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What is the largest number of different dominoes that can be created with six distinct numbers on the faces? | nvidia |
Find all possible values of $a$ and $b$ such that the area of a triangle with side lengths $a$, $b$, and $a+b$ is equal to its perimeter. | nvidia |
Let $T$ be the set of all positive integers. The function $g : T \to T$ satisfies the following two properties:
(i) First,
\[g \left( x + 1 \right) = g(x) + x + 1\]for all $x \in T.$
(ii) Second,
\[g \left( xy \right) = g(x) \times g(y)\]for all $x \in T$ and $y \in T$ such that $xy \in T.$
Let $m$ be the number of... | nvidia |
What is the sum of all positive integer divisors of 91? | nvidia |
The graphs of two functions, $f(x)$ and $g(x)$, are shown below.
Let $h(x) = f(g(x))$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, and $b$ is the number of points where the graphs of $y=g(x)$ and $y=h(x)$ intersect, then what is $a+b$?
[asy]
size(150);
real ticklen=3;
real tick... | nvidia |
Let $g$ be a function that takes nonnegative integers to real numbers, such that $g(1)=1$ and for any nonnegative integers $a$, $b$, and $c$,
\[g(a)g(b)+g(b)g(c)+g(c)g(a)=g(a+b+c)+g(a-b+c)+g(a+bc),\]provided that $a-b+c\geq 0$. Find the sum of all possible values of $g(10)$. | nvidia |
What is the smallest positive integer that has exactly eight distinct positive factors? | nvidia |
In a triangle $ABC$, the points $D$, $E$, and $F$ are the midpoints of the sides $BC$, $CA$, and $AB$, respectively. The point $G$ is the centroid of the triangle. Find the ratio of the area of the triangle $DGF$ to the area of the triangle $EFG$ if $AE$ is perpendicular to $BC$. | nvidia |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}^n = \mathbf{I}.\] | nvidia |
Six standard six-sided dice are rolled. We are told that exactly three dice show the same number, two other dice show the same different number, and the last die shows a third different number. These six dice are arranged randomly in a row. What is the probability that the three dice that show the same number are cons... | 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 |
Consider the function $g$ satisfying $g(x+y) = g(x) + \frac{y}{x}$ for all positive real numbers $x$ and $y$. If $g(2) = 5$, find the value of $g(7)$. | nvidia |
Consider the sequence defined by $a_n = \frac{n!}{(n+2)!}$ for $n \geq 1$. Find the sum of the first $k$ terms of this sequence, where $k$ is a positive integer. | nvidia |
Find the value of $x$ such that $.\overline{8} \cdot x =.\overline{2} \cdot (1-x)$ | nvidia |
The units digit of the product $1! \times 2! \times 3! \times \cdots \times 2006!$ is equal to $1$. Determine the last two digits of the factorial of the smallest integer $n$ such that $n!$ is divisible by $1000!$ and $n>1000$. | nvidia |
What is the sum of the last two digits of the least number that is divisible by the first three positive prime numbers? | nvidia |
Determine the degree of the polynomial $$\left(\frac{x^2+ax+b}{x^3+cx^2+d}\right)\left(x^4+ex^3+f\right)$$ where $a,b,c,d,e,f$ are all nonzero constants. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f((x - y)^2) = f(x)^2 - 2xf(y) + y^2\]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 |
Determine all possible real values of $a$ for which the expression $\frac{1}{\sqrt{a - x}} + \frac{1}{\sqrt{x - a}}$ can be simplified into a single function for all real values of $x$. | nvidia |
Let $k$ be a real number such that $k > 1$ and
\[\sum_{n=1}^{\infty} \frac{5n-1}{k^n} = \frac{13}{4}.\]Find $k.$ | nvidia |
Determine all possible values of $n$ such that the polynomial $x^3 + 2x^2 - 3x + 5$ can be expressed as the difference of two polynomials, one of degree $n$ and the other of degree $n+2$. | nvidia |
In an ellipse with semi-major axis $a$ and semi-minor axis $b$, the foci lie on a line that also contains two points on the ellipse whose distances from the center of the ellipse are consecutive whole numbers. If $a + b = 19$, what is the sum of the distances from the center of the ellipse to these two points? | nvidia |
Any five points are taken inside or on a square with side length $1$. Let a be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than $a$. Then $a$ is:
$\textbf{(A)}\ \sqrt{3}/3\qquad \textbf{(B)... | nvidia |
Find all real values of $a$ for which the equation $\frac{a}{x-1} + \frac{2}{x-2} = 3$ has two distinct solutions, and then determine the positive difference between these solutions for $a = 1$. | nvidia |
Given a set of whole numbers between $\sqrt[4]{7}$ and $\sqrt[4]{1500}$,
find the sum of the square of the largest whole number smaller than $\sqrt[4]{1500}$
and the square of the smallest whole number greater than $\sqrt[4]{7}$. | nvidia |
A table with three rows and three columns has nine squares. Five of these squares contain one of the integers 1, 2, 3, 4, 5, and the remaining four squares are empty. For each row and each column, the sum of the numbers in the filled squares equals the number of empty squares. Determine the value of the product of the ... | nvidia |
Let $\mathbf{u}$ be a vector with magnitude $1$ and direction $40^\circ$ counterclockwise from the positive $x$-axis, and let $\mathbf{v}$ be a vector with magnitude $1$ and direction $120^\circ$ counterclockwise from the positive $x$-axis. Find the dot product of $\mathbf{u}$ and $\mathbf{v}$. | nvidia |
The product of the positive integer divisors of a positive integer $n$ is 729. Find $n$. | nvidia |
A set of $n$ points in the plane are given, such that no three points are collinear. What is the minimum number of lines that can be drawn through pairs of these points, such that every point lies on at least one of these lines? | nvidia |
Let $g(x)$ be a function such that $g(2x+1) = 4x^2+2x+1$ for all real numbers $x$. Find $g(x^2+2)$. | nvidia |
Problem:
Let $S$ be the set of all positive integers $n$ such that $n$ is a divisor of 3240 and $n$ is not a multiple of 3. Find the number of elements in $S$ that are not prime numbers.
Note: I've tried to create a problem that requires a different approach to solve, focusing on the properties of divisors and prime n... | nvidia |
Find all real solutions to $x^3+(x+1)^3+(x+2)^3=(x+3)^3$. Enter all the solutions, separated by commas. | nvidia |
Let g be a function defined on the positive integers, such that
(i) g is a function which returns a positive integer
(ii) g(a + b) = g(a) + g(b) for all positive integers a and b, and
(iii) if a and b are positive integers satisfying a + b = g(a)g(b), then g(a) = b or g(b) = a.
Find the product of all possible values... | nvidia |
Given that the average of three consecutive integers is less than 25, what is the largest possible integer value among them such that their sum does not exceed 70? | nvidia |
What is the smallest positive integer $n$ such that the sum of its positive integer divisors is exactly twice the number of its positive integer divisors? | nvidia |
Find the smallest positive integer k such that 12! + k \* 11! is divisible by 13. | nvidia |
Determine all real values of x such that the equation $$(x^2 + 4x - 7)^2 = (2x + 5)^2$$ has no real solutions for x. | nvidia |
How many different arrangements of the digits 1, 2, 3, 4, and 5 can be made such that the resulting 5-digit number has exactly three digits that are multiples of 3? | nvidia |
Let $\mathbf{B}$ be a $3 \times 3$ matrix, with real entries, such that $\mathbf{B}^4 = \mathbf{0}.$ Find the number of different possible matrices that $\mathbf{B}^3$ can be. If you think the answer is infinite, then enter "infinite". | nvidia |
A five-digit number is chosen at random from the set of all five-digit numbers. What is the probability that the number chosen has at least one digit that appears more than once? Express your answer as a common fraction. | nvidia |
Let $N$ be the product of the positive integers from 1 to 12 inclusive. What is the largest number $m$ such that $N$ is divisible by $3^m$? | nvidia |
Find the monic cubic polynomial, in $x$, with real coefficients, which has roots that are the cube roots of unity. | nvidia |
Find all binary operations ⋆ that satisfy the properties a⋆(b⋆c)=(a⋆b)+c and a⋆a=0 for all nonzero real numbers a, b, and c. (Here + represents addition). Determine the value of 2023⋆(7⋆x) when x is equal to 5. | nvidia |
Find all complex numbers $z$ such that $z^4 = 16i$. Enter the solutions, separated by commas. | nvidia |
A circle is inscribed in a triangle with sides of lengths 5, 12, and 13, and its center is equidistant from all three sides of the triangle. Find the radius of the circle. | nvidia |
Given that $h(x)$ and $j(x)$ are rational functions such that $h(x) \cdot j(x) = \frac{x^2+1}{x^2-1}$, find $j(x)$ if $h(x) = \frac{x+1}{x-1}$. | nvidia |
What is the least four-digit positive integer, with all different digits, that is divisible by each of its digits? | nvidia |
Find the minimum value of $\ds\frac{x^2y^2+y^2z^2+z^2x^2}{x^2y^2z^2}$, given that $x,y,z$ are positive integers satisfying $\ds\frac{1}{x}+\ds\frac{1}{y}+\ds\frac{1}{z}=1$. | nvidia |
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?
$\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$
| nvidia |
How many different positive integers can be represented as the sum of two different integers between -10 and 10, inclusive? | nvidia |
Given the inequality $\left(\dfrac{3}{4}\right)^x<0.2$, find the smallest integer value of $x$ for which the inequality holds. | nvidia |
Find the smallest positive integer k such that 3715k is divisible by 1000000. | nvidia |
Tom is writing down all the positive integers that are divisible by 15 and have at least 3 different prime factors. What is the smallest number that Tom will write down? | nvidia |
Suppose that $a$ and $b$ are positive integers such that $(a-bi)^2 = 8-6i$. What is $a-bi$? | nvidia |
In the equation
$$5x^2-kx+1=0$$
determine $k$ such that the difference of the roots be equal to unity.
New problem:
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 $n$ be a positive integer. Find the sum of the squares of the $n$ roots of the equation $x^n = 1$, that is, the roots of unity. Express your answer in terms of $n$. | nvidia |
Find all rational numbers $x$ such that $\lfloor x\rfloor=x^2$, where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. | 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 |
The number of divisors of a positive integer $n$ is 16. Find the largest possible value of $n$ such that $n$ is less than 1000. | nvidia |
Given a positive integer $n$, find the maximum value of $\frac{x_1^{n-1} + x_2^{n-1} + \cdots + x_n^{n-1}}{x_1 + x_2 + \cdots + x_n},$ where $x_1,x_2,\ldots,x_n$ are positive real numbers such that $x_1x_2\cdots x_n=1$. | nvidia |
Let $S$ be a set of integers from 1 to $n$, where $n$ is a positive integer. Find the smallest value of $n$ 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 $a+b=c$. | nvidia |
Three complex numbers $a+bi$, $c+di$, and $e+fi$ are related such that $d=f$, $e=a+c$, and the product of the numbers is a real number. Determine $a+b+c$. | nvidia |
Let $g(n)$ return the number of distinct ordered pairs of positive integers $(a, b)$ such that for each ordered pair, $a^3 + b^3 = n$. What is the smallest positive integer $n$ for which $g(n) = 2$? | nvidia |
Determine the sum of the positive integer divisors of $97$ which are not multiples of $3$ and have the property that the sum of their digits is a divisor of $12$. | nvidia |
Find the difference between the sum of the greatest common divisor and the least common multiple of $a$ and $b$, and the sum of $a$ and $b$, where $a$ and $b$ are positive integers such that $\text{lcm}(a,b)+\text{gcd}(a,b)=42$ and $\text{lcm}(a,b)-\text{gcd}(a,b)=18$. | nvidia |
Let $S$ be the sum of the first $50$ positive even integers. Find the largest integer $k$ such that $S$ is divisible by $2^k$. | nvidia |
Let $x$ and $y$ be positive real numbers such that
\[x^3 + 3x^2y + 3xy^2 + y^3 = 64.\]Let $p$ and $P$ be the minimum and maximum values of $x^2 + xy + y^2,$ respectively. Find the value of $\frac{P}{p}$. | nvidia |
Let $x$ be the number of positive multiples of 5 that are less than 75, and $y$ be the number of positive integers that are less than 75 and multiples of both 5 and 3. Compute $(2x - 3y)^2$. | nvidia |
Find all values of $x$ that satisfy the equation $|x-2|=3x-1$. Express your answers in simplest fractional form. | nvidia |
In a geometric sequence, the sum of the first 1234 terms is 150, and the sum of the first 2468 terms is 300. Find the sum of the first 3702 terms. | nvidia |
Solve for the sum of all possible values of $x$ when $3^{x^2+4x+4}=9^{x+2}$. | nvidia |
What is the least four-digit positive integer, with all different digits, that is divisible by each of its digits? | nvidia |
The value of
\[\frac{n}{2} + \frac{18}{n}\]is smallest for which positive integer $n$? | nvidia |
Let $S_n$ be the sum of all positive three-digit integers that are divisible by $n$ and whose digits are in geometric progression. For which value of $n$ is $S_n$ maximized? | nvidia |
The quadratic equation $ax^2+bx+c=0$ has roots $x_1$ and $x_2$. Given that the product of the roots $x_1x_2=-1$, and the sum of the roots $x_1+x_2$ is a prime number $p$, find the smallest possible value of the difference between the roots $|x_1-x_2|$.
(Note: This problem requires a different approach to solve, using ... | nvidia |
Problem:
Determine the value of $a$ such that the coefficient of $x^5$ in the expansion of $(x^2-ax+1)^3$ is equal to $-3$.
[Finish here.] | nvidia |
Given that the point $(9,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=\frac{f(2x)}2+2$. What is the sum of coordinates of that point? | nvidia |
Find a prime number that is a factor of every sum of four consecutive integers. | nvidia |
Let $T_n$ denote the sum of the first $n$ terms of a geometric sequence with common ratio 2. If $\frac{T_{2n}}{T_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term. | nvidia |
Let $\{a_n\}$ be a sequence of consecutive integers such that the average of the sequence is 45. If the largest number in the sequence is 60, what is the smallest integer in the sequence? | nvidia |
Find the length of the side of the largest square that can be inscribed in a circle of radius 5. | nvidia |
Determine the value of $n$ such that $19n+13$ and $23n+19$ have the same remainder when divided by 7. | nvidia |
A triangular array of numbers is constructed with the following rule: each number is the product of the two numbers directly above it. The top row consists of the numbers 2, 3, and 5. What is the sum of the numbers in the fourth row of this triangular array?
\begin{tabular}{rccccccccccc}
Row 1:& & & & & & 2 & 3 & 5\\\... | nvidia |
Consider the operation $\Delta$ defined by $a\Delta b=a^2b+a-1$. If $x$ is a number such that $x\Delta (2x)=x\Delta (2x+1)$, find the value of $x$. | nvidia |
What is the maximum number of points that can be placed on a sphere of radius 1 such that no two points are closer than 1 unit apart? | nvidia |
A sequence of positive integers $b_1,b_2,\ldots$ is defined by the rule $b_n=2b_{n-1}-1$ if $b_{n-1}$ is even and $b_n=\frac{b_{n-1}+1}{2}$ if $b_{n-1}$ is odd. For how many positive integers $b_1 \le 1000$ is it true that $b_1$ is greater than each of $b_2$, $b_3$, and $b_4$? | nvidia |
Find the sum of all two-digit positive integers that are not multiples of 3 or 5. | nvidia |
A set of 5 consecutive odd numbers starts with a prime number. The sum of the squares of these numbers is 855. Find the middle number. | nvidia |
Determine $k$ such that the difference of the roots be equal to unity in the equation
$$5x^2-kx+1=0$$
given that the roots of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$. | nvidia |
Find the number of positive integers less than 10,000 that can be formed using the digits 1, 2, 3, and 4 if the number must contain at least one 1 and exactly two 2s. | nvidia |
In a Cartesian coordinate system, find the smallest positive integer $n$ such that the points $(n, n)$, $(n^2, n^3)$, and $(n^3, n^2)$ form a triangle whose perimeter is greater than 2015. | 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 |
Find the sum of the even integers from 20 through 58, inclusive. | nvidia |
Let $a$, $b$ and $c$ be positive real numbers satisfying $a+b+c=3$ and $\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}=5$. Compute $a^2+b^2+c^2$. | nvidia |
Given a positive integer $x$ that is a multiple of 4, find the greatest possible value of $x$ such that $x^2$ is less than 1000. | nvidia |
Find the sum of all positive integer values of $a$ such that the equation $3x^2 - 7x + a = 0$ has two real roots. | nvidia |
Find the values of the constant $a$ for which the graph of the function $f(x) = \frac{x^2+3x+a}{x^2-4x-21}$ has exactly one vertical asymptote.
Enter all possible values, separated by commas. | nvidia |
Suppose that $a$ and $b$ are positive integers such that $a^2+b^2=1162$ and $ab=4$. Determine $a+b$. | nvidia |
Consider the equation $2x + 3y = 5xy = 11$, where $x$ and $y$ are real numbers. Given that the solutions for $x$ are in the form $x = \frac{p \pm q\sqrt{r}}{s}$, where $p$, $q$, $r$, and $s$ are positive integers and the expression is completely simplified, find the value of $p + q + r + s$. | nvidia |
Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. | nvidia |
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