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Determine a value of $a$ such that the equation
$$x^2+2ax+12=0$$
can be easily solved in your head by factoring. | nvidia |
Determine the number of distinct integers $n$ that can be expressed in the form
\[\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor + \cdots + \lfloor kx \rfloor\]where $x$ is a real number, $k$ is a positive integer, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$. | nvidia |
Let $a$, $b$, $c$ be real numbers such that $a+b+c=0$. Determine all possible values of the parameter $k$ for which any root $x$ of the equation
$$x^3+ax^2+bx+c=0$$
satisfies the inequality
$$|x|\le k.$$ | nvidia |
Find all real numbers $a$ such that the inequality $\frac{x^2-ax+a^2}{(x-1)^2(x+3)} < 0$ holds for exactly two intervals of $x$. | nvidia |
Consider the set of all positive integers from 1 to 100 that are not divisible by 3 or 7. Let N be the number of elements in this set. Find N. | nvidia |
How many nonnegative integers can be written in the form\[a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,\]where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$?
$\textbf{(A) } 512 \qquad \textbf{(B) } 729 \qquad \textbf{(C) } 1094 \qquad \textbf{(D) } 3281 \qquad \textbf{(E)... | nvidia |
Let $x$ be a real number such that $x^3+3x^2-7x-11=0$. If $a$ and $b$ are the roots of the equation $x^2+px+q=0$, where $p$ and $q$ are constants, find the value of $|a-b|$. | nvidia |
Determine the smallest positive integer $n$ such that $n^2+1$ is divisible by a prime number greater than $n$. | nvidia |
Determine the largest positive integer that is not divisible by any prime greater than 5. | nvidia |
Find the sum of all real numbers $x$ satisfying the equation $\sqrt{x^2+1} + x = \frac{5}{\sqrt{3}}$. | nvidia |
Find the smallest positive integer that has exactly 10 positive integer factors, with the additional constraint that its prime factorization contains only two distinct prime numbers. | nvidia |
Find the smallest positive five-digit number that is divisible by each of the five smallest prime numbers. | nvidia |
Let $a$ and $b$ be two positive real numbers such that $a+b=1$. Find the sum of the infinite series $a+ab+ab^2+ab^3+\cdots$. | nvidia |
A fair coin is flipped 7 times. What is the probability that the number of consecutive heads is greater than the number of consecutive tails or the number of consecutive tails is greater than the number of consecutive heads? | nvidia |
Find the smallest positive integer $n$ such that the sum of $n^3$ and $n^2$ is a perfect square. | nvidia |
Count the number of positive integers less than 1000 that are multiples of 9 and have exactly two digits equal to 7. | nvidia |
Let $\mathcal{T}$ be the set of all possible triangles with vertices on the coordinate plane whose side lengths are positive integers less than or equal to 20. Determine the number of triangles in $\mathcal{T}$ that have exactly one right angle, and express your answer modulo 250. | nvidia |
Point $P$ is located inside triangle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15.$ Find the radius of the circle that passes through points $A, B,$ and $C$ and is tangent to line $BP$. | nvidia |
Let $\triangle ABC$ be a right triangle with integer side lengths such that the area of the circle inscribed within its boundaries is 121. Given that the perimeter of $\triangle ABC$ is 840, what is its area? | nvidia |
Find the value of $a$ such that the cubic polynomial $x^3 + ax^2 + 1$ can be factored as the product of a linear and a quadratic polynomial, where the quadratic polynomial is a perfect square. | nvidia |
Find the value of $x$ such that the sum of the squares of four consecutive integers, starting from $x$, is equal to 450. | nvidia |
Determine the maximum possible number of points of intersection that can occur when 3 different parabolas and 1 straight line are drawn on the same coordinate plane. | nvidia |
Let $f$ be a function from the set of real numbers to itself, such that $f(f(f(x)))=x$ for all real numbers $x$. Find the number of possible values of $f(0)$ if it is given that $f(0) \neq 0$. | nvidia |
Determine the value of $\sqrt{\frac{1}{a^3}} \cdot \sqrt{a^5}$, where $a$ is a positive real number. | nvidia |
Given that the point $(4,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=3f(4x)+5$. What is the sum of the coordinates of that point? | nvidia |
What is the maximum number of zeros that can be formed by multiplying three different positive integers less than 1000? | nvidia |
Find the sum of all positive integers such that their representation in binary digits is a palindrome, and when converted to base 4, the resulting representation is a palindrome with a different number of digits. Express your answer in base 10. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f((x - y)^2) = f(x)^2 - 2xf(y) + y^2\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(1),$ and let $s$ be the sum of all possible values of $f(1).$ Find $n \times s.$ | nvidia |
Find all positive integers $m$ for which there exists a positive integer $n$ such that the equation $$n\sin(m\cos(\pi nx))=x$$ has infinitely many real solutions. | nvidia |
Determine the value of $y$ such that the number $\underline{77y}$ is divisible by 55. | nvidia |
What is the sum of all the numbers in the row of Pascal's triangle that has 20 numbers? | nvidia |
What is the greatest whole number that must be a divisor of the sum of any three consecutive positive integers? | nvidia |
Tom and Alex are two friends who have a combined total of 36 stickers. The number of stickers that Alex has is a positive multiple of the number of stickers that Tom has. If Tom gives some of his stickers to Alex, then the ratio of the number of stickers that Alex has to the number that Tom has becomes 5:1. What is the... | nvidia |
The function $g(x)$ satisfies
\[g(x+y) = g(x) + g(y)\]
for all real numbers $x$ and $y$, and $g(x) \neq 0$ for at least one real number $x$.
Find $g(4)$, given that $g(1) = 2$. | nvidia |
A polynomial $p(x)$ of degree 4 has roots $-2,$ $5,$ $9,$ and $-1/3$ and satisfies $p(0) = e$ and $p(1) = f$, where $e$ and $f$ are positive integers. Determine the minimum possible value of $e + f$ such that the coefficients of $p(x)$ are all integers. | nvidia |
A triangle with vertices $A$, $B$, $C$ is given in the complex plane, where $|AB|=|BC|=|CA|=6$ and $|A|=|B|=|C|$. Find the length of the shortest line segment that connects one of the vertices of the triangle to the circle $|z|=2$. | nvidia |
Let $x$ be a positive integer that has only even digits in its base-5 representation. What is the smallest possible value of $x$ that is one less than a multiple of 25? | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a + b + 2c = 2$ and $ab + 2ac = 1$. Find the maximum value of $c$. | nvidia |
Find the difference between the sum of the greatest common divisor and the least common multiple of $a$ and $b$, and the sum of $a$ and $b$, where $a$ and $b$ are positive integers such that $\text{lcm}(a,b)+\text{gcd}(a,b)=42$ and $\text{lcm}(a,b)-\text{gcd}(a,b)=18$. | nvidia |
Find the number of ways to arrange the numbers 1, 2, 3, 4, 5, and 6 in a row such that the sum of any two consecutive numbers is not a multiple of 3. | nvidia |
Find the greatest integer value of $b$ for which the expression $\frac{9x^3+4x^2+11x+7}{x^2+bx+8}$ has a domain of all real numbers. | nvidia |
Find the sum of the decimal equivalents for the fractions $\frac{1}{5}$ and $\frac{1}{11}$. What is the smallest number of digits after the decimal point that is needed to be sure that the obtained number is closer to this sum than to the sum of the decimal equivalents for the fractions $\frac{1}{5}$ and $\frac{1}{12}$... | nvidia |
Let $f(x)$ be a polynomial of degree $n$ such that
$$\lim_{x \to \infty} \frac{f(x)}{x^{n+1}} = 1.$$
Find the largest possible number of distinct roots that $f(x)$ can have. | nvidia |
Let A represent the set of positive integers less than 500 divisible by 17, B represent the set of positive integers less than 500 divisible by 19, and C represent the set of positive integers less than 500 divisible by both 17 and 19.
Find the number of elements in the set \( A \cup B \), i.e., the set containing the... | nvidia |
Find the greatest common divisor of all numbers that can be expressed as the difference of two consecutive perfect cubes. | nvidia |
Find a positive integer that is divisible by 14 and has a square root between 25 and 25.3. | nvidia |
Find the value of $b$ such that the equation $\ln{(x^2+bx+c)} + \ln{(x^2-bx+c)} = \ln{(x^4+1)}$ has at least one real solution. | nvidia |
How many positive integers $M$ less than $1000$ are there such that the equation $x^{2x} = M$ has a solution for $x$? | nvidia |
A quadrilateral is made of four rigid rods of lengths 10, 12, 15, and 18 inches joined end-to-end in a specific order. The quadrilateral is then placed on a flat surface, and it is found that the opposite sides do not intersect. Determine the minimum number of inches that must be added to one of the rods so that the qu... | nvidia |
The number 2021 can be expressed as the sum of two positive integers in several ways, but only one of these sums has a product of its terms that is the largest possible. What is the sum of the digits of the product of the terms of this sum? | nvidia |
Count the number of positive integers less than 100 that have no repeating digits. | nvidia |
Determine the number of sequences of six positive integers, each less than 150, that form a strictly increasing geometric sequence. | nvidia |
Determine the value of x such that the expression (2x+1)^4 \cdot (3x+1)^4 is equal to 16 times the product of the first four positive even numbers. | nvidia |
Find the smallest four-digit number that appears in the Fibonacci sequence.
Note: The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding numbers, starting from 0 and 1. | nvidia |
The sum of the interior angles of a polygon with 12 sides is equal to the sum of the exterior angles of a polygon with 25 sides. Determine the number of acute angles in the 12-sided polygon. | nvidia |
Problem:
Find the values of $a$ and $b$ for which the equation $ax^2+bx+2=0$ has two distinct real solutions and the sum of the squares of the roots is equal to 14. It is given that $a>0$ and $a+b<0$.
Determine the ordered pair $(a,b)$. | nvidia |
Determine the number of positive integers such that the difference between their square roots and their cube roots is greater than $\frac{1}{2}$. | nvidia |
Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than $30$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$. | nvidia |
Let $a$ and $b$ be positive integers such that $3^a + 3^b = 90$. Find the value of $a + b$. | nvidia |
A function f(x) is defined for all real numbers x except x=0 and x=1. For all nonzero real numbers a such that 0<|a|<1, the value of the function f(a) can be calculated using the formula:
f(a) = (1/(1-a)) + (1/(1+a)) + (1/(1-(1/a)))
Given that f(a) = 7/3, determine the value of a. | nvidia |
Find the sum of the values of $x$ which satisfy $x^2 +1992x = 1993$.
Consider the polynomial equation $x^2 + ax + b = 0$. Determine the sum of the values of $x$ that satisfy this equation, given that the product of the roots is equal to the sum of the coefficients $a$ and $b$. | nvidia |
Given a point (x, y) on the coordinate plane, find the number of points (x', y') that satisfy the condition that the distance between (x, y) and (x', y') is equal to the distance between (x', y') and the origin (0, 0). | nvidia |
Let $f(n)$ be the number of natural numbers $x$ such that $\lfloor \log_2{x}\rfloor = n$, where $\lfloor x \rfloor$ denotes the largest integer that is less than or equal to $x.$ Find the sum $f(1) + f(2) + f(3) + \cdots + f(10).$ | nvidia |
What is the smallest positive integer $n$ such that $n^3+2n^2+3n+5$ is divisible by 73? | nvidia |
The equation $x^3 + 2x^2 - 7x - 12 = 0$ has three distinct solutions. Determine the value of the sum of the squares of the solutions, given that the sum of the solutions is -2. | nvidia |
Find the sum of the first five prime numbers that have a units digit of 3. | nvidia |
Find the value of $n$ such that the equation $2^n - 2^{n-2} = 12$ is satisfied. | nvidia |
Find the smallest integer $n$ such that $2^{2n}-1$ is divisible by $7^2$. | nvidia |
Let $x$ be a positive number such that $x^2=3^3$. Calculate the value of $x^5$. | nvidia |
Find the sum of the series $1\cdot2 + 2\cdot3 + 3\cdot4 + \dots + n(n+1)$, where $n$ is a positive integer. | 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 |
In a quadrilateral $ABCD$, the length of the diagonal $AC$ is 12, and the medians from points $B$ and $D$ intersect at a point $M$ such that the ratio of $BM$ to $MD$ is 2:3. If the length of the median from point $C$ is 8, find the length of the diagonal $BD$. | nvidia |
Problem:
Let $ABCD$ be a regular tetrahedron with $AB = 12$, and let $M$ be a point on $\overline{AB}$ such that $AM = 3$. A solid $T$ is obtained by drilling a hole through the tetrahedron, where the sides of the hole are planes parallel to $\overline{AC}$ and containing the edges $\overline{AM}$ and $\overline{MB}$. ... | nvidia |
The six vertices of a regular hexagon are labeled with distinct integers from 1 to 6. The sum of the integers at any two adjacent vertices is the same, and the sum of the integers at any two vertices separated by one vertex is also the same. Find the sum of the integers at the vertices that are diagonally opposite. | nvidia |
For how many three-digit positive integers is the sum of the digits equal to $5?$ | nvidia |
Suppose that $(u_n)$ is a sequence of real numbers satisfying
\[u_{n+2}=2u_{n+1}+u_n\]and that $u_3=9$ and $u_6=128$. What is $u_5$? | nvidia |
If $m$ is the product of all positive integers less than $101$ that are multiples of $7$, and $n$ is the product of all positive integers less than $101$ that are multiples of $11$, find the greatest common divisor of $m$ and $n$. | nvidia |
Find the smallest positive integer n such that n! is a multiple of 11^3. | nvidia |
Find the number of prime divisors of 50! that are greater than 20. | nvidia |
Given that $a$, $b$, and $c$ are positive integers, simplify $\sqrt[4]{a^6b^8c^{10}}$ such that the sum of the exponents of the variables outside the radical is minimized. What is the minimum sum? | nvidia |
Consider the polynomial equation $x^6 - 2x^3 + 2 = 0$. Determine the smallest positive integer $m$ such that the sum of the $m^{\text{th}}$ powers of the roots of the given equation equals 0. | nvidia |
An equilateral triangle is drawn such that one of its vertices lies on the line $y = x$, and so that the other two vertices lie on the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$. What is the length of the side of the triangle? | nvidia |
Find the value of $x$ such that $\frac{3}{5}$ of $x$ is equal to $\frac{2}{3}$ of $x+2$. | nvidia |
Find the greatest integer value of $a$ for which the expression $\frac{4x^3-2x^2-5x+3}{x^2+ax+6}$ has a domain of all real numbers. | nvidia |
Let $a$, $b$, and $c$ be the solutions of the cubic equation $4x^3-13x^2+9x-3=0$. Find the sum of the squares of the pairwise differences $(a-b)^2+(a-c)^2+(b-c)^2$. | nvidia |
Find the value of $x$ such that $\log_2(x^4) = -2$. | nvidia |
Determine the smallest positive integer $n$ such that $11^n+1$ is divisible by a perfect square other than 1. | nvidia |
Problem:
Points $A$ and $B$ are symmetric with respect to the line $y = x$. The coordinates of point $A$ are $(x, y)$. Find the coordinates of point $B$ if the area of the rectangle formed by the points $(0,0)$, $(0,y)$, $(x,y)$, and $(x,0)$ is equal to the area of the rectangle formed by the points $(0,0)$, $(0,x)$, $... | nvidia |
Let $P(x)$ be a polynomial with integer coefficients. Suppose that the roots of the equation $P(P(x))=0$ are exactly the same as the roots of the equation $P(x)=0$. What is the smallest possible number of distinct roots that $P(x)$ can have? | 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 |
What is the largest value of $x$ that satisfies the equation $\sqrt{2x}=4x$?
Consider the equation $\sqrt{x^2+1}=x^2$. Find the largest value of $x$ that satisfies this equation. | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $abc = 27.$ Find the minimum value of $a + 3b + 9c.$ | nvidia |
The number $2.\overline{517}$ can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Determine the sum of the digits of $a$. | nvidia |
A square is inscribed in a triangle with side lengths 5, 12, and 13. Find the number of ways to draw a segment inside the square so that it divides the square into two smaller squares. | nvidia |
If $m$ and $n$ are positive integers such that $\gcd(m,n) = 12$, then what is the smallest possible value of $\gcd(10m,15n)$? | nvidia |
Using only the digits 7, 8 and 9, how many positive seven-digit integers can be made that are palindromes? | nvidia |
In a regular hexagon $ABCDEF$ inscribed in a circle of radius $R$, let $P$ be a point on the circle such that $AP = BP$. Let $X$ be the midpoint of $AB$, and let $Q$ be the foot of the perpendicular from $P$ to $AB$. Prove that $XQ^2 + AP^2 \leq 2R^2$ for all permissible positions of $P$. | nvidia |
Given positive real numbers $a,$ $b,$ and $c,$ determine the maximum value of the expression
\[\frac{a^2b^2c^2(a + b + c)^2}{(a^2 + b^2)^2 (b^2 + c^2)^2}.\] | nvidia |
Find all real values of $x$ that satisfy the equation
$$\frac{x^2+4x-8}{x-2} + \frac{x^2-3x-4}{x+1} + \frac{3x^2-4x-3}{x-3} + \frac{x^2+2x-3}{x+3} = 8.$$ | nvidia |
Let $x$ and $y$ be real numbers such that $x+y$, $x^2+y^2$ and $x^3+y^3$ form an arithmetic progression. Determine all possible values of $xy$. | nvidia |
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