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The vertices of a triangle are located at $A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)$. Given that the equation of the line that contains the side $\overline{AB}$ of the triangle is $(y_1-y_2)x-(x_1-x_2)y+(x_1y_2-x_2y_1)=0$, find the coordinates of the center of gravity (centroid) of the triangle.
Note: If a triangle has v... | nvidia |
For all prime numbers $p$, what is the smallest integer that always divides into the sum of $p$ and the square of $p$? | nvidia |
Find the number of distinct four-digit positive integers whose digits sum up to 10, where each digit is greater than or equal to 2. | nvidia |
Determine $x+yi$ such that $(x+yi)^2=7+24i$. | nvidia |
Let $m$ and $n$ be positive integers such that the product $mn$ can be expressed as a sum of consecutive integers from 1 to $k$. Determine the number of such pairs $(m,n)$, if $k=15$. | nvidia |
Let $g$ be a function defined on the positive integers, such that
\[g(x+y) = g(x) \cdot g(y)\]for all positive integers $x$ and $y.$ Given $g(2) = 3$ and $g(6) = 27,$ find $g(12).$ | nvidia |
Find the smallest positive integer that is a perfect cube and also a sum of consecutive squares. | nvidia |
A number $x$ is equal to $7\cdot24\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? | nvidia |
In a right-angled triangle, the sum of the squares of the three side lengths is 1800. What is the length of the hypotenuse of this triangle?
Another problem:
A right-angled triangle has two sides of lengths $a$ and $b$. The area of the triangle is $\frac{1}{2}ab$. Find the length of the hypotenuse of the triangle if... | nvidia |
In a rectangle, two congruent circles are drawn such that they are tangent to each other and to the sides of the rectangle. A line is drawn through the center of the circles and the points where the circles touch the sides of the rectangle, forming two smaller right triangles. The area of one of the right triangles is ... | nvidia |
Given a set of points in the plane, find the point that minimizes the sum of distances to all other points in the set. Specifically, consider the set of points {(1,1), (3,4), (6,2), (8,7)}. Find the point (x, y) that minimizes the sum of distances to all other points in the set. | nvidia |
Let $x$ and $y$ be two positive real numbers such that $x + y = 12$ and $x^2 + y^2 = 120$. Find the value of $xy$. | nvidia |
Find the product of $10101_2$ and $101_2$. Express your answer in base $2$.
Another problem:
Find the sum of all the odd numbers from $1$ to $100$ that can be expressed as the product of two distinct prime numbers. | nvidia |
Find the sum of all positive integral values of $n$ for which $\frac{n+6}{n}$ is an integer. | nvidia |
Find the equation of the line that passes through the midpoint of the line segment with endpoints (2, 3) and (8, 15), and is perpendicular to the line that passes through the points (1, 2) and (4, 5). | nvidia |
In a convex quadrilateral, the lengths of three sides are given as 8 cm, 10 cm, and 12 cm. The length of the fourth side can be expressed as x. Determine the maximum possible value of x such that the quadrilateral can be inscribed in a circle with an integer radius. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 3.$ Find the minimum value of
\[\frac{4}{x} + \frac{9}{y} + \frac{16}{z}.\] | nvidia |
In triangle $ABC$, $D$ is a point on side $BC$ such that $BD:DC = 2:3$. If $AE$ is parallel to $BC$ and intersects $BD$ at point $F$, find the ratio of the area of triangle $BEF$ to the area of triangle $ABC$. | nvidia |
Given that $x^3-9x^2+11x-1=0$ has roots $a,b,c$, find the value of $(a+b+c)^2-(ab+bc+ca)+2abc$. | 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 |
Let $P$ be a set of $2000$ points, each assigned a weight of $1$ or $-1$, such that the sum of the weights is $0$. Find the probability that there exists a linear functional $f$ on $\mathbb{R}^2$ and a number $s$ such that the set of points where $f(x,y) < s$ contains exactly half of the weight of $P$. | nvidia |
Problem:
How many positive integers $n$ satisfy $200 < n^2 < 900$?
New Problem:
How many positive integers $n$ satisfy $200 < n^2 < 900$ and have exactly one of their digits equal to 5? | nvidia |
If $z$ is a complex number such that $|z|=5$, find $z\overline{z}$. (Note that $\overline{z}$ denotes the conjugate of $z$.) | nvidia |
Find the number of ways to color the vertices of a regular hexagon with six different colors such that no two adjacent vertices have the same color and the resulting coloring is not a rotation of another coloring. | nvidia |
What is the smallest positive integer $n$ such that $3n$ is a perfect square and $2n$ is a perfect cube? | nvidia |
In a right triangle with legs parallel to the coordinate axes, the vertices are $(0,0)$, $(0,4)$, and $(a,0)$ for some positive integer $a$. The medians of the triangle intersect at a point with coordinates that are integers divided by $5$. Find the value of $a$. | nvidia |
Find the largest value of $a$ for which there exist real numbers $b$ and $c$ such that the parabola $y = ax^2 + bx + c$ passes through the points $(0, 0)$, $(1, 1)$, and $(2, 4)$, and has a vertex with non-negative $x$-coordinate. | nvidia |
Let $x$ be a real number such that $x^3+4x=8$. Determine the value of $x^7+64x^2$.
Another problem inspired by this one:
Find all real values of $x$ that satisfy the equation $x^5+4x^2+2x=10$, and then calculate the value of $x^9+16x^4$ for each solution. | nvidia |
The smallest integer greater than 1 that has exactly 12 divisors can be written in the form $p^a \cdot q^b$, where $p$ and $q$ are prime numbers and $a$ and $b$ are non-negative integers. Find $p + q$. | nvidia |
How many different positive integers can be represented as the sum of two distinct members of the set $\{1, 2, 4, 8, 16, 32\}$? | nvidia |
Let $a$, $b$, $c$ be positive real numbers such that $a+b+c=10$ and $ab+bc+ca=25$. What is the smallest possible value of $a^2+b^2+c^2$? | 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 |
There are values $C$ and $D$ such that
\[\frac{Dx+15}{x^2+8x+12}=\frac{2}{x+2}+\frac{C}{x+6}.\]
Find $C+D$. | nvidia |
Find all possible values of $a$ such that the equation $x^3 + ax + 27 = 0$ has at least one real root in the interval $(-3, 3)$. | nvidia |
Find the sum of all possible positive integer values of $c$ such that the quadratic equation $3x^2 + 7x + c = 0$ has rational roots. | nvidia |
What is the sum of all integer values of n such that 20/(2n-1) is an integer?
New Problem:
Determine the sum of all integer values of m such that (3m-2)/(m+1) is an integer. | nvidia |
Find all real numbers $x$ such that $[x] + x = 10$, where $[x]$ is the greatest integer less than or equal to $x$. Express your answer as a decimal. | nvidia |
Let $x$ be a real number such that $x^3+4x=8$. Determine the value of $x^7+64x^2$.
Another problem inspired by this one:
Find all real values of $x$ that satisfy the equation $x^5+4x^2+2x=10$, and then calculate the value of $x^9+16x^4$ for each solution. | nvidia |
In a regular hexagon, a circle is inscribed and six smaller circles are inscribed in the triangles formed by the sides of the hexagon and the radii of the large circle. If the radius of the large circle is 12 cm, find the area of the region inside the hexagon and outside all seven circles. | nvidia |
Find the value of $n$ such that $x^2 + nx + 72$ can be factored as the product of two linear factors with coefficients that are consecutive integers. | nvidia |
Find the sum of the infinite series $\ds\f{1}{3}+\ds\f{1}{6}+\ds\f{1}{12}+\cdots$. | nvidia |
What is the sum of the last two digits of the sum of all the even positive integers between 100 and 200? | nvidia |
A circle of radius $5$ has its center at $(3,4)$. Points $A$ and $B$ are on the circle, and the line through $A$ and $B$ intersects the circle at $C$. If the line through $A$ and $B$ has slope $2$, find the length of segment $AB$. | nvidia |
Find the value of $b$ so that the lines with the given equations are parallel.
\begin{align*}
y &= 4x-7 \\
5y-bx &= 5.
\end{align*} | nvidia |
Determine the smallest positive integer n such that the sum of the cubes of the first n positive integers is a multiple of 144. | nvidia |
Find the number of real solutions to the equation $$|x^3+2x^2-7x-12|^{x^2-4}=1.$$ | nvidia |
Given that $a$ and $b$ are positive integers, find the number of solutions to the equation $a! \cdot b! = (a + b)!$. | nvidia |
Let $S$ be the sum of the first $n$ positive integers that are divisible by $4$. Find the largest integer $m$ such that $S$ is divisible by $2^m$. | nvidia |
Three players simultaneously toss three coins each. The probability that at least one player gets all his coins the same face (either all heads or all tails) is $\ds\frac mn$, where m and n are relatively prime positive integers.
Find $\ds m+n$. | nvidia |
Find the smallest positive integer $n$ such that $2^{2n} + 2^{n+3} + 1$ is a perfect square. | nvidia |
The sum of the squares of two consecutive positive integers is 650. What is the sum of their reciprocals? | nvidia |
Find the number of different positive integers that can be expressed as the sum of two distinct elements from the set $\{1, 2, 4, 8, 16, 32\}$. | nvidia |
Find the largest three-digit number that is not divisible by any of the three smallest prime numbers. | nvidia |
A mapping $f$ is defined on the positive integers by the following rule: If the integer $n$ is written in decimal notation as $a_1 a_2 \dots a_p$, then $f(n)=a_1+a_2+\cdots+a_p$.
At how many positive integers $n$ is the sequence $(n,f(n),f(f(n)),\dots)$ purely periodic for $k$ terms? | nvidia |
Consider an infinite geometric sequence whose terms are all positive integers. If the square of the fifth term equals the sum of the first four terms, determine the value of the common ratio. | nvidia |
The product of the first 1000 digits of the decimal expansion of $\frac{1}{7}$ is a number that ends with exactly $n$ zeros. Find $n$. | nvidia |
How many four-digit positive integers have at least one digit that is a 2 or a 3? | nvidia |
Find the sum of the first seven composite numbers that have a units digit of 3. | nvidia |
For all prime integers $p$, what is the smallest integer that always divides into the sum of $p$ and the square of $p$? | nvidia |
The sum of the digits in a 5-digit number is equal to the sum of the digits in a 3-digit number. The 5-digit number is 12345, and the 3-digit number has distinct digits. What is the largest possible value of the 3-digit number such that the sum of its digits is equal to the sum of the digits in the 5-digit number? | nvidia |
Find the number of positive integers whose cube roots are within the range of 10.2 and 10.3. | nvidia |
Find the integer $n,$ $-180 < n < 180,$ such that $\cos n^\circ = \cos 240^\circ.$ | nvidia |
Find the smallest four-digit palindrome number that is divisible by each of its distinct, non-zero digits and has the sum of its digits equal to 18. | nvidia |
Given that $x$ is a positive integer, find the number of real numbers $y$ such that $x^2 - 4xy + 4y^2 = 1$. | nvidia |
The two positive integers a and b such that a + b + ab = 137 are represented by two distinct prime numbers. Calculate the positive difference between the numbers a and b. | nvidia |
There are three positive integers that are divisors of each number in the list $$24, 48, -8, 80, 60.$$Find the sum of these three positive integers. | nvidia |
Consider a quadratic equation of the form $ax^2+bx+c=0$.
Find $k$ such that the equation $ax^2+(b+k)x+c=0$ has two distinct solutions when the original equation has a double root. | nvidia |
Determine the number of ways to paint a cube with 6 different colors, such that no two adjacent faces have the same color. | nvidia |
In a triangle, the length of one side is 8 units, and the lengths of the other two sides are $x$ and $3x-2$ units. If the sum of the squares of the lengths of all three sides is 336, what is the length of the side opposite the largest angle? | nvidia |
Let $w$ and $z$ be two complex numbers such that $|w|=|z|=1$ and $|w-z|=1$. Determine the number of possible values of $\ds\left|\frac{w+z}{w-z}\right|$.
Please note that this is just one possible problem inspired by the given one. There may be many other problems that can be created depending on the approach and the ... | nvidia |
The number $1478$ can be expressed as $23m + n$, where $m$ and $n$ are positive integers. Find the greatest possible value of $m - n$. | nvidia |
Consider the function $$f(x) = \frac{ax-b}{x^2-c},$$ where $a,$ $b,$ and $c$ are constants. Find all values of $x$ such that $f(f(x)) = x.$ | nvidia |
Find the sum of all three-digit positive integers that are perfect cubes and have a digit sum equal to 18. | nvidia |
Given the problem:
Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a^2 + b^2 + c^2 = 1.$ Find the maximum value of
\[2ab \sqrt{2} + 2bc.\]
Here's another problem inspired by this one:
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^3y^2z = 1.$ Find the minimum value of
\[ \frac{x^2}{y} + yz... | nvidia |
Find all real numbers $x$ that satisfy the equation
$$\left|\frac{x+3}{x-1}\right|^{x-1}=1.$$
Be sure to consider all possible cases. | nvidia |
Given positive integers $m$ and $n$, determine the smallest integer $k$ greater than 1 such that $k^{-1} \pmod{m}$ and $k^{-1} \pmod{n}$ are both defined and $k^{-1} \pmod{m} + k^{-1} \pmod{n}$ is also an integer. | nvidia |
Determine the least positive integer $n$ such that $\left\lceil\sqrt{n}\right\rceil$ is a perfect square and $n$ is a prime number. | nvidia |
The altitude drawn from the vertex $C$ of an acute triangle $ABC$ intersects side $AB$ at point $D$.
Let $\mathbf{c}$ be the vector from $C$ to $D$, and let $\mathbf{a}$ and $\mathbf{b}$ be the vectors from $C$ to $A$ and $B$ respectively.
Given that the area of triangle $ABC$ is 21, find the magnitude of $\mathbf{c}... | nvidia |
Determine the value of the following expression:
\[\begin{vmatrix} a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix},\]
where $a,b,c$ are positive integers such that $a<b<c$. | nvidia |
Let $x$ and $y$ be real numbers such that $2(x^2 + y^2) = x + y.$ Find the maximum value of $x - y.$ | nvidia |
Let $a$ be a real number, $|a| < 1,$ and let $w$ be a complex number such that
\[w - \frac{1}{w} = a.\]Find $|w|.$ | nvidia |
In a triangle with side lengths 5 cm, 12 cm, and 13 cm, find the radius of the incircle, given that the area of the triangle is 30 square centimeters. | nvidia |
Determine the number of integers less than or equal to 50 that have a unique prime factorization with only 2s and 3s. | nvidia |
Find the sum of the squares of the solutions to the equation $x^4-17x^3+26x^2-24x-60=0$. | nvidia |
Let $a$, $b$ be positive integers such that $a-b=6$. Denote by $p$ the smallest prime divisor of $b$. Determine $p$ given that $\text{gcd}\left(\frac{a^3+b^3}{a+b}, ab\right) = p^2+1$. | nvidia |
A sphere with center $(-2, 5, 7)$ and radius $r$ passes through the point $(0, 0, 0)$. What is the sum of the $x$-coordinates of the two points on the sphere that are also on the plane $x = y$? | nvidia |
Find the smallest positive six-digit number that is divisible by each of the six smallest prime numbers. | nvidia |
Suppose that $x$ and $y$ are positive integers such that $x^2-xy+y^2=2008$. Find the greatest possible value of $x-y$. | nvidia |
An octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 108 centimeters. What is the number of square centimeters in the area of the octagon? | nvidia |
$n$ dice are rolled simultaneously. The probability that at most two of them show a 6 is $\frac{3}{8}$. Find $n$. | nvidia |
What are the last two digits in the sum of factorials of the first 100 positive integers? | nvidia |
What is the smallest positive integer that has exactly four distinct positive divisors, and whose prime factorization has at least two distinct prime factors? | nvidia |
What positive real number $x$ has the property that
$\ds x \cdot \lfloor x \rfloor = 6.99$?
Note that here $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. | nvidia |
Find a cubic function that has zeroes at $x=1$, $x=2$, and $x=3$, and that takes the value $12$ when $x=4$. | nvidia |
The sum of two positive integers is 24 and the sum of their squares is 556. What is the product of the integers? | nvidia |
The vertices of a triangle are $(2,4)$, $(7,6)$, and $(5,1)$. Reflect this triangle across the line $y=x$. Find the sum of the coordinates of the point which is the image of the vertex $(2,4)$, after this reflection. | nvidia |
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 3. If $\frac{S_{3n}}{S_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term. | nvidia |
Let $\mathbf{A}$ be a $3 \times 3$ matrix, with real entries, such that $\mathbf{A}^2 = \mathbf{A}.$ Find the maximum possible rank of matrix $\mathbf{A}$. | nvidia |
A small town has three different types of streetlights: ones that turn on every 3 nights, ones that turn on every 5 nights, and ones that turn on every 7 nights. The town has exactly one streetlight of each type at each intersection, and all streetlights are turned on tonight. In how many nights from tonight will all i... | nvidia |
Find the sum of all the odd integers from 1 to 99, excluding those that end with the digit 5. | nvidia |
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