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Let ∘ denote a binary operation on the set of all positive real numbers, satisfying the following properties:
1∘x=x
(x∘y)∘z=(x∘z)∘y
x∘(y∘x)=x
For any positive real number x, find the value of x∘(1∘x). | nvidia |
Let $f(x)=x^2+1$. Find the product of all $x$ that satisfy the equation $f^{-1}(x)=f(x^2)$. | nvidia |
Let $a$ and $b$ be two positive integers such that $a^2 - b^2 = 2019$. Find the minimum value of $a + b$. | nvidia |
Find the smallest positive integer whose sum of divisors is a perfect square. | nvidia |
Determine the value of $x$ for which the expression $\frac{x+5}{2x}$ is equal to zero. | nvidia |
Given that a real number $x$ satisfies \[\sqrt{64-x^2}-\sqrt{36-x^2}=4.\]Find the value of $\sqrt{64-x^2}+\sqrt{36-x^2}$. | nvidia |
Given that $m$ is an integer and $0 < 3m < 24$, what is the sum of all possible integer values of $m$? | nvidia |
Given that a triangle's vertices are $A(0,1)$, $B(5,0)$, and $C(a,a+1)$, determine $a$ such that line $AC$ is perpendicular to line $BC$. | nvidia |
Solve the inequality $(2x-1)(x+1)^2<0$. What are the values of $x$ that satisfy this inequality? | nvidia |
Let's generate a new math problem inspired by the given one.
Let $A$, $B$, and $C$ be points chosen uniformly at random on the boundary of a circle with center $O$ and radius $1$. The probability that the area of the triangle $ABC$ is greater than or equal to $\frac{\pi}{4}$ can be written as $\frac{p}{q}$, where $p$ ... | nvidia |
A set of distinct positive integers is given by $\{a, a^2, a^3, \ldots, a^8, a^9, a^{10} \}$. Determine the number of different positive integers that can be represented as the sum of two distinct members of the set. | nvidia |
In triangle $ABC$, the altitudes $AD$ and $BE$ intersect at $H$. Given that $HD=8$ and $HE=12$, find the area of triangle $ABC$. | nvidia |
Find the sum of all the positive divisors of 55. | nvidia |
Lily's favorite numbers are those that are divisible by 4. How many different last two digits are possible in numbers that Lily likes? | nvidia |
Find the smallest positive integer that is divisible by four different prime numbers. | nvidia |
Given that $\frac{1}{3} \equiv 5 \pmod{7}$, find the integer $n$, $1 \le n \le 6$, such that $n \equiv \frac{1}{5} \pmod{7}$. | nvidia |
Find the largest integer $n$ such that the product of the first $n$ terms of the sequence
$\sqrt{1+\frac{1}{k}}$ is less than $2$.
I hope this problem is to your liking. Please let me know if you need anything else. | nvidia |
A deck of cards contains an infinite number of cards, numbered 1, 2, 3, and so on, such that each card's number is equal to twice the number on the previous card. If the sum of the numbers on all the cards in the deck is 63, what is the number on the first card? | nvidia |
Find the product of all integer divisors of $105$ that also divide $14$. (Recall that the divisors of an integer may be positive or negative.) | nvidia |
Compute the number of ordered triples of integers $(x,y,z)$ with $1\le x<y<z\le 100$ such that $x^2+y^2+z^2$ is a perfect square. | nvidia |
Find the number of whole numbers $n$ such that $n^2$ lies between the decimal representations of the fractions $\frac{2023}{100}$ and $\frac{4047}{200}$. | nvidia |
Let $C_i$ be the set of all integers $n$ such that $10^i\leq n < 10^{i+1}$. For example, $C_2$ is the set ${100,101,102,\ldots,999}$. How many of the sets $C_0, C_1, C_2, \ldots, C_{5}$ do not contain a number that is a sum of two perfect squares? | nvidia |
How many positive integers less than $1000$ can be written as the sum of two positive perfect fourth powers? | nvidia |
Let g ( t ) = t ^{2} . The sequence ( g _{n} ) of functions is defined by g _{1} = g and g _{n} = g ∘ g _{n−1} for all n ≥ 2. Let T be the set of all real numbers t such that g _{n} ( t ) = t for some positive integer n . Find the number of elements in T . | nvidia |
Given that $3^{15}=10^x \cdot 5^y$, find the value of $xy$. | nvidia |
Find \[\dfrac{3}{\,\,\frac{6}{11}\,\,}.\]
Find the sum of all the numbers between 10 and 50 that are not divisible by 7. | nvidia |
Let's consider a set of four numbers. The sum of the first two numbers is 27, the sum of the second and third numbers is 41, and the sum of the third and fourth numbers is 23. What is the sum of all four numbers? | nvidia |
Find the smallest positive integer whose digits form an arithmetic sequence and whose sum of digits is 12. | nvidia |
How many positive four-digit integers less than 3000 have at least two digits that are the same? | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x^2 + yf(z)) = xf(x) + zf(y)\]for all real numbers $x,$ $y,$ and $z.$
Let $n$ be the number of possible values of $f(5),$ and let $s$ be the sum of all possible values of $f(5).$ Find $n \times s.$ | 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 $y=g(1/x)+2$. What is the product of the coordinates of that point? | nvidia |
Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| nvidia |
Find the sum of all positive integers less than 20 that are not divisible by 2 or 3. | nvidia |
What is the greatest integer less than or equal to\[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?\]
$\textbf{(A) }80\qquad \textbf{(B) }81 \qquad \textbf{(C) }96 \qquad \textbf{(D) }97 \qquad \textbf{(E) }625\qquad$
| nvidia |
Let $g(x)$ be a function defined on the set of all positive real numbers, such that $g(a) + g(b) = g(ab)$ for all positive real numbers $a$ and $b.$
If $g(3) = 2,$ then find $g(9),$ $g\left(\frac{1}{3}\right),$ and $g\left(\frac{1}{9}\right).$ | nvidia |
Find the minimum value of the expression $x^2 + y^2 + 2xy + 2x + 2y + 1$ over all real numbers $x$ and $y.$ | nvidia |
Let $g$ be a function taking the positive integers to the positive integers, such that
(i) $g$ is increasing (i.e. $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^n + n^m = m \cdot n,$ then $g(m) = n$ or $g(n) = m.$
Find t... | nvidia |
Find the sum of all positive integers less than 100 that are cubes of perfect cubes. | nvidia |
Let $x$ and $y$ be real numbers such that
$\frac{x}{y}+\frac{y}{x}=1$,
and $x^2+y^2=7$.
Find the value of $x+y$. | nvidia |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | nvidia |
The integers $P$ and $Q$ are chosen such that
\[\frac{P}{x+1}+\frac{Q}{x^2-x+6}=\frac{x^2+7x+12}{x^3+7x^2-6x+6}\]for all real values of $x$ except $-1$, $\frac{1-\sqrt{23}}{2}$, and $\frac{1+\sqrt{23}}{2}$. Find $Q-P$. | nvidia |
What is the sum of the first ten positive multiples of $13$?
Determine $n$ such that the sum of the first $n$ positive multiples of $7$ is equal to the sum of the first $n$ positive multiples of $11$. | nvidia |
Let $a$, $b$, and $c$ be three positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime to both $a$ and $b$.
Compute $\gcd(a^b+c^b,a^b+c^b+c^2+1)$. | nvidia |
Count the number of 10-digit positive integers that have all digits either 1 or 2 and have at least one sequence of three consecutive 2's. | nvidia |
The product of two positive integers plus their sum is 95. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | nvidia |
Solve the system of equations:
\begin{align*}
x^y - y^x &= 2,\\
x^{2y} + y^{2x} &= 14.
\end{align*}
Find all real solutions $(x, y)$ such that $x$ and $y$ are not both equal to 1. | nvidia |
Find the smallest positive integer $a$ such that $a^2 \equiv 13 \pmod{36}$. Express your answer as an integer from $0$ to $35$. | 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 |
Let $x$ and $y$ be nonzero real numbers such that
\[(3 + 5i)(x + yi)\]is real. Find $\frac{x}{y}.$ | nvidia |
Find the integer $n$, $0 \le n \le 7$, such that \[n \equiv -3737 \pmod{8}.\] | nvidia |
Find the number of integers $m$ that satisfy
\[25 < m^2 < 121.\] | nvidia |
Find the number of positive integers less than 1000 that have a remainder of 7 when divided by 11. | nvidia |
Find the value of $x$ such that $\sqrt{3x + 7} = 10$. | nvidia |
Find all real values of $p$ for which the system of equations
$$\begin{cases}
y=\frac{x^2}{p} \\
x^2 + y^2 = 4
\end{cases}$$
has exactly one solution. | nvidia |
Find the value of $k$ such that
$\dfrac{1}{(x^2-4)(x^2+9)}=\dfrac{k}{x-2}+\dfrac{1}{x+2}+\dfrac{1}{x^2+9}$. | nvidia |
Let $f(x)=ax^2+bx+c$, where $a$, $b$ and $c$ are positive integers.
Given that $f(1)+f(2)+\cdots+f(n)=2017n$, determine
$f(1)+f(2)+\cdots+f(2017)$. | nvidia |
Let $P(x)$ be a polynomial of degree $n$ with integer coefficients, where $n$ is a positive integer. The polynomial has the property that $P(k)$ is a perfect square for every integer $k$. Determine the maximum possible number of distinct integer roots of $P(x)$. | nvidia |
What is the smallest positive integer that is a perfect square and is divisible by each of the first six positive integers that are perfect squares? | nvidia |
Four standard six-sided dice are rolled. The sum of the squares of the numbers on the upper faces of three of them is 45, and the sum of the numbers themselves is 15. What is the smallest possible sum of the numbers on the upper faces of all four dice? | nvidia |
In a triangle, the length of the base is (x-5), and the height is (x+2). The area of this triangle is equal to the area of a circle with radius (x-1). Determine the value of x for which this equality holds true. | nvidia |
Determine the maximum number of points that can be placed on a circle of radius 1 such that any two points are at least 1 unit apart. | nvidia |
Let $A = \begin{pmatrix} 0 & 2 \\ 1 & 1 \end{pmatrix}$. Find all real numbers $c$ such that the matrix equation $A^2 = cA$ holds. | nvidia |
If $x$, $y$, and $z$ are digits and $0.xyz$ can be expressed as $\frac{1}{p}$ where $p$ is an integer such that $0<p\le8$, then what is the largest possible value of $x+y+z$? | nvidia |
Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{9}{10}.\] | nvidia |
Find the largest positive real number $d$ such that for all nonnegative real numbers $a$ and $b$,
\[\frac{a + b}{2} + d \sqrt{ab} \leq \max \{a, b\}.\] | nvidia |
Let
\[g(x) =
\begin{cases}
-3x^2+1 &\text{if }x<0, \\
4x^2-1&\text{if }x\ge0.
\end{cases}
\]
Find all possible values of $a$ such that the equation $g(x)=a$ has exactly one real solution. | nvidia |
Determine the smallest number that can be added to the sum of the four smallest prime numbers so that the result is divisible by 6 and is not a perfect square. | nvidia |
The figure shows a square with side length 8 cm and four congruent circles with radius 3 cm, centered at the vertices of the square. Each circle intersects the square and two neighboring circles. Find the area of the region inside the square and outside all four circles.
[asy]
fill((0,0)--(8,0)--(8,8)--(0,8)--cycle,g... | nvidia |
If $x$ and $y$ are positive real numbers such that $5x^2 + 10xy = x^3 + 2x^2 y,$ what is the value of $x$? | nvidia |
Suppose that $n$ is a positive integer such that in base $7$, then $n$ can be expressed as $\overline{ABC}_7$, and in base $11$, then $n$ can be expressed as $\overline{CBA}_{11}$. Find the largest possible value of $n$ in base $10$. | nvidia |
Simplify the expression $\ln \frac{a^x}{b^y}$, where $a$ and $b$ are positive constants and $x$ and $y$ are real numbers. | nvidia |
How many natural numbers, consisting only of digits 1 and 2, are there, such that they are divisible by the sum of their digits? | nvidia |
The area of a square equals the square of a length of the side of the square. The perimeter of a square equals the sum of the lengths of all four sides. The sum of the areas of two squares is 65, while the difference in their areas is 33. Find the sum of their perimeters. | nvidia |
Find a positive integer that is divisible by 18 and whose cube root is a number between 8 and 8.1. | nvidia |
Let $a^2=\frac{16}{44}$ and $b^2=\frac{(2+\sqrt{5})^2}{11}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a+b)^3$ can be expressed in the simplified form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$? | nvidia |
Let $z$ be a complex number such that $z^2 + z = -1$. Determine the sum of the squares of the real and imaginary parts of $z$. | nvidia |
Determine all values of $x$ for which the expression $\frac{x^2+5x+6}{(x^2-4)(x^2+2x-3)}$ is defined. | nvidia |
The four sides of a quadrilateral have lengths of $a$, $b$, $c$, and $d$. The midpoints of the sides form a smaller parallelogram with sides of length $m$ and $n$, where $m=\frac{b+d}{2}$ and $n=\frac{a+c}{2}$. Given that the area of the parallelogram is 30 and the perimeter of the quadrilateral is 44, find the value o... | nvidia |
Compute the sum of all the solutions to the equation $z^6 = -1$, where $z$ is a complex number. | nvidia |
Find a positive integer whose square can be represented as the sum of the cubes of its digits. | nvidia |
Given that the function f(x) = x^2 + kx + l has a graph that passes through the points (0, 3) and (4, 15), find the value of k such that the function is maximized at x = 2. | nvidia |
Find $x$ such that $\log_x 81=\log_2 16$. | nvidia |
Let $m \ge 3$ be an integer and let $S = \{3,4,5,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $a+b = c$ and $a+c = b$. | nvidia |
What is the sum of all integer values $n$ for which $\binom{20}{n}+\binom{20}{10}=\binom{21}{11}$? | nvidia |
In the multiplicative group of integers modulo 28, find the order of the element 27, i.e., the smallest positive integer n such that 27^n ≡ 1 (mod 28). | nvidia |
Find the number of positive integers that can be expressed as the sum of a perfect square and a perfect cube, where the sum is less than 800. | nvidia |
Let $T$ be the set of rational numbers $r$ that have a decimal representation of the form $r = p.q_1q_2q_3q_4\ldots$, where $p$ is an integer and $q_i = q_{i+8}$ for all positive integers $i$. Suppose that $10001$ is a divisor of $77001$. Determine the number of elements in $T$ that are not integers. | nvidia |
Find the smallest positive integer $n$ such that
$\sqrt{\left\lfloor \frac{n(n+1)}{2}\right\rfloor^2 - 1}$
is also an integer. | nvidia |
In the complex plane, the vertices of a regular polygon are represented by complex numbers $z_1, z_2,..., z_n$. The center of the polygon is the origin, and the distance from the origin to each vertex is 1. Let $a_n$ be the sum of the products of all possible pairs of these complex numbers, and let $b_n$ be the sum of ... | nvidia |
A hyperbola and a circle share a center, and the transverse axis of the hyperbola is the line containing the diameter of the circle. The hyperbola and circle intersect at four points. Given that the equation of the circle is $x^2 + y^2 = 4,$ find the sum of the distances from the center to each of the four intersection... | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x + g(y)) = g(x) + g(y) + g(g(x) + y)\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(0),$ and let $t$ be the product of all possible values of $g(0).$ Find $m \times t.$ | nvidia |
Find the smallest positive integer $n$ such that the sum of its divisors (excluding $n$ itself) is equal to the number of its divisors. | nvidia |
Determine the greatest integer $n$ such that $20!$ contains at least $n$ factors of 6. | nvidia |
Find the largest possible value of $x$ such that $3^x$ divides evenly into $13^{12}$. | nvidia |
Let $x$ be a real number such that $|x|<1$ and let $n$ be a positive integer.
Determine the smallest positive integer $n$ such that
\[\left|\sum_{k=1}^{n} \frac{x^{2k-1}}{2k-1}\right|<|x|." | nvidia |
There are two pairs $(x,y)$ of real numbers that satisfy the equation $x+y = 3xy = 4$. Given that the solutions $x$ are in the form $x = \frac{a \pm b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, what is the value of $a + b + c + d$? | nvidia |
Let $P$ be the set of points $(x, y)$ in the plane, such that $\frac{x}{2} + y \leq 1$. Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a point $(x, y) \in P$ such that $n(x^2 + y^2) = 1.$ | nvidia |
Count the total number of different integers that can be written in the form $a^2 \cdot b^3$, where $a$ and $b$ are positive integers less than or equal to $4$. | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-2ax+a^2=0$$
are such that
$x_1^2+x_2^2=4$.
Determine $a$. | nvidia |
Given a function $f(x)=|x^2-1|$, find the set of all $x$ for which $f(x)>f(3)$. | nvidia |
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