24$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
Find the ordered pair $(a,b)$ of real numbers such that
\[\cos 3x = a + b \sec x,\]
for all real $x$ satisfying $\cos x > 0.$ | nvidia |
Determine the smallest value of $x$ such that the inequality $|x^2-3x+2|<5$ is satisfied. | nvidia |
If $a$ and $b$ are positive integers such that $2a+7b=140$, what is the greatest possible value of $ab$? | nvidia |
Let $f(x)=a\cos(bx+c)+d$ where $a,b,c,d$ are real numbers with $a>0$. If $a+b+c+d=6$ and the maximum value of $f(x)$ is 4, determine $a$.
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pen axispen=black+1.3bp;
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real t... | nvidia |
Let $a$, $b$, and $c$ be real numbers such that $a+b+c=0$. Find the smallest positive integer $n$ such that all the roots of $x^3 + ax^2 + bx + c = 0$ are $n^{\text{th}}$ roots of unity. | nvidia |
Find all real values of $a$ for which the equation $x^3 - ax^2 + (a+1)x - 2 = 0$ has at least one real root that is an integer. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 1.$ Find the maximum value of
\[x + y + z.\] | nvidia |
Let $S$ be the sum of the roots of all the polynomials of the form
$x^2+px+q$, where $p$ and $q$ are positive integers that satisfy
$p+q=2003$.
Determine the value of
$\left\lfloor \frac{S}{100000}\right\rfloor$ | nvidia |
Find the number of positive, three-digit integers that have exactly one 7 as a digit and do not contain a 4 as a digit.
(Hint: Instead of counting the numbers with at least one 7, count the numbers with no 7's and subtract from the total count of three-digit numbers without 4's.) | nvidia |
An $\textit{even abundant number}$ is a positive integer that is divisible by 2 and the sum of its distinct proper factors that are multiples of 2 is greater than the number itself. Determine the largest even abundant number less than 50. | nvidia |
Determine all values of $n$ for which $2^n$ can be expressed in the form $3m+1$, where $m$ is an integer. | nvidia |
Find the sum of all positive integers $a$ such that for every integer $n$, the number $2an^2 + 3$ is divisible by $a+1$. | nvidia |
Find the smallest positive integer k such that the decimal expansion of (2k+1)/7 contains only digits 0, 1, 2, and 3. | nvidia |
Find $x$ such that $\log_x 4=\log_{27} 3$. | nvidia |
The point $(2,5)$ is on the graph of $y=f(x)$, and $f(x)=g(2x)$ for all $x$. What is the sum of the coordinates of a point that must be on the graph of $y=g(x)$? | nvidia |
Given a sequence of numbers defined by $a_n = \sqrt[n]{n}$, find the value of $n$ for which $a_n$ is maximized. | nvidia |
Let $f(x)$ be a polynomial function of degree $n$, where $n$ is a positive integer. It is known that $f(x)$ has $n+1$ distinct roots $r_1, r_2, ..., r_{n+1}$ such that $f(r_i) = 0$ for all $i$. If the polynomial $g(x) = f(x^2)$ has exactly $2n+1$ distinct roots, determine the number of distinct roots of the polynomial ... | nvidia |
Find the number of real values of $k$ for which the circle $x^2 + y^2 = 25$ intersects the line $y = x^2 + k$ at exactly one point. | nvidia |
Let $S(n)$ be the sum of the digits of a positive integer $n$. For example, $S(189) = 18$. For a particular positive integer $n$, $S(n) = 234$. Which of the following could be the value of $S(n+2)$?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 245\qquad\textbf{(E)}\ 268$ | nvidia |
Let $x$ and $y$ be real numbers such that $x^2 + y^2 \le 1$ and $x + y \ge 1.$ Find the largest possible value of $3x + 4y$. | nvidia |
A function $f(x)$ is defined for all positive integers such that $f(1)=5$, and for $n>1$,
$f(n)=f(n-1)+2$ if $n$ is a prime number,
$f(n)=f(n-1)-1$ otherwise.
Compute $f(f(f(f(10))))$. | nvidia |
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