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The sum of the squared deviations of five numbers from their mean is 40. If two numbers $a$ and $b$ are added to the list, the sum of the squared deviations of the seven-member list from its mean becomes 120. What is the sum of the squared deviations of $a$ and $b$ from the mean of the seven-member list? | nvidia |
Let $a$, $b$, $c$ be positive integers such that $a^2b^3c^4 = 6^9$.
Find $a+b+c$. | nvidia |
Find the sum of all the roots of the equation $x^{17}-x^{15}+x^{13}-x^{11}+\cdots +x^3-x = 0$. | nvidia |
The product of two positive integers is 120, and their sum is 30. What is the smallest positive integer that can be added to each of the two integers such that the new sum is a perfect square? | nvidia |
Find the smallest integer that is the sum of three distinct prime numbers, each greater than 5. | nvidia |
Let $a$, $b$, $c$ be three distinct real numbers such that
$$\begin{vmatrix}
0 & 2b & 1\\
1 & b+c & c\\
1 & bc & a
\end{vmatrix}=0.$$
Find the value of $a$. | nvidia |
What is the sum of all integer values $n$ for which $\binom{20}{n}+\binom{20}{10}=\binom{21}{11}$? | nvidia |
For how many positive real values of $y$ is $\sqrt[3]{216-\sqrt{y}}$ a positive integer? | nvidia |
Find the smallest four-digit number that is divisible by the sum of its distinct, non-zero digits. | nvidia |
Determine the value of $b$ such that the graphs of the equations $y=x^2-2x+1$ and $y=-bx^2+5x-2$ intersect at exactly one point. | nvidia |
Find the sum of all possible values of $x$, where $x$ is a digit that makes the number $4x23$ a multiple of 11 and $x23$ a multiple of 5. | nvidia |
Let $n$ be a positive integer such that $n^2+1$ is divisible by the sum of the first $n$ positive integers. Determine $n$. | nvidia |
Determine the sum of the series $\sum_{k=2}^{\infty} \frac{2k-1}{k(k-1)}$ if it is known that it converges. | nvidia |
The quotient of two positive integers is $\frac{5}{2}$ and their product is 160. What is the value of the larger of the two integers? | nvidia |
Find the number of lattice points that satisfy the equation $xy = 2001^2$, where $x$ and $y$ are integers with the same parity. | nvidia |
It is known that, for all positive integers $k$,
$1^2+2^2+3^2+\ldots+k^{2}=\frac{k(k+1)(2k+1)}6$.
Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\ldots+k^2$ is a multiple of $200$.
| nvidia |
Find a cubic polynomial with integer coefficients such that $2+\sqrt{3}$ is a root, and all coefficients are less than 10 in absolute value. | nvidia |
What is the smallest positive integer that can be the sum of the first ten terms of an arithmetic sequence whose terms are positive integers, if the sequence's common difference is a prime number? | nvidia |
Find the area of the triangle with vertices $(6,5,3),$ $(3,3,1),$ and $(15,11,9).$ | nvidia |
A solid right prism with height $h$ and a regular polygon base with $n$ edges (each side having length $s$) is sliced by a plane passing through the center of the prism and two adjacent vertices of the polygon base.
Determine the value of $n$ such that when the solid is rotated around the line passing through the midp... | nvidia |
Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and
\[f(x - y) = \sqrt{f(xy) + 4}\]for all $x > y > 0.$ Determine $f(2023).$ | nvidia |
Find the smallest positive integer $n$ such that $n^2 \equiv 15 \pmod{17}$. | nvidia |
Find the number of positive integers that are less than or equal to 100 and have exactly three positive factors. | nvidia |
Find the value of $m$ such that the graph of the equation $y = mx^2 + 2x - 1$ is tangent to the circle centered at the origin with radius 2. | nvidia |
Let $a,$ $b,$ and $c$ be three positive real numbers whose sum is 2. If no one of these numbers is more than three times any other, then find the maximum value of the expression $ab + bc + ca.$ | nvidia |
Determine the number of different 7-digit positive integers that can be formed using each of the digits 1, 2, 3, 4, 5, 6, and 7 exactly once, such that the digit in the units place is greater than the digit in the tens place. | nvidia |
Let $f(x)=x^2-x-2$, and let $g(x)$ be a function such that
$g(g(x))=f(x)$ for all real numbers $x$. How many different functions $g$ satisfy this condition? | nvidia |
Find the smallest positive integer value of $c$ such that the equation $x^3 - cx + 5 = 0$ has exactly one real solution. | nvidia |
If $f(x)$ is a function whose domain is $[0,5]$, and $f(g(x))=x^2$ is defined for all real numbers $x$, what is the minimum number of distinct intervals that the domain of $g(x)$ can be divided into? | nvidia |
The parabola $y=x^2+ax+b$ has vertex $(h,k)$, where $h$ and $k$ are integers. After reflecting the parabola in the $y$-axis, the new parabola has vertex $(h',k')$, where $h'$ and $k'$ are also integers. Find $a+b$ if the area of the triangle formed by the two vertices and the origin is 32 square units. | nvidia |
Find the product of all positive integral values of $n$ such that $n^2-35n+306= p$ for some prime number $p$. Note that there is at least one such $n$. | nvidia |
What is the sum of all integer values $n$ for which $\binom{32}{16}+\binom{32}{n}=\binom{33}{17}$? | nvidia |
A number $x$ is such that $\cos^3 x = \frac{1}{8}$. Find the value of $\sin^2 x$. | nvidia |
The difference between the squares of two consecutive positive integers is 97. What are these numbers? | nvidia |
What is the least positive integer $n$ such that the sum of the digits of $3^n$ is greater than the sum of the digits of $2^n$?
(Remember, this problem should be solvable and reasonable) | nvidia |
In a regular hexagon $ABCDEF$, the length of each side is $1$ unit. A point $P$ is located inside the hexagon such that $\triangle PAB$ is an equilateral triangle. Determine the perimeter of the pentagon $PCDEF$. | nvidia |
Let n be a positive integer such that the sum of the squares of the digits of n is 50. What is the remainder when n is divided by 9? | nvidia |
A rectangular box has width $12$ inches, length $16$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$... | nvidia |
Find the smallest value of $y$ that satisfies the equation $|y+2|=9$. | nvidia |
A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that for each complex number $z$, $f(z)$ is equidistant from both $z$ and the origin. Given that $|a+bi|=8$, find $b^2.$ | nvidia |
Let $x$ and $y$ be real numbers. Given that $\frac{x+2y}{2x+y}=\frac{5}{6}$ and $x^2+xy+y^2=1$, find $\frac{x}{y}$. | nvidia |
Find the sum of all positive integers less than 100 that are cubes of perfect cubes. | 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 |
Consider a function $h$ defined on the positive integers, with the property that $h(x) + h(y) > x^3$ for all positive integers $x$ and $y.$ Let $k$ be such a function $h$ for which the sum $k(1) + k(2) + \dots + k(30)$ is minimized. Find the minimum possible value of $k(25).$ | nvidia |
In the figure, $ABCD$ is a rectangle and $M$, $N$ are the midpoints of $AB$ and $CD$, respectively. Find the maximum area of the rectangle such that a circle of radius 5 can be inscribed in quadrilateral $BMDN$. | nvidia |
Two real numbers $x$ and $y$ satisfy $x-y=4$ and $x^3-y^3=28$. Compute $xy$. | nvidia |
A polynomial equation of degree $n$ has $n$ real solutions. What is the sum of all the products of these solutions taken two at a time? | nvidia |
Find the number of positive divisors of 36. | nvidia |
Eight positive integers are written on the faces of an octahedron. Each vertex is labeled with the product of the four numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to 2401, then what is the sum of the numbers written on the faces? | nvidia |
Choose the smallest of the following products, and express it as a fraction in simplest form:
$$\frac{2}{3} \cdot \frac{3}{4}, \ \ \frac{2}{3} \cdot \frac{4}{5}, \ \ \frac{2}{3} \cdot \frac{5}{6}, \ \ \frac{2}{3} \cdot \frac{6}{7}, \ \ \frac{2}{3} \cdot \frac{7}{8}$$ | nvidia |
Find the largest integer $k$ such that $3^k$ is a divisor of $2019!$. | nvidia |
Problem:
Consider a polynomial equation of degree 6 with real coefficients, having two pairs of conjugate complex roots and two real roots. The sum of one pair of conjugate roots is equal to $2 + 3i$, and the product of the two real roots is equal to $-5$. Find the coefficient of the $x^4$ term in the polynomial.
Note... | nvidia |
Find the least positive integer $n$ such that $5616n$ is a multiple of $11609$. | nvidia |
Find the sum of the squares of the roots of the equation $x^6-3x^4+2x^2-1=0$ that satisfy $|x|>1$. | nvidia |
Find all real values of $x$ that satisfy $\frac{x(x+1)}{(x-4)^2} \ge 12.$ (Give your answer in interval notation.) | nvidia |
Find the number of integers $k$ such that the polynomial $x^3 - kx^2 + 2kx - k$ has three distinct roots that form an arithmetic progression. | nvidia |
Find the number of positive integers $n$ such that the product of $n$ and its mirror image (the number formed by reversing the digits of $n$) is a perfect square. | nvidia |
A square is inscribed in a circle of radius $2\sqrt{3}$ such that one vertex of the square lies on the diameter of the circle. Find the perimeter of the square. | nvidia |
Find the number of integers that can be expressed as the difference of two perfect squares, given that the integers are between 20 and 100. | nvidia |
Determine the smallest positive integer $n$ such that $n^3 + n^2 - n - 1$ is divisible by 13. | nvidia |
The numbers $x$ and $y$ are positive integers such that $x + y = k$, where $k$ is a positive integer. Find the maximum value of $x \cdot y$. | nvidia |
Find the number of real solutions to the equation $\sqrt{x^2+1} = \sqrt{x}+1$ where $x$ is a positive number. | nvidia |
The sum of two positive integers is 50 and their difference is 12. What is the value of the positive difference of the squares of the integers? | 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 |
Let $a$ and $b$ be positive real numbers. If the polynomial $x^2+ax+b$ has exactly two distinct real roots and $a+b=9$, find the value of the sum of the reciprocals of all possible values of $b$. | nvidia |
Find the sum of all values of $y$ for which the expression $\frac{y^2-4y+4}{y^2-4y-5}$ is undefined. | nvidia |
Find the smallest positive integer $n$ such that the number formed by concatenating $n$ with its cube ($n^3$) is a palindrome. | nvidia |
A circle passes through the points $(2,5)$, $(6, k)$, and $(0,9)$. Determine the value of $k$ such that the circle has a diameter of 10 units. | nvidia |
Consider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| nvidia |
Find the largest power of 2 that divides the product of all positive integers from 1 to 30. | nvidia |
A bag contains ten balls, some of which are red and the rest of which are yellow. When two balls are drawn at random at the same time, the probability that both balls are red is $\frac{1}{15}$. How many balls in the bag are red? | nvidia |
Let $T(n)$ denote the sum of the number of positive divisors of all positive divisors of $n$, including 1 and $n$. For example, $T(4)=1+2+2+3=8$. Find the number of positive integers $n \leq 1000$ for which $T(n)$ is a perfect square. | nvidia |
Two positive integers p and q are chosen such that p is the smallest positive integer with exactly four positive divisors and q is the smallest integer greater than 200 with only two positive divisors. What is p + q? | nvidia |
For all composite integers $n$, what is the largest integer that always divides into the difference between $n$ and the square of $n$? | nvidia |
Given that $x$ and $y$ are positive integers, find the number of solutions to $x + y = 133$ where $x \leq y$. | nvidia |
Given a nonzero number x such that $\{x\}, \lfloor x \rfloor,$ and $x$ form an arithmetic sequence, find the least positive integer $n$ such that $x^n$ is an integer. | nvidia |
Find the sum of all positive integers less than 100 that are perfect fourth powers and also perfect squares. | nvidia |
Let $f(x)$ be a function such that $f(0)=1$ and for all positive integers $x$, $f(x) = x \cdot f(x-1)$. Determine the value of the following expression: $\frac{f(x)}{f(x+2)}$. | nvidia |
Two fair six-sided dice are rolled. What is the probability that the sum of the numbers showing on the two dice is less than 6, given that at least one of the dice shows an even number? | nvidia |
Determine the number of non-degenerate triangles that can be formed with two sides of lengths 5 and 7 units and a third side that is also an integer length. Consider that some triangles may have the same set of side lengths but with a different arrangement of sides. Count each unique triangle only once. | nvidia |
In a rectangle $ABCD$ with integer dimensions, points $E$ and $F$ are chosen on the sides $BC$ and $AD$, respectively, such that the area of the triangle $AEF$ is half the area of the rectangle. If the perimeter of the rectangle is $48$ units, what is the maximum possible area of the rectangle? | nvidia |
Let \(x, y,\) and \(z\) be three positive real numbers such that \(x^2+y^2+z^2=3.\) Find the maximum value of the expression
\[\frac{x^2y^2+y^2z^2+z^2x^2}{xyz}.\] | nvidia |
Problem:
Find the smallest positive integer $n$ such that the difference between $\frac{1}{n}$ and $\frac{1}{n+1}$ is less than $\frac{1}{10}$.
Solution:
We want to find the smallest positive integer $n$ such that $\frac{1}{n} - \frac{1}{n+1} < \frac{1}{10}$.
We can rewrite the inequality as:
$$\frac{1}{n} - \frac{1... | nvidia |
Prove that $\sum_{k=0}^{n}\binom{n}{k}x^ky^{n-k}=(x+y)^n$, and then use this formula to find the value of $\sum_{k=0}^{49}\binom{99}{2k+1}$. | nvidia |
In a right pyramid with square base, the vertices of the base are the points (0,0,0), (1,0,0), (1,1,0), and (0,1,0). Find the coordinates of the vertex of the pyramid, given that its height is 2 units and the perpendicular from the vertex to the base intersects the base at the point (1/2,1/2,0). | nvidia |
Find the set of values that the function
\[g(x) = \frac{2 \cos x + \sin x}{\cos x - \sin x}\]
can take on as $x$ ranges over all real numbers such that $\cos x \neq \sin x$. Enter your answer using interval notation. | nvidia |
What is the smallest positive integer n such that the sum of the digits of n^3 is a multiple of 10? | nvidia |
What is the largest positive integer that is not divisible by any of the prime numbers less than 10? | nvidia |
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 $x$ be a real number such that $x^3+4x=8$. Determine the value of $x^7+64x^2$. | nvidia |
Determine the number of positive integers less than 70 that are not a perfect square, cube or fifth power. | nvidia |
Let $g(x)$ be a function such that $g(1)=1$ and $g(2x)=g(x)+1$ for all $x$. Find the value of $x$ such that $g(x)=7$. | nvidia |
Let $P$ be a point inside a regular tetrahedron $ABCD$ such that its distances from the centroids of the faces of $ABCD$ are all equal. If the volume of $ABCD$ is 10 cubic units, determine the volume of the solid obtained by connecting the centroids of the faces of $ABCD$ with point $P$. | nvidia |
Given a number $x$ that is equal to the product of two prime numbers $p$ and $q$, $x = p^m \cdot q^n$, what is the smallest positive integer $y$ such that the product $xy$ is a perfect square and also divisible by 5? | nvidia |
Find the maximum value of the sum of the four products $xy$, $yz$, $zw$ and $xw$, where $x$, $y$, $z$ and $w$ are distinct integers such that the sum of $x$, $y$, $z$ and $w$ is 24. | nvidia |
Find the maximum possible value of $x_1x_2+x_2x_3+\cdots+x_{n-1}x_n$ where $x_1, x_2, \ldots, x_n$ are positive integers with $x_1+x_2+\cdots+x_n=2010$ and $n$ is a positive integer greater than 1. | nvidia |
The polynomial $x^4 + x^2 + 1$ is factored as
\[x^4 + x^2 + 1 = q_1(x) q_2(x) \dotsm q_m(x),\]
where each factor $q_i(x)$ is a non-constant polynomial with integer coefficients. Find the smallest possible value of $m$ such that all the roots of the polynomial are real. | nvidia |
Given a differentiable function f ( x ) defined for all real numbers x , find f (0), if for all real numbers x
f ( x ^{2} ) + x ^{2} f ′( x ^{2} ) − xf ′( x ) = 1 − 2 x . | nvidia |
Find the number of distinct natural numbers from 1 to 1000 whose digits are distinct and form a geometric progression. | nvidia |
Find the largest integer $n$ such that $n^2 \leq 1234$. | nvidia |
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