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Determine the smallest value of $b$ such that the polynomial $p(x) = x^3 + bx + 1$ has at least one real root. | nvidia |
Find the largest possible value of x such that 18x and 24x are both factors of 432. | nvidia |
Determine the value of $x$ such that the expression $\left(\dfrac{3x+5}{x-2}\right)^0$ is undefined. | nvidia |
Find the number of different positive integers that can be represented as a sum of two distinct members of the set {1, 2, 3,..., 12, 13, 14}. | nvidia |
Find the sum of all the products of two different factors that can be formed using the numbers 5, 6, and 7, if the product of the two factors is even.
Can you determine the total sum of all these possible products? | nvidia |
Find the largest positive real number $d$ such that for all nonnegative real numbers $a$ and $b,$
\[\sqrt{ab} - d |a - b| \le \frac{a + b}{2}.\] | nvidia |
Find the smallest positive integer that can be written in the form $a^2 + b^3$, where $a$ and $b$ are integers. | nvidia |
Point $P$ is located inside triangle $ABC$ such that the distances from $P$ to the sides $AB$, $BC$, and $CA$ are in the ratio $2:3:4$. If the lengths of the sides of the triangle are $AB=10$, $BC=12$, and $CA=15$, find the distance from $P$ to the side $AB$. | nvidia |
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\[{n\choose k} : {n\choose k+1} : {n \choose k+2}\]Find the sum of all possible values of $n+k$.
| nvidia |
Let $f(x)$ be a quadratic function such that $f(0) = -24$, and the sum of the squares of the solutions of the equation $f(x) = 0$ is equal to 100. Determine the product of the solutions of this equation. | nvidia |
Find the smallest positive integer $n$ such that $n^2+7n$ is $1$ less than a perfect square. | nvidia |
Find all $t$ such that $x-t$ is a factor of $6x^2+13x-5.$
Find the value of $k$ such that the equation $x^2 + 3x + k = 0$ has exactly one solution. | nvidia |
Find a nonzero $p$ such that $px^2-12x+4=0$ has only one solution. | nvidia |
Find the largest positive integer $n$ such that $n^3$ divides $3^4\cdot5^2\cdot7^3$. | nvidia |
Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 11? | nvidia |
Find the number of non-similar right triangles with integer side lengths whose area is equal to 30 square units. | nvidia |
Find the number of nonnegative integers $n$ such that the expression $2^n+1$ is a perfect cube. | nvidia |
What is the largest integer n such that the product of any five consecutive integers is divisible by n? | nvidia |
Find the positive difference between the sum of the squares of the first 10 positive integers and the sum of the cubes of the first 5 positive integers. | nvidia |
In the geometric sequence $x, x^2, 256$, determine the value of $x$ if the sum of the terms is equal to $320$. | nvidia |
Find the smallest positive integer n that can be represented as a sum of two consecutive integers in base a and as a sum of three consecutive integers in base b, where a and b are any bases larger than 2. | nvidia |
Find all roots of the polynomial $x^3+x^2-4x-4$. Enter your answer as a list of numbers separated by commas.
Problem:
Find all values of $a$ such that the equation $x^3 - 2ax^2 + (a^2 + 1)x - 2a = 0$ has three distinct integer roots. | nvidia |
The points $(2, 8), (15, 20)$ and $(7, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum? | nvidia |
Find the smallest positive integer $m$ such that all the roots of the equation $z^6 + z^3 + 1 = 0$ are $m^{\text{th}}$ roots of unity. | nvidia |
Find the product of the roots of the equation $(x^2+1)(x-2)-(x-2)(x^3+1)=0$. | nvidia |
Determine all real numbers $a$ such that for every positive integer $n$, $n + a$ is composite. | nvidia |
Let
\[f(x) = x^2 + ax + b.\]
Find all real numbers $a$ and $b$ such that $f(x) = f(f(x))$ for all real numbers $x$. | nvidia |
Determine the number of real solutions to the equation
$$ \left(\frac{x^{n-1}-1}{x-1} \right)^2 - \left(\frac{x^n-1}{x-1} \right)^2 = \left(\frac{x^{2n}-1}{x-1} \right)^2 - n $$ for any positive integer $n>1$.
Also find all solutions if $n$ is an odd number.
As follows we assume that $x \neq 1.$ | nvidia |
Let $r$ be the root of a given polynomial that has exactly two different integer roots. Determine $r$ such that the number of polynomials that have $r$ as a root and that satisfy the given constraints is maximal. | nvidia |
Find the number of positive three-digit integers such that the product of their digits is equal to the sum of the squares of their digits. | nvidia |
What is the sum of all integer values $n$ for which $\binom{26}{13}+\binom{26}{n}=\binom{27}{14}$?
Write another problem inspired by this one:
Find all integers $m$ such that the equation $\binom{m+1}{m} + \binom{m+1}{m-1} = 3^m$ has a unique solution. | nvidia |
Let $C$ be the number of three-digit numbers that have at least one digit repeated. Let $D$ be the number of three-digit numbers that are divisible by 3. Find $C+D$. | nvidia |
Find the value of $a$ so that the lines with the given equations are perpendicular. \begin{align*}
y &= 2x+13 \\
6y+ax &= 6.
\end{align*} | nvidia |
The graphs of the functions $f(x)=3x^2-2x-6$ and $g(x)=-x^2+7x-3$ intersect at two points. Determine the product of their $x$-coordinates if the sum of their $y$-coordinates is 6. | nvidia |
What fraction of the positive integers less than or equal to 1000 have exactly three positive integer divisors? | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a + b + c = 12.$ Find the minimum value of $a^2 + b^2 + c^2.$ | nvidia |
What is the minimum number of days that must pass so that the same day of the week falls on the same date of the month as it does today? | nvidia |
The squares of a chessboard are coloured in a specific way such that each row and each column forms an arithmetic sequence. If the colours are represented by integers, the numbers on the main diagonal are given by the arithmetic sequence $a, a+d, a+2d, \ldots, a+(n-1)d$ where $n$ is the number of squares on the diagona... | nvidia |
Given a set of positive real numbers $\{a_1, a_2,..., a_n\}$, define the function $$\text{{K}}(a_1, a_2,..., a_n) = \frac{a_1}{a_2} + \frac{a_2}{a_3} +... + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}.$$Let $\text{{K}}_m$ be the minimum value of $\text{{K}}$ for a set of $m$ distinct positive real numbers. Find the value of ... | nvidia |
Consider a 5-digit palindrome number, where the first and last digits are the same, and the second and fourth digits are the same. How many 5-digit palindrome numbers can be formed such that at least one of the digits is even? | nvidia |
Find the value of $x$ such that the product of $3^x$ and $2^x$ is equal to the square root of $2^{24} \cdot 3^{24}$. | nvidia |
Find the number of positive integers $n$ such that $2^n$ is between $\sqrt{10}$ and $\sqrt{1000}$. | nvidia |
Find the sum of all the elements in the 15th row of Pascal's triangle that are divisible by 7. | nvidia |
Find the largest, positive three-digit number that is a multiple of 7 and has exactly three distinct digits. | nvidia |
Count the number of positive integers $n$ such that $n!$ is a perfect square, where $n!$ is the product of all positive integers up to $n$. | nvidia |
In a triangle with vertices A(2,3), B(5,7), and C(-1,4), find the equation of the line that passes through the point which is equidistant from all three vertices. | nvidia |
Given the function $f(x) = \frac{1}{x^2 + 1}$, whose domain is all real numbers, find the range of the function and explain why it has that range. | nvidia |
In the three-digit integer $AB2$, what is the largest digit $B$ so that the three-digit integer will be divisible by 11 and the sum of the digits will be greater than 10? | nvidia |
Determine the domain of the function $y=\sqrt{|\sin(\log_3 x)|}$, where $x$ is a positive real number. | nvidia |
Find the area of the triangle with vertices $(-1,4),$ $(7,0),$ and $(11,5).$ | nvidia |
What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions? | nvidia |
Two functions $f(x)$ and $g(x)$ have the property that $f(2x) = g(x)$ for all real numbers $x.$ Given that the graphs of $y=f(x)$ and $y=g(x)$ intersect at $(1,1),$ $(2,2),$ and $(3,3),$ find the sum of the $x$-coordinates of all points where the graphs of $y=f(2x)$ and $y=2g(x)$ intersect. | nvidia |
Let $f(x)$ be a function that satisfies
$$f(x+y)=f(x)+f(y)$$
for all real numbers $x$ and $y$, and $f(1) = 3$. Find $f(5)$. | nvidia |
Two coins are tossed simultaneously, and the probability that at least one of them shows a head is equal to the probability that they both show tails plus the probability that only one of them shows a head. What is the probability that each of the coins shows a head? | nvidia |
Suppose that $a$ and $b$ are positive integers such that $(a-bi)^2 = 8-6i$. What is $a-bi$? | nvidia |
Given that $A$ is a $2\times 2$ matrix such that $A^{2013} = A$, find $A$ if it is known that the trace of $A$ is equal to zero and $A$ is not the zero matrix.
Prove that a matrix satisfying these conditions does exist and that it is unique up to multiplication by a constant. | 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 |
Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. | nvidia |
The least value of x such that x^2 + (4 + i)^2 is a real number. | nvidia |
A drawer contains a mixture of red and blue socks, where the number of red socks is three times the number of blue socks. If 5 socks are randomly drawn from the drawer, the probability that exactly 2 of them are blue is $\frac{1}{5}$. Find the total number of socks in the drawer. | nvidia |
Find all solutions $x$ (real and otherwise) to the equation
\[x^4+64=0.\]Enter all the solutions, separated by commas. | nvidia |
In a circle with center $O$, $AB$ is a diameter, $C$ is a point on $AB$, and $D$ is a point on the circle such that $\triangle ACD$ is isosceles. Let $E$ be the point of intersection of $AD$ and $BC$, and $F$ be the point of intersection of $BD$ and $AC$. If the area of $\triangle CEF$ is $\frac{1}{4}$ the area of $\tr... | nvidia |
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n$. For how many values of $n$ is $n+S(n)+S(S(n))=2007$? | nvidia |
Find all positive real numbers $x$ that satisfy
\[x \sqrt{12 - x} + \sqrt{12x - x^3} \ge 12.\]Enter all solutions, separated by commas. | nvidia |
A cone and a cylinder have the same height and share the same base. The volume of the cylinder is three times as large as the volume of the cone. Find the sum of the radius of the cone's base and the slant height of the cone. | nvidia |
Find all real numbers that are tripled when added to their cubes. | nvidia |
Find the sum of all positive integers less than 20 that are not divisible by 2 or 3. | nvidia |
Consider a sequence of 15 distinct integers such that the sum of any 3 consecutive integers in the sequence is positive, and the sum of any 4 consecutive integers in the sequence is negative.
How many such sequences are there? | nvidia |
$n$ coins are simultaneously flipped. The probability that at most one of them shows tails is $\frac{3}{16}$. Find $n$. | nvidia |
A circle with radius $r$ is inscribed in a triangle $\triangle ABC$ such that the area of the sectors of the circle enclosed by the sides of the triangle are $3$, $6$, and $12$, respectively. Find the area of $\triangle ABC$. | nvidia |
Given that $\sum_{i=1}^{n}x_i=\prod_{i=1}^{n}x_i=120$, where $x_i$ are positive integers, find the largest possible value of $n$ such that each $x_i$ is a non-constant expression in terms of a variable $t$, where the expressions are distinct. | nvidia |
Determine the largest integer value of $m$ such that the inequality
\[\tan^m x + \cot^m x \ge 2\]
holds true for all real numbers $x$ in the interval $[0, \frac{\pi}{2})$. | nvidia |
What is the largest positive integer $n$ such that $\frac{10^n - 1}{9}$ is divisible by 25? | nvidia |
Determine the largest prime factor of the product of all positive integers less than or equal to 30 that are relatively prime to 30. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a^2 + b^2 = 9.$ Find the minimum value of $a^3 + b^3.$ | nvidia |
Suppose that $f(x)$ is a function such that
\[f(xy) + x = xf(y) + f(x)\]for all real numbers $x$ and $y.$ If $f(-1) = 5$ then compute $f(-1001).$ | 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 |
On a rectangular grid, a square is drawn so that its vertices coincide with the grid points. If two opposite vertices are located at points (5,7) and (11,13), find the side length of the square. | nvidia |
The sum of the digits of a positive integer is 17, and the sum of the squares of its digits is 365. If the integer has at most three digits and is divisible by 11, what is the sum of its digits? | nvidia |
Given that the equation $ax^2 + bx + c = 0$ has two real solutions $x_1$ and $x_2$, prove that the inequality $ac > \frac{b^2}{16}$ holds if $x_1$ and $x_2$ are both between $-\frac{1}{2}$ and $\frac{1}{2}$. | nvidia |
What is the smallest positive integer that is not the sum of two prime numbers, and is also not the sum of a prime number and a composite number? | nvidia |
Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of
\[\frac{1}{a} + \frac{2}{b}.\] | nvidia |
What is the probability that the sum of the numbers shown on two 6-sided dice, one red and one green, is a multiple of 3, given that the red die shows a number greater than 3? | nvidia |
The sequences of positive integers $1,a_2, a_3, \dots$ and $1,b_2, b_3, \dots$ are an increasing arithmetic sequence and a decreasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$. | nvidia |
Given that $y$ is a positive real number, find $y$ in the equation $\frac{30}{50}=\sqrt{\frac{50}{y}}$. | nvidia |
Find the ratio of the sum of the measures of the interior angles of a polygon to the number of its sides. | nvidia |
Find the maximum value of $\cos^2 x - \sin^2 x + \sin x \cos x$ over all real numbers $x.$ | nvidia |
What is the sum of all positive integers less than 1000 that are cubes of perfect cubes minus the sum of their proper divisors? | nvidia |
The number $n$ is strictly between $-5$ and $2$, and the denominator of the simplified fraction $\frac{n}{15}$ is $3$. Find the sum of all such possible values of $n$. | nvidia |
Let $f(x)$ be a non-constant monic polynomial with real coefficients such that for some real number $a$,
\[f(x) = \left(x^2 + ax + \frac{a^2}{2}\right) g(x)\]
for some monic polynomial $g(x)$ with real coefficients. Find the largest possible number of complex roots of $f(x)$ with imaginary part 1. | nvidia |
Consider a regular hexagon with side length 1, whose vertices are labeled A, B, C, D, E, and F. Three points are randomly chosen from the vertices of the hexagon. Let b be the smallest possible number such that it is always possible to select one pair of points from these three such that the distance between them is eq... | nvidia |
The function
$$t(z)=\sqrt[3]{\frac{1}{(z-4)^2}}$$
is defined on a subset of the real numbers. Determine the set of all $z$ such that $t(z)$ is defined, and express the answer in interval notation. | nvidia |
Suppose that $f(x)$ is a function such that
\[f(xy) + x = xf(y) + f(x)\]for all real numbers $x$ and $y.$ If $f(-1) = 5$ then compute $f(-1001).$ | nvidia |
Positive integers $c$ and $d$ are chosen with $2<c<d$ such that no triangle with positive area has side lengths $2, c,$ and $d$ or $\tfrac{2}{d}, \tfrac{2}{c},$ and $2$. What is the smallest possible value of $d$? | nvidia |
What is the sum of all of the two-digit primes that are greater than 12 but less than 99 and are still prime when their two digits are interchanged? | nvidia |
Determine the set of all real numbers $x$ such that
$x^3 + 4x^2 - 6x - 2 < \ln(x^2 - 4)$ | nvidia |
Find the largest integer $n$ such that $6!$ is divisible by $7^n$. | nvidia |
Find the greatest possible three-digit number that is a multiple of 5, has the sum of its digits equal to 18, and is a palindrome. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a + b + c = 1$. Find the maximum value of
\[(a^2 + 2b^2)(b^2 + 2c^2)(c^2 + 2a^2).\] | nvidia |
In a set of five consecutive integers, the sum of the smallest and largest numbers is equal to twice the middle number. If the second smallest number is 17, find the product of the three largest numbers. | nvidia |
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