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Given a polynomial $P(x) = x^3 + ax^2 + bx + c$ with roots $r_1$, $r_2$, and $r_3$, find the value of $a$ such that the expression $\frac{1}{r_1^2r_2} + \frac{1}{r_2^2r_3} + \frac{1}{r_3^2r_1}$ is equal to a rational number $m/n$, where $m$ and $n$ are relatively prime positive integers. Determine the value of $m+n$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 4.$ Find the minimum value of
\[x + \frac{1}{y}.\] | nvidia |
Given that $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is a $2\times2$ matrix such that $A^2=5A-6I$, find $\det A$. | nvidia |
The function $f(x)$ satisfies
\[f(2^x) + xf(2^{-x}) = 1\]for all real numbers $x.$ Find $f(2).$ | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 100.$ Find the maximum value of $xyz$. | nvidia |
A circle centered at $(h,k)$ intersects the line $x = y$ at two points: $(n,n)$ and $(m,m)$. The circle also intersects the $x$-axis at two points: $(-5,0)$ and $(7,0)$. Given that $n$ and $m$ are integers, find $k$. | nvidia |
A four-digit number begins with a digit $a$, and ends with the digit $b$. The number formed by reversing the digits $a$ and $b$ is $90$ less than the original number. Given that the sum of $a$ and $b$ is $9$, find the value of $a$. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 12.$ Find the maximum value of $ab + ac + bc.$ | nvidia |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 1000^\circ.$ | nvidia |
Find the sum of the squares of the first 10 positive integers that are congruent to 1 modulo 3. | nvidia |
There are four points that are $3$ units from the line $y=7$ and $7$ units from the point $(5,7)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? | nvidia |
Given that $\sqrt{n} - \sqrt{n-1} > \frac{1}{5}$, what is the greatest possible positive integer value of $n$ such that this inequality holds true? | nvidia |
Given a number that is divisible by 9 and has the property that the difference between its last two digits is 5, what is the sum of its last two digits? | nvidia |
The integer $n$ is the largest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\frac{n}{15}$.
| nvidia |
Find the number of solutions of the equation $10 < -x + 13$ that are positive integer values.
New Problem:
Consider a non-decreasing sequence of positive integers $a_1, a_2, \ldots, a_n$. Determine the number of terms, $n$, if the sequence satisfies the condition: $\sum_{i=1}^{n} a_i < 15$. | nvidia |
The number $a$ is a three-digit number that can be expressed as the sum of three consecutive integers. If the units digit of $a$ is 5, find the units digit of the middle integer. | nvidia |
What is the largest whole number $n$ that can be added to 6531 such that the resulting number is still divisible by 9? | nvidia |
Find the maximum value of $5^x - 25^x,$ over all real numbers $x.$ | nvidia |
Find the volume of the region $B$ in the three-dimensional space that consists of all points $(x,y,z)$ such that $x^2+y^2+z^2 \leq 1$, $x \geq 0$, $y \geq 0$, and $z \geq 0$, and the projection of $(x,y,z)$ onto the $xy$-plane lies within the region bounded by the curves $y=x^2$ and $y=\sqrt{x}$. | nvidia |
How many distinct sequences of four integers can be made from the set $\{-3,-2,-1,0,1,2,3\}$ if each sequence must begin with a negative integer, end with a positive integer, and no integer can appear in a sequence more than once? | nvidia |
The area of a right-angled triangle is $\frac{1}{8}$. Determine the sum of the squares of its legs. | nvidia |
A regular octahedron has vertices at coordinates $(x_i,y_i,z_i)$ for $i=1,2,...,6.$ If the sum of the squares of the $x$-coordinates is $100,$ find the sum of the squares of the $z$-coordinates. | nvidia |
Three complex numbers, $p+qi$, $r+si$, and $t+ui$, satisfy the conditions $q=2$, $t=2p-r$, and the sum of the numbers is $3+5i$. Determine the value of $p-s-u$. | nvidia |
Find the greatest common divisor (GCD) of all positive integers $k$ for which $\mathop{\text{gcd}}[k,24]=8$. | nvidia |
Find the number of positive integers less than 100 that are divisible by 3 but not by 7. | nvidia |
Find the product of three consecutive positive integers if their sum is equal to their product minus 16. | nvidia |
Determine the smallest positive integer $k$ such that the $k$th digit after the decimal point of the decimal equivalent of $\frac{2}{11}$ is greater than the corresponding digit of $\frac{3}{13}$. | nvidia |
In a regular polygon, the sum of all exterior angles is equal to the sum of all interior angles. Find the number of sides of this polygon. | nvidia |
Let $g(x)$ be a function such that
$$\sum_{i=1}^x g(i)=xg(x),$$
for all positive integers $x$. If $g(10)=19$, calculate $g(12)$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $y^2 = x - 2$ and $y = \frac{2}{x}$. Find the value of $x$. | nvidia |
If $a$ and $b$ are two positive numbers such that their sum is equal to the sum of their squares, find the value of $\frac{1}{a^3b^4} + \frac{1}{a^4b^3}$. | nvidia |
Let $A$ be a $2\times 2$ matrix such that $A^2=I$, and the entries of $A$ are positive integers. What is the largest possible value of the determinant of $A$? | nvidia |
Find all values of $a$ such that the equation $(x^2-a)(x-a^2) = x^3 + 2$ has exactly two real solutions.
Note: This problem requires a different approach, focusing on the conditions for a quadratic equation to have exactly two real solutions, rather than the original problem's focus on the discriminant of a quadratic ... | nvidia |
The integer $n$ is the largest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\frac{n}{15}$.
| nvidia |
Find the sum of the first ten perfect squares that can be expressed as the sum of two consecutive integers. | nvidia |
Consider the sequence of numbers defined by $a_n = \frac{n!}{2^n}$, where $n$ is a positive integer. Find the largest value of $n$ such that $a_n$ can be expressed as a sum of distinct powers of $3$. | nvidia |
A sequence of seven positive integers has all of the following properties:
$\bullet$ The sum of the terms is 28,
$\bullet$ the sum of the squares of the terms is 120,
$\bullet$ the sum of the cubes of the terms is 400,
$\bullet$ the sum of the fourth powers of the terms is 1200.
What is the smallest possible v... | nvidia |
A rectangle has a perimeter of 24 cm. A circle with center at the rectangle's center is tangent to one of the long sides and one of the short sides. If the long side of the rectangle is reduced by 4 cm, then the area of the rectangle becomes an integer number of square centimeters. Find the radius of the circle. | nvidia |
Given that $\frac{1}{x-1} - \frac{1}{x+1} < \frac{1}{12}$, find the greatest possible integer value of $x$ such that $x$ is the hypotenuse of a right-angled triangle with integer legs. | nvidia |
Let c be a real number randomly selected from the interval $[0, 100]$. Then, p and q are two relatively prime positive integers such that p/q is the probability that the inequality $x^2+16x+|c|<0$ has at least one real solution. Find the value of $p+q$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 1.$ Find the minimum value of
\[\frac{1}{x} + \frac{1}{\sqrt{xy}} + \frac{1}{\sqrt[3]{xyz}}.\] | nvidia |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | nvidia |
Find the sum of all prime numbers that can be obtained by rearranging the digits of the number 1234. | nvidia |
What is the sum of all positive integer values of $n$ such that $\frac{n+18}{n}$ is an integer? | nvidia |
Compute the number of ways to arrange the letters in the word "CRICKET" such that the vowels are together and the consonants are together, but the vowels and consonants can be in any order among themselves. | nvidia |
Let $N$ be the smallest positive integer with exactly 15 positive divisors such that $N=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$, where $p_1,p_2,\dots,p_k$ are distinct primes and $e_1,e_2,\dots,e_k$ are positive integers. What is the largest possible value of $e_1+e_2+\cdots+e_k$? | nvidia |
Suppose $x$ and $y$ are integers such that $xy+5x+4y=-5$. Find the greatest possible value of $y$. | nvidia |
Let $m$ and $n$ be positive integers such that $m-n=3$. Determine the number of positive integers $k<1000$ such that $k\equiv n \pmod{m}$. | nvidia |
Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N.$ What is the perimeter of $\triangle AMN?$
$\textbf{(A)}\ 27 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 33 \qqua... | nvidia |
Consider the function $f(x) = \arctan\left(\frac{x^n - n}{\ln (mx)}\right)$ where $m$ and $n$ are positive integers, $m>1$. Find the range of values for $m$ and $n$ such that the domain of $f(x)$ is a single closed interval. | nvidia |
Find the smallest perfect number that is not a multiple of 4. A perfect number is a positive integer that is equal to the sum of its proper divisors, excluding the number itself. For example, 6 is a perfect number because 1 + 2 + 3 = 6. | nvidia |
Determine the number of pairs $(a,b)$ of real numbers such that the polynomial
\[x^4 + ax^2 + b = 0\]has exactly one real solution. | nvidia |
A set of 10 different integers is randomly arranged. What is the probability that the first two numbers are both odd and that the third number is greater than both of them? | nvidia |
Let $\alpha$ be a real number, $|\alpha| < \sqrt{3},$ and let $\beta$ be a complex number such that
$$\beta^2 + \frac{1}{\beta^2} = \alpha.$$
Find $\operatorname{Re}(\beta^2).$ | nvidia |
Find the number of positive integers $m$ such that the difference between the number of positive integer divisors of $m^2$ and the number of positive integer divisors of $m^3$ is equal to 10. | nvidia |
Find the sum of the LCM and GCD of two numbers a and b, where a and b are both prime numbers greater than 10, and their sum is 20. | 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 value of $b$ for which the graphs of $y=x^2-4x+b$ and $y=|x-2|$ intersect at exactly two points. | nvidia |
In a city, there are $n$ schools, each with $m$ students. The students are randomly assigned to $n$ different extracurricular activities, with each student attending only one activity. What is the probability that for every positive integer $k<\frac{m}{2}$, no collection of $k$ activities contains exactly $k$ students ... | nvidia |
Given that $x$ is real and $x^4-\frac{1}{x^4}=8x^2+\frac{8}{x^2}+14$, find $x^2+\frac{1}{x^2}$. | nvidia |
Let $x$ be a positive real number. Consider a sequence of consecutive positive integers $\{a_1,a_2,...,a_n\}$ such that for every $3$-element subset $\{a_i,a_j,a_k\}$ of this sequence, $x^2<a_i^2+a_j^2+a_k^2<2x^2$ holds. Find the maximum possible value of $n$ for $x=2\sqrt{3}$. | nvidia |
Let $a$, $b$, and $c$ be three positive real numbers such that $a+b+c=4$. Find the maximum value of $\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+a^2}$. | nvidia |
The product of two positive integers plus their sum is 103. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | nvidia |
Find the number of positive integers less than 20 that have exactly 4 distinct positive divisors. | nvidia |
Determine all real numbers $y$ such that the equation $\sqrt{y^2 + 2y - 1} = y + 3$ holds true. | nvidia |
Find the value of $a$ for which the area of the triangle whose vertices are the points $(0, 0)$, $(0, a)$, and $(a, a^2)$ is 5 square units. | nvidia |
Let $a$ and $b$ be real numbers such that $\sqrt{ab}=1$ and $a+b=-3$. If $\left|\frac{a}{b}+\frac{b}{a}\right|=4$, find the product of all possible values of $a$.
Can I help you with anything else? | nvidia |
A positive real number $x$ is such that $\sqrt[3]{1-x} + \sqrt[3]{1+x} = \sqrt[3]{2}.$ Find $x^3.$ | nvidia |
Let $x_1, x_2, \ldots, x_n$ be real numbers such that $x_1 + x_2 + \cdots + x_n = 0$. Find the largest real number $c$ such that
\[|x_1|^3 + |x_2|^3 + \cdots + |x_n|^3 \geq c|x_1 + |x_2| + \cdots + |x_n||^3\]
for all $n$ and $x_1, x_2, \ldots, x_n$. | nvidia |
Find all values of $x$ such that $\arctan x < \arccot x$. | nvidia |
Let $n$ be the largest integer with 2020 digits that is not divisible by 3. What is the smallest positive integer $m$ such that $n^2 + m$ is divisible by 9? | nvidia |
The Fibonacci sequence is defined by the recurrence relation $F_n=F_{n-1}+F_{n-2}$, where $F_1=1$ and $F_2=1$. Find the least positive integer $k$ such that $F_k$ is divisible by 17. | nvidia |
Find the length of the shortest path that connects the point $(-2,3)$ to the line $y=2x-1$ and then to the point $(4,1)$, given that the path must form a right angle where it meets the line. | nvidia |
Let $f(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. Given that $n$ is a multiple of 24, what is the smallest value of $n$ such that $f(n) > 20$? | nvidia |
Find the number of positive integers that divide 84 without leaving a remainder and are also multiples of 7. | nvidia |
Find the number of positive integers that are common factors of $32$ and $54$, excluding $1$ and the greatest common factor of the two numbers. | nvidia |
Let $x$ and $y$ be positive real numbers such that $7x + 5y = 70.$ Find the maximum value of $xy^2.$ | nvidia |
Find the equation of the surface that is the locus of points equidistant from the two spheres given by the equations
$(x-2)^2+(y+3)^2+(z-1)^2=9$
and
$(x+2)^2+(y-3)^2+(z+1)^2=16$. | nvidia |
Determine the minimum possible value of the sum
\[\frac{a}{2b} + \frac{b}{4c} + \frac{c}{8a},\]where $a,$ $b,$ and $c$ are positive real numbers. | nvidia |
Let $f$ be a function defined on the positive integers, such that
\[f(xy) = f(x) + f(y)\]for all positive integers $x$ and $y.$ Given $f(10) = 14$ and $f(40) = 20,$ find $f(500).$ | 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 |
Compute $1-2+3-4+ \dots -98+99$ . | nvidia |
What is the smallest positive integer $n$ such that the repeating decimal representation of $\frac{1}{n}$ contains the digit 4? | nvidia |
How many natural-number factors does $n$ have, where $n$ is the product of all the positive integers from 1 to 12? | nvidia |
Find the number of distinct triples $(a,b,c)$ of positive real numbers satisfying $a+b+c=10$ and $\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca} = \frac{1}{3}$. | nvidia |
Find the number of real 2x2 matrices A satisfying det(A)=1 and ||A+A^(-1)||=1, where ||.|| denotes the matrix norm. | 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 |
A tetrahedron $ABCD$ has vertices $A(1,0,0)$, $B(0,1,0)$, $C(0,0,1)$, and $D(3,2,1)$. A point $E$ is chosen inside the tetrahedron such that the four smaller tetrahedra $ABDE$, $ACDE$, $BCDE$, and $ABC$ all have equal volumes. Find the value of $2x + 3y + z$, where $(x,y,z)$ are the coordinates of point $E$. | nvidia |
The sum of the squares of three consecutive positive integers is 590. Find the smallest of these integers. | nvidia |
Find the sum of all divisors of 540 that are perfect squares. | nvidia |
Let $x$ and $y$ be positive real numbers such that
\[\sin (x + y) = \sin x + \sin y.\]Find the minimum value of $\tan x.$ | nvidia |
Find all real numbers $a$ such that the roots of the polynomial
$$x^3 - 6x^2 + 21x + a$$form an arithmetic progression and are not all real. | nvidia |
The sum of the angles between two adjacent vertices in a random permutation of a regular polygon with 12 vertices is computed. The average value of all such sums can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | nvidia |
Find the number of positive integers less than or equal to 150 that are not divisible by 3 and have exactly two positive divisors. | nvidia |
Let $a$ and $b$ be positive integers such that $a+b=ab$. Determine the number of distinct prime factors of the sum of the positive divisors of $a^b$. | nvidia |
What is the largest four-digit number that can be formed using the digits 2, 4, 6, and 8, with the restriction that the sum of the first two digits is equal to the sum of the last two digits? | nvidia |
Let $x$, $y$ be positive real numbers such that $x^{\frac{1}{2}}+y^{\frac{1}{2}}=1$ and $x^{\frac{3}{2}}+y^{\frac{3}{2}}=2$.
Find $x^2+y^2$. | nvidia |
Let $p$ and $q$ be two distinct prime numbers such that $p^2-pq+q^2$ is a perfect square. Determine the largest possible value of $p+q$. | 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 |
In the expression $\left(3x^2+2x+5\right)\left(2x-1\right)$, find the coefficient of the $x^3$ term and the constant term, and then find the sum of these two coefficients. | nvidia |
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