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Find the value of the expression \[ (m-2) \cdot (m-1) \cdot m \cdot (m+1) \cdot (m+2), \] where $m=3$. | nvidia |
Find all values of $x$ such that the equation
$$\frac{x^2+2x+1}{x^2+3x+1} + \frac{x^2+4x+2}{x^2+5x+2} + \frac{x^2+6x+3}{x^2+7x+3} = 3$$
is true.
Can you find all solutions to this equation? | nvidia |
A girl has $n$ consecutive positive integers that she wants to divide into three groups. Each group must have exactly one number that is a multiple of 3. The sum of the numbers in each group must be the same. What is the smallest value of $n$ for which this is possible? | nvidia |
Let $x$ be the number of positive integers less than $100$ that are multiples of $4$ but not multiples of $6$. Let $y$ be the number of positive integers less than $100$ that are multiples of $2$ but not multiples of $3$. Compute $(x - y)^2$. | nvidia |
Let $C$ be a circle centered at $(1,1)$ with radius $2$. Consider the set of all points inside $C$ for which the coordinates are integers. Find the number of points in this set that are not collinear with the center of the circle. | nvidia |
Find the sum of the infinite geometric series $$\frac{1}{2}+\frac{x}{2}+\frac{x^2}{4}+\frac{x^3}{8}+\cdots$$ where $x$ is a number such that $|x| < 2$. Express your answer in terms of $x$. | nvidia |
Find the number of positive integers $n$ less than 120 for which there exists an integer $m$ divisible by 4, such that the roots of the quadratic equation $x^2 - nx + m = 0$ are consecutive even positive integers. | nvidia |
Given the equation $z = \frac{x}{y}$, where $x$ and $y$ are complex numbers, find the positive value of $t$ such that $|x| = t$ and $|y| = \frac{1}{t}$, given that $|z| = 3$ and $\text{arg}(z) = \frac{\pi}{4}$. | nvidia |
Determine all real numbers $t$ that satisfy the inequality
$$\frac{2t}{\sin^2 t} < \frac{1}{t^2} + \frac{1}{\ln t}.$$
Express your answer in interval notation. | nvidia |
Find the sum of all positive integers less than 100 that are cubes of perfect cubes. | nvidia |
Problem:
If the sum of the squares of the real and imaginary parts of a complex number $z$ is equal to 10, find the modulus of $z^2$. | nvidia |
If $n$ is a positive integer, compute
\[
\left\lfloor \sqrt[n]{\frac{1}{n}}\cdot \frac{1}{\sqrt[n]{\frac{1}{n}}} \right\rfloor.
\] | nvidia |
Find the largest four-digit palindrome number that is divisible by 15 and has a sum of digits equal to 18. | nvidia |
A sequence of 20 consecutive positive integers is arranged in a 4x5 grid, such that the sum of the numbers in each row is the same. What is the smallest possible sum of the numbers in the grid? | nvidia |
Find the sum of the cubes of the solutions of $x^3-4x^2+3x-5=0$. | nvidia |
Find all perfect cubes between 100 and 200 that can be expressed as the sum of two perfect squares. | nvidia |
Let $a_n$ be the number of divisors of $2^n \cdot 3^{2n} \cdot 5^{3n}$ that are not divisible by $3$, and let $b_n$ be the number of divisors of $2^n \cdot 3^{2n} \cdot 5^{3n}$ that are not divisible by $5$. For how many positive integers $n$ do we have $a_n = b_n$? | nvidia |
Find the largest positive integer that is less than 1000 and whose digits sum to a multiple of 3. | nvidia |
How many positive integers have a square that lies between 4000 and 5000? | nvidia |
Find the sum of the positive whole number divisors of 300. | nvidia |
Consider a polynomial $P(x)$ satisfying the condition
\[P(x)-P(y)=(x-y)P(xy)\]for all real numbers $x$ and $y$. Given that $P(0)=1$, find $P(3)$. | nvidia |
The hyperbola
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$
has two foci that lie on the line $x=2$ and has an asymptote $y=4x$.
Find the equation of the other asymptote of the hyperbola in the form "$y=mx+b$". | nvidia |
Two geometric sequences, $A$ and $B$, both begin with 100 and have common ratios of 0.5 and 2 respectively. Determine the value of the index $n$ for which the sum of the first $n$ terms of sequence $A$ is equal to the sum of the first $n$ terms of sequence $B$. | nvidia |
In a triangle $ABC$, three ellipses are drawn, each with its major axis along one of the sides of the triangle and its foci at the vertices. A line passing through the midpoint of side $BC$ is such that the total length of the parts of the three ellipses to one side of the line is equal to the total length of the parts... | nvidia |
Consider a set of four integers $\{a, a+2, a+4, a+6\}$ for some integer $a$. We want to find the smallest positive integer $a$ such that each element in the set is divisible by a distinct perfect square ($p^2$, where $p$ is prime), and the squares are greater than or equal to $2^2$. | nvidia |
Find the sum of all real numbers $x$ satisfying the equation:
$\cos(2x) + 2\sin(x) + 5 = 0$
(Note: This problem requires a different approach, using trigonometric identities and properties, to find the sum of all real solutions.) | nvidia |
Let $a$, $b$, and $c$ be nonzero real numbers such that
$$(a+bi)(c+2i)$$is a real number. Find $\frac{b}{a}+\frac{2}{c}$. | nvidia |
Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $3(6x+1)\equiv 4\pmod p$. | nvidia |
Let $p(x)$ be a polynomial such that the equation $\frac{(x-3)(p(x)-2)}{(x+2)(x-1)} = 3$ has infinitely many solutions for $x.$ For this polynomial $p(x),$ it turns out that there are only finitely many values of $x$ which are not solutions to the equation. Find the product of these values of $x.$ | nvidia |
Given that $|2z-1|+|2z+1|=2|z-3i|$, determine the geometric shape that the complex number $z$ traces in the complex plane. | nvidia |
Consider a function $g$ defined on the set of integers, such that $g(0)=0$ and $g(n)=1+g(n-1)$ for all integers $n>0$. Also, $g(n)=g(-n)$ for all integers $n$. Find the value of $\sum\limits_{i=1}^{10} g(g(i))$. | nvidia |
A fair die is rolled twice, and the two numbers are added together. What is the probability that the sum of the numbers is either a prime number or a multiple of 3? | nvidia |
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed.
| nvidia |
The sum of the squares of three distinct integers is equal to the sum of the cubes of the same integers. Find all such triples of integers. | nvidia |
Evaluate the expression \[ \frac{a+2}{a+1} \cdot \frac{b-1}{b-2} \cdot \frac{c + 8}{c+6} , \] given that $c = b-10$, $b = a+2$, $a = 4$, and none of the denominators are zero. | nvidia |
Determine the smallest positive integer n such that the product of its proper divisors (excluding n itself) is equal to 2^n. | nvidia |
Find the sum of all positive integers $k$ that have a unique expansion of the form
\[k = (a_3a_2a_1a_0)_i\]
where $a_3 \neq 0$ and the base $i$ satisfies $i^2 = -1$. | nvidia |
The number 30 can be expressed as the sum of two positive integers in 29 different ways. How many ways can 60 be expressed as the sum of two positive integers? | nvidia |
Find the sum of all possible values of $n$ for which the expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $1:2:3$ that can be written in the form $\left\lceil\frac{n!}{k!(n-k)!}\right\rceil : \left\lceil\frac{n!}{(k+1)!(n-k-1)!}\right\rceil : \left\lceil\frac{n!}{(k+2)!(n-k-2)!}\right\rceil... | nvidia |
In a certain base $b$, the product of the two numbers $\overline{AB}$ and $\overline{BA}$ is equal to the sum of the two numbers $\overline{AA0}$ and $\overline{111}$. If $A$ and $B$ are distinct non-zero digits, find the base $b$. | nvidia |
Find the least positive integer $n$ such that $5616n$ is a multiple of $11609$. | nvidia |
Find the sum of all the divisors of 24 that are not multiples of 3. | nvidia |
The area of a triangle with vertices at $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ is given by the formula:
$$\text{Area} = \frac{1}{2} |x_1y_2 + x_2y_3 + x_3y_1 - x_1y_3 - x_2y_1 - x_3y_2|$$
Find the largest possible value of $n$ such that the area of the triangle with vertices at $(1,1)$, $(2,2^n)$, and $(3,3^n)$ is... | nvidia |
Let $f(x)$ be a monic polynomial of degree $n$ such that $f(x) = x$ for all $x \in \{-n, -n+1, \ldots, n-1, n\}$. Find the value of $f(n+1) + f(-n-1)$. | nvidia |
Find all real numbers $x$ which satisfy \[\frac{x-1}{x-3} \ge 2.\](Give your answer in interval notation.) | nvidia |
Given that $f(x) = \frac{x^2 - 1}{x^2 + 1}$, find the value of
\[ f(a)f(a^2)f(a^3)...f(a^n) \]
where $a$ is a real number such that $a^n = 1$ and $n$ is an odd positive integer. | nvidia |
Find the value of $x$ such that $\frac{x^3}{6} = 5!$. | nvidia |
A circle and a parabola intersect in four points. Prove that the line segments joining these points form a rhombus. | nvidia |
Find the sum of all positive integers n such that n is a divisor of at least one of the numbers 15, 16, or 17, but n is not a divisor of any of the numbers 20, 21, or 22. | nvidia |
A sequence of squares is formed by connecting the midpoints of the sides of a larger square. The resulting sequence of squares has side lengths that form a geometric series with common ratio $1/2$. If the area of the largest square is 16 square units, what is the sum of the areas of all the squares in the sequence? | nvidia |
Find the number of unordered pairs of distinct positive integers that have a product of 100 and a sum that is a perfect square. | nvidia |
Consider a 3D cube with 8 vertices. The sum of the x-coordinates of these vertices is 12, the sum of the y-coordinates is 8, and the sum of the z-coordinates is 4. The midpoints of the edges of the cube form the vertices of a second polyhedron, a rhombic dodecahedron. Find the sum of the x, y, and z-coordinates of the ... | nvidia |
Problem:
Consider a cube with side length 6 units, positioned so that one vertex is at the origin and the adjacent vertices lie on the positive coordinate axes.
Find the area of the triangle formed by connecting the points (3,0,0), (0,3,0), and (0,0,6). | nvidia |
Four red and four blue balls are to be arranged in a row such that no two balls of the same color are adjacent. How many different arrangements are possible? | nvidia |
A bag contains 20 identical marbles, each labeled with a unique number from 1 to 20. Three marbles are randomly drawn from the bag and their labels are noted. Then, these marbles are returned to the bag, and two new marbles are randomly drawn. What is the probability that at least one of the two new marbles has a label... | nvidia |
A point (x,y) is chosen at random inside the triangle with vertices (0,0), (0,1), and (1,0). Determine the probability that $x+y<\frac{1}{2}$ | nvidia |
Find the largest possible value of $x+y+z$, given that $x$, $y$, and $z$ are positive integers and that $xyz$ is a divisor of $120$. | nvidia |
Given an infinite sequence of numbers, for every positive integer $n$, it holds that
the average of the first $n$ terms of the sequence equals $n^2$.
Find the first three terms of the sequence. | nvidia |
Find the largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$. | nvidia |
Given that
\begin{eqnarray*}&(1)& a\text{ and }b\text{ are both integers between 1000 and 9999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& b\text{ is the number formed by reversing the digits of }a\text{; and}\\ &(3)& c=|a-b|. \end{eqnarray*}
How many distinct values of $c$ are possible? | nvidia |
Find the sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$. | nvidia |
Let $b_1 , b_2 , \dots$ be a sequence for which $b_1=3$ , $b_2=5$, and $b_n = b_{n-1} + b_{n-2}$ for each positive integer $n \ge 3$. Find the remainder when $b_{2006}$ is divided by 7. | nvidia |
Find all real values of $y$ that satisfy $\sqrt{y+4} - \frac{24}{\sqrt{y+4} - 2} = 7$.
Enter all the solutions, separated by commas. | nvidia |
Four standard six-sided dice are rolled. We are told there is no three-of-a-kind, but there is a pair of dice that show the same number. These two dice are set aside, and the other two dice are re-rolled. What is the probability that after re-rolling these two dice, at least three of the four dice show the same value? | nvidia |
Let $a$ and $b$ be positive real numbers such that $a^2 + b^2 = 1.$ Find the maximum value of
\[a + 2b.\] | nvidia |
A bookshelf has 5 shelves, and each shelf can hold 8 rows of books. If the bookshelf is currently empty, what is the 12th positive integer that represents the total number of books that can be placed on the bookshelf such that each row contains the same number of books, which is a multiple of 5? | nvidia |
Find the sum of all integers $k$ such that $\binom{23}{4} + \binom{23}{5} = \binom{24}{k}$. | nvidia |
If $m$ is the smallest positive integer for which there exist positive real numbers $x$ and $y$ such that
\[(x + yi)^m = (x - yi)^m,\]compute $\frac{y}{x}.$ | nvidia |
Find the smallest four-digit positive integer that has four distinct digits and its digits' product is divisible by 24. | nvidia |
Find the smallest positive integer $x$ such that $x$ can be expressed as the sum of two distinct prime numbers, and $x$ can also be expressed as the product of two distinct composite numbers. | nvidia |
Find the number of ways to arrange $n$ identical white balls and $n$ identical black balls into a sequence of $2n$ balls such that no two adjacent balls are white. | nvidia |
In the set of all positive integers whose prime factorization contains only the primes 2, 3, and 5, a number is chosen at random. What is the probability that it is a perfect square? | nvidia |
Determine the maximum value of $k$ such that the equation $x^2 + kx + 1 = 0$ has real roots. | nvidia |
Let $T$ be the set of all possible binary strings of length $n$, where $n$ is a positive integer. Three strings are chosen independently and at random with replacement from the set $T$ and labeled $s_1, s_2,$ and $s_3$ in the order they are chosen. The probability that both $s_1$ is a prefix of $s_2$ and $s_2$ is a pre... | nvidia |
The graph shown represents the equation $y = ax^2 + bx + c$. Find $c$.
[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 tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=t... | nvidia |
Determine which of the following products has the largest value and express it as a decimal number:
$$0.25 \times 0.75, \ \ 0.33 \times 0.66, \ \ 0.4 \times 0.6, \ \ 0.5 \times 0.5$$ | nvidia |
There are two numbers, $x$ and $y$, such that when $x$ is placed to the left of $y$, the number formed is exactly four times the sum of $x$ and $y$. If the difference between $x$ and $y$ is 25, what is the sum of $x$ and $y$? | nvidia |
Find the sum of the $x$-coordinates of the points of tangency of the parabola $y=x^2+3x-7$ with the lines passing through the point $(2,-5)$. | nvidia |
Find the largest value of x such that
$$\frac{x^2+1}{x^2-1}>2$$ | nvidia |
Semiprime numbers are numbers that are the product of exactly two prime numbers. What is the sum of all two-digit semiprime numbers less than 60? | 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 $4y=2g(3x)-1$. What is the product of the coordinates of that point? | nvidia |
Find the sum of all positive integral values of $m$ such that $m^2-18m+77= q$ for some prime number $q$. Note that there is at least one such $m$. | nvidia |
In the equation
$$2x^2-kx+3=0$$
determine $k$ such that the product of the roots be equal to twice their sum. | nvidia |
In a regular polygon with 24 sides, all the angles are equal in measure. How many degrees are in the sum of the measures of all the angles in the polygon? | nvidia |
Find the largest prime divisor of the product of all positive integers less than or equal to 20 that are relatively prime to 20. | nvidia |
Find the largest number $k$ such that the equation
$$x^2-kxy+y^2=1$$
has no integer solutions for $x$ and $y$. | nvidia |
Find the set of values for $x$ for which the function $\frac{x^3-2x^2+4x+7}{x^2+2x-3}$ takes only positive values. | nvidia |
A number is randomly selected from the set {1,2,3,...,n}, and then another number is randomly selected from the remaining numbers in the set. What is the probability that the two numbers are relatively prime? | nvidia |
Find all values of $x$ such that $[x] + [2x] + [3x] = 12$, where $[x]$ denotes the greatest integer less than or equal to $x$. | nvidia |
A regular icosahedron is formed by joining the centers of adjacent faces of a dodecahedron. The ratio of the volume of the icosahedron to the volume of the dodecahedron is
$\mathrm{(A) \frac{\sqrt{5}}{20} } \qquad \mathrm{(B) \frac{\sqrt{10}}{32} } \qquad \mathrm{(C) \frac{1}{10} } \qquad \mathrm{(D) \frac{\sqrt{3}}{16... | nvidia |
Let $m$ be a positive integer, and let $b_1, b_2, \ldots, b_m$ be distinct integers such that
\[b_1 4^{2008} + b_2 4^{2007} + \cdots + b_m 4^{2009-m} = 2^{2008}.\]Find the smallest possible value of $m$ such that the above equation holds for some sequence of integers $b_1, b_2, \ldots, b_m$. | nvidia |
A regular hexagon is inscribed in a circle. The radius of the circle is 8 cm. Find the area of the hexagon if the ratio of the area of the hexagon to the area of the circle is 3/4. | nvidia |
A polynomial $f(x)$ of degree $4$ is chosen randomly and uniformly from the set of all monic polynomials with integer coefficients, such that the sum of the absolute values of the coefficients is less than or equal to $10$. Find the probability that $f(x)$ has at least one real root. | nvidia |
Let c be a real number randomly selected from the interval $[-25,25]$. Then, p and q are two relatively prime positive integers such that p/q is the probability that the equation $x^4+36c^2=(9c^2-12c)x^2$ has $\textit{at least}$ two distinct real solutions. Find the value of $p+q$. | nvidia |
What is the smallest positive integer $n$ such that there exist two different convex polygons, each having $n$ sides, with integer side lengths and equal perimeter? | nvidia |
In the equation $x^2 + kx + 1 = 0,$ determine $k$ such that the sum of the roots is equal to $-3$. | nvidia |
What is the maximum possible value of $n$ such that, given a set of $n$ non-collinear points on the unit circle, the centroids of all possible triangles formed by choosing three points from the set are distinct? | nvidia |
Determine $a$ such that the quadratic equation $x^2+ax+1=0$ has real roots whose reciprocals are integers. | nvidia |
Determine the last two digits of the number $7^{7^{2007}}$. | nvidia |
Let $z$ be a complex number such that $z^3 = -8i$. Find the maximum value of $|z-2+3i|$ | nvidia |
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