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Let's construct an equilateral triangle using three circular arcs. The center of each arc will be the midpoint of the respective sides of an equilateral triangle with a side length of 6 cm, and the radius of each arc will be equal to half of the side length of the triangle. What will be the length of the arc that forms... | nvidia |
What are the last two digits in the sum of factorials of the first 100 positive integers? | nvidia |
In the system of equations
$$\begin{cases}
2x+y=a^2\\
4x+2y=2a^2
\end{cases}$$
find $a$ such that the system has exactly two solutions $(x,y)$. | nvidia |
If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2=\log_{10} b$, what is the median of the set $\{0, 1, a, b,
1/b\}$? | 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 |
The blue graph shown represents the function $f(x) = x^3 + bx^2 + cx + d$. Find $d$. (The grid lines are spaced one unit apart.)
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdown... | nvidia |
Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?
$\textbf{(A)}\ \sqrt{15} \qquad \textbf{(B)}\ \sqrt{21} \qquad \textbf{(C)}\ 2\sqrt{6} \qquad \textbf{(D)}\ 5 \qquad \textbf{(... | nvidia |
Let $a_0$ be the area of an equilateral triangle with side length 1, and let $a_n$ be the area of the equilateral triangle formed by connecting the midpoints of the sides of the triangle with area $a_{n-1}$. Find the smallest positive integer $n$ such that $a_n < \frac{1}{10}a_0$. | nvidia |
What is the smallest positive integer $n$ such that the sum of the first $n$ odd integers is greater than $1000$? | nvidia |
Let $\omega$ be a complex number such that $\omega^3 = 1.$ Determine the smallest positive integer $n$ such that when $\omega$ is raised to the power of $n$, the imaginary part of the result is $1/29$ of the imaginary part of $\omega$. | nvidia |
The function $f(x)$ satisfies
\[f(xy) = \frac{f(x)}{y}\]for all positive real numbers $x$ and $y.$ If $f(30) = 20,$ find $f(40).$ | nvidia |
Determine the number of ordered pairs (x, y) where x and y are positive integers such that x divides y and y divides 6!. | nvidia |
Find the maximum value of $\frac{x^3}{y} + \frac{y^3}{x}$ for positive real numbers $x$ and $y$ such that $x + y = 1$. | nvidia |
What is the sum of all the numbers in the row of Pascal's triangle that has 20 numbers? | nvidia |
What is the largest number of non-overlapping unit squares that can be placed on a 5 by 5 grid of lattice points, such that no square shares a side with another square? | nvidia |
Given that the Fibonacci sequence has the property that the ratio of consecutive terms tends to the golden ratio, find the value of the infinite product \prod_{n=2}^{\infty} \left( \frac{F_{n+1}}{F_n} - \frac{F_{n-1}}{F_n} \right) and determine its convergence. | nvidia |
Find the smallest positive integer $n$ such that the difference between $n$ and its reverse is a multiple of 7. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $$a > b+c$$ and $$\frac ab + \frac bc > \frac 43.$$ Determine the least possible value of $b$ if $$a+b+c=\frac{13}4.$$ | nvidia |
Find the sum of all the odd numbers between 2.5 and 20.8 that are not multiples of 3. | nvidia |
Let $g$ be a function defined on the real numbers, such that
\[g(x+y+z) = g(x)g(y)g(z)\]for all real numbers $x$, $y$, and $z.$ Given $g(1) = 2$ and $g(2) = 4,$ find $g(5)$. | nvidia |
Let $a_1, a_2, \dots, a_n$ be positive real numbers whose sum is 10, and let $b_1, b_2, \dots, b_n$ be positive real numbers whose product is 27. Find the minimum value of
\[\frac{a_1}{b_1} + \frac{a_2}{b_2} + \dots + \frac{a_n}{b_n}.\] | nvidia |
A " shuffle" is a sequence of $n$ consecutive integers such that every second integer is divisible by 3. Determine the number of shuffles of length $n=2000$ that start with an integer not divisible by 3 and end with an integer divisible by 3. Express your answer as the remainder when divided by $10^5$. | nvidia |
A circle of radius $r$ is inscribed in a trapezoid with bases of length $a$ and $b$. The trapezoid is constructed such that the area of the trapezoid is equal to the area of the circle. Find the height of the trapezoid, $h$, in terms of $r$, $a$, and $b$. | nvidia |
The geometric mean of an even number of consecutive positive even integers is $x$. Find the product of the smallest and largest of the integers in terms of $x$. | nvidia |
Given that $f(x) = ax^3 + bx^2 + cx + d$ is a cubic function with real coefficients, find the value of $a$ such that $f(x)$ has exactly two real roots, one of which is a double root. | nvidia |
Find the number of positive integers less than or equal to 180 that have at least three different positive divisors. | nvidia |
In the equation $x^2-kx+1=0$, determine $k$ such that the difference of the roots be equal to unity.
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 $a$. | nvidia |
Let $A$ be the area of the triangle with vertices $(0,0)$, $(1,0)$, and $\left(\frac{1}{2},\frac{1}{k}\right)$, where $k$ is a positive integer. Compute the smallest value of $k$ such that $A<\frac{1}{2023}$. | nvidia |
Determine the maximum possible value of $y$ for which the equation $$\left(\frac{y+1}{y-2}\right)^3-\left(\frac{y+1}{y-2}\right)=6$$ has a real solution. | nvidia |
Find the sum of all the even factors of 240 that are not divisible by 4. | nvidia |
Find the sum of all the coefficients in the expansion of $(1+x)^{40}$, where the coefficients are taken from the row of Pascal's triangle with 41 numbers, but only consider the coefficients that correspond to even powers of x. | nvidia |
Let $a$ and $b$ be real numbers. Consider the system of equations:
\begin{align*}
18a + 73b &= k_1\\
a - b &= k_2,
\end{align*}
where $k_1,k_2 \in \mathbb{R}.$ Find all values of $k_1$ and $k_2$ for which the system has at least one solution. | nvidia |
For how many positive integers $x$ is $300 \leq x^2 \leq 400$? | nvidia |
For positive integers $n$, let $f(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point. How many positive integer divisors does $f(2010)$ have? | nvidia |
What is the smallest whole number that must be a multiple of the sum of any two consecutive positive integers? | nvidia |
Solve for the sum of all possible values of $x$ when $3^{x^2+4x+4}=9^{x+2}$. | nvidia |
The sequence of positive integers that are either powers of 2 or sums of distinct powers of 2 is given. Find the largest term in this sequence that is less than 1000. | nvidia |
In a certain triangle, the sum of the squares of the lengths of the sides is equal to 361. The triangle is also known to be a right triangle, with the length of the hypotenuse being 6 units longer than one of the other sides. Determine the length of the hypotenuse.
(Note: I've created a problem that requires a differe... | nvidia |
A sequence of squares is inscribed in a circle with radius 10. The area of the first square is 36 square units, and the area of each subsequent square is 75% of the area of the previous square. What is the total area of the infinite sequence of squares? | nvidia |
The expression $n^{\sqrt{2n-1}}$ is an integer for some positive integer $n$. What is the smallest value of $n$ that satisfies this condition? | nvidia |
Count the number of positive integers less than 100 that have an odd number of positive divisors.
Note that these divisors include 1 and the number itself. | nvidia |
The equation $x^2 - 6x + 2 = 0$ has two real roots $x_1$ and $x_2$. Determine the value of $a$ such that the product of the roots of the equation $(x - x_1)(x - x_2)(x - a) = 0$ is equal to the sum of the roots of the same equation. | nvidia |
Find the range of the function $$f(x)=\sqrt[3]{1+\sqrt[3]{2-\sqrt[3]{x}}}$$ where $x$ is a real number. | nvidia |
Name the greatest whole number less than $100$ that has an odd number of positive factors. | nvidia |
Find the number of positive integer divisors of $2^3\times3\times5^3\times7$ that are multiples of 15. | nvidia |
Given that $f(3x+2)=x^2+1$ for all real $x$, determine the value of $f(5)$. | nvidia |
Find the smallest three-digit palindrome number that is divisible by the sum of its digits. | nvidia |
Determine the values of $m$ for which the inequality $mx^2-12x+m+3\geq0$ holds true for all real values of $x$. | nvidia |
Find the integer $n$, $0 \le n \le 23$, such that
$5n\equiv13\pmod{24}$ | nvidia |
Let $f(x)$ be a function from the set of positive integers to itself, defined such that for all $x$ we have that $f(f(x))=x$. Find the smallest possible number of elements in the domain such that we can find a nonnegative integer $k$ for which the equality $$\sum_{i=1}^k f(i)= \sum_{i=k+1}^{2k} f(i)$$ holds. | nvidia |
Let $a$ be an integer such that $2a \equiv 1 \pmod{99}$. Find the smallest positive value of $b$ such that $ab \equiv 1 \pmod{99}$. | nvidia |
Consider the expression $(a + b)^2 - (a^2 + b^2)$ where $a$ and $b$ are positive integers. Prove that the result is always an even number, regardless of the values of $a$ and $b$. | nvidia |
When two fair 6-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that the two digit number $ab$ (where $a$ and $b$ are digits) and $a$ and $b$ are all divisible by 3? | nvidia |
Determine the largest positive integer $a$ such that the value of
$$\frac{\sqrt{a-1}}{\sqrt{a+1}}$$
is in the form
$$\frac{\sqrt{b}}{\sqrt{b+1}}$$
where $b$ is also a positive integer. | 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 |
What is the sum of all integer values of $n$ such that $\frac{20}{2n - 1}$ is an integer? | nvidia |
How many of the natural numbers from 1 to 600, inclusive, contain the digit 5 at least once? (The numbers 152 and 553 are two natural numbers that contain the digit 5 at least once, but 430 is not.) | nvidia |
Find the largest integer $n$ for which there exists a spacy set $S \subseteq \{1, 2, 3, \dots, n\}$ such that the sum of the elements in $S$ is equal to $n(n+1)/2$. | nvidia |
In the expansion of $(1+x)^n$ using the binomial theorem, where $n$ is a positive integer and $x$ is a real number, what is the sum of the coefficients of the even-powered terms? | nvidia |
Consider a positive integer $a$ greater than 1. Suppose that a prime number $p$ satisfies the condition $a^{p-1}-1=a^k$ for some positive integer $k$. Prove that $k$ is always even. | nvidia |
Let $\alpha$ and $\beta$ be the roots of the equation $x^2+x-1=0$. In the expansion of $(x+\alpha)^7$, determine the number of terms that contain $\beta$. | nvidia |
There are infinitely many positive integers $k$ which satisfy the equation
\[\cos^2 (k^2 + 6^2)^\circ = 1.\]Enter the two smallest solutions, separated by commas. | nvidia |
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of the possible values of $a?$
$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 18$
| nvidia |
Let $P$ be a point inside a regular pentagon such that the distance from $P$ to each vertex of the pentagon is greater than 1. Find the minimum value of the sum of the squares of the distances from $P$ to the vertices of the pentagon. | nvidia |
Determine the range of values of $c$ such that the function $f(x) = \frac{3x^2+5x-2}{x^2-4x+c}$ has exactly one vertical asymptote. | nvidia |
Find all prime numbers $p$ such that the decimal representation of $p^2$ contains only the digits 1 and 5. | nvidia |
The integers $A$ and $B$ are chosen such that
\[\frac{A}{x^2+3x}+\frac{B}{x-2}=\frac{x^2-2x-4}{x^3+x^2-6x}\]for all real values of $x$ except $0$, $-3$, and $2$. Find $B/A$. | nvidia |
A circle with center $(2,5)$ passes through the point $(5,1)$. Find the coordinates of the point where the circle intersects the positive branch of the $y$-axis. | nvidia |
Let $f(x)$ be a function defined on the real numbers such that $f(x) = x^2$ if $x$ is irrational, and $f(x) = x + 1$ if $x$ is rational. Find $f(f(f(f(2))))$. | nvidia |
Let $x$ be a positive real number and let $A$, $B$ and $C$ be defined as follows: $x+\frac{1}{x}=A$, $x^2+\frac{1}{x^2}=B$, $x^3+\frac{1}{x^3}=C$. Prove that $C$ can be expressed as a function of $A$ and $B$, and determine the maximum value of $\frac{C}{AB}$. | nvidia |
The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $c$. (The grid lines are spaced one unit apart.)
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownle... | nvidia |
Find all real numbers $a$, $b$, and $c$ such that
$$\frac{a^{x^2}+b^{x^2}}{c^{x^2}+(ab)^{x^2}}=\frac{a+b}{c+ab}$$for all real numbers $x$.
Note: This new problem requires a different approach than the original, as it involves finding values for multiple variables ($a$, $b$, and $c$) that satisfy an equation with a va... | nvidia |
Let $ABC$ be a triangle with $\angle BAC = 60^\circ$. A circle is tangent to the sides $AB$ and $AC$ at $X$ and $Y$ respectively, such that the points on the circle diametrically opposite $X$ and $Y$ both lie on the side $BC$. The area of the portion of the circle that lies outside the triangle is known to be $1/4 \p... | nvidia |
The product of two numbers is 12. The sum of their squares is 34. What is the positive difference of the squares of the reciprocals of the two numbers? | nvidia |
Find the number of ordered pairs of positive integers (a, b) such that the least common multiple of a and b is 60. | nvidia |
A cylindrical container is placed so that one of its bases lies on the $xy$-plane with the origin at its center. The equation of the cylinder is $x^2 + y^2 = 25$. Find the length of the segment of the $z$-axis that lies inside the cylinder. | nvidia |
A right triangle with legs of length $a$ and $b$ is inscribed in a circle. If the area of the circle is $16\pi$ square units, find the maximum possible value of the area of the right triangle, in square units. | nvidia |
Find the area of the quadrilateral with vertices $(2,3),$ $(6,-2),$ $(10,7),$ and $(1,8).$ | nvidia |
A square piece of paper has sides of length $x$ cm. Three smaller squares with side lengths 5 cm, 9 cm, and $n$ cm are cut out from its corners. What is the sum of all possible whole number values of $n$ such that the remaining shape is a rectangle? | nvidia |
Find the smallest positive four-digit number that is divisible by each of the four smallest prime numbers. | nvidia |
In a triangular array of numbers, each number is the sum of the two numbers directly above it. The first row has only one number, 1. The second row has two numbers, 2 and 3. The third row has three numbers, 4, 5, and 6. Find the smallest positive integer n such that the nth row contains three numbers that are in geomet... | nvidia |
Find the largest possible value of $n$ for which $2^{15}$ can be expressed as the sum of $n$ consecutive positive even integers. | nvidia |
Given that $$(a+b+c)(ab+ac+bc)=48$$and that $$a^2(b+c)+b^2(a+c)+c^2(a+b)=8$$for real numbers $a$, $b$, and $c$, find the value of $abc$. | nvidia |
Find the integer $k$, $0 \le k \le 7$, such that \[k \equiv -2192 \pmod{8}.\] | nvidia |
Let $f(x) = \tan(\arcsin(\cos(\arctan x)))$. Find all real numbers $x$ such that $f(x) = f^{-1}(x)$. | nvidia |
Let $n$ be a positive integer greater than 1. Find the number of unordered tuples of positive integers $(a_1,a_2,...,a_k)$ satisfying the following conditions:
1. The sum of the components in the tuple is $n$, that is, $a_1+a_2+\cdots+a_k=n$;
2. The product of the components in the tuple is $n$, that is, $a_1a_2\cdots ... | nvidia |
In a circle, two chords intersect at a point. The lengths of the segments of one chord are 8 cm and 10 cm, while the lengths of the segments of the other chord are 5 cm and x cm. Find the value of x such that the two chords bisect each other. | nvidia |
Find the number of ways to arrange 16 books on a bookshelf such that 5 of the books are red and the remaining 11 books are blue, and no two red books are adjacent. | nvidia |
Let $a$ and $b$ be real numbers. If the polynomial $x^2 + ax + b$ has exactly one real root and $a = 2b - 1$, find the value of the sum of all possible values of $b$. | nvidia |
Find all real numbers $m$ for which there exists a nonzero, 2-dimensional vector $\mathbf{w}$ such that
\[\begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix} \mathbf{w} = m \mathbf{w}.\]
Enter all the solutions, separated by commas. | nvidia |
Find all complex numbers $z$ such that
\[z^2 = -77 - 36i.\]Enter all complex numbers, separated by commas. | nvidia |
Let $g(n)$ be the product of all the proper divisors of a positive integer $n$. If $g(g(n)) = n$, then call $n$ self-sustaining. How many self-sustaining positive integers are there that have exactly three distinct prime factors? | nvidia |
Find the value of $y$ such that the three-digit number $\underline{2y6}$ is divisible by 9. | nvidia |
Let $C_i$ be the set of all integers $n$ such that $100i\leq n < 100(i + 1)$ and $n$ is a perfect cube. For example, $C_0$ is the set ${0, 1, 8, 27, 64}$. How many of the sets $C_0, C_1, C_2, \ldots, C_{999}$ contain at least one perfect square that is not a perfect cube? | nvidia |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/29 of the original integer. | nvidia |
Find the number of positive integers $n$ such that the product of $n$ and $n+2$ is between 120 and 180. | nvidia |
Solve for the sum of all possible values of $x$ when $2^{x^2-2x+1}=4^{x-1}$. | nvidia |
Find the number of triples (x,y,z) of nonnegative integers that satisfy the equation
x+2y+3z=6. | nvidia |
Find the number of integer values of $a$ for which the inequality $ax^2 + 2x + 3 \ge 0$ has exactly two integer solutions. | nvidia |
Consider a circle centered at the origin with a radius of 1. Let $T$ be a transformation that reflects any point $P$ across the line $y = \sqrt{3}x$ and then reflects the resulting point across the line $y = \frac{1}{\sqrt{3}}x$. Given that the point $A$ has coordinates $(1, 0)$, find the smallest positive integer $n$ ... | nvidia |
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