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Let $f(x)$ be a monic quartic polynomial with real coefficients such that the equation $f(x)=0$ has four distinct complex solutions: $x_1$, $x_2$, $\overline{x_1}$, and $\overline{x_2}$. Given that $$|x_1|^2+|x_2|^2=6$$ and $$|x_1|+|x_2|=2$$ Determine the constant term of $f(x)$. | nvidia |
In a certain type of digital lock, the key is a 3-digit number consisting of different digits from 0 to 9. The lock opens if the product of the three digits equals the sum of the numbers formed by arranging the three digits in all possible ways. Find the greatest possible value of the key (as a 3-digit number) that can... | nvidia |
Find the number of three-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6, such that the sum of the first and last digits is a multiple of 3. | nvidia |
Let $g$ be a function defined on the positive real numbers, such that
$$g(x+y)=g(x)g(y)$$
for all positive real numbers $x$ and $y.$ Given that $g(2)=3$ and $g(4)=9$, find the value of $g(5)$. | nvidia |
Find the sum of all positive integer values of $b$ such that $6x^2 - 17x + b = 0$ has two real roots. | nvidia |
Given that $a+b+c=1$ and $ab+bc+ca=\frac{1}{2}$, find the value of $a^2b^2+b^2c^2+c^2a^2$. | nvidia |
The graph of $y=g(x)$ is shown below, with $2$ units between grid lines. Assume $g(x)$ is defined only on the domain shown.
What is the sum of all integers $d$ for which the equation $g(x)=d$ has exactly $4$ solutions?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14c... | nvidia |
What is the maximum number of consecutive even positive integers that can be added together before the sum exceeds 600? | nvidia |
Find all perfect square numbers that end with the digit 6. | nvidia |
Simplify $$\frac{11!}{9! + 2\cdot 8!}$$
A similar problem:
Simplify $$\frac{11!}{9! + 2\cdot 8!}$$
Here is a new problem inspired by this one:
The product of all positive integers from 1 to 11 is equal to 39916800. Determine the sum of all positive integers from 1 to 11. | nvidia |
Determine the day of the week on which the first day of the next year occurs, given that the first day of the current year is a Sunday and that the year is a leap year. | nvidia |
What is the sum of all of the solutions of the equation $\frac{4x}{20}=\frac{5}{x}$? | nvidia |
Find the least positive integer $n$ such that $2^{2007} + n$ is divisible by 7. | nvidia |
Find the smallest positive integer $n$ such that $n^4$ is divisible by 12960000. | nvidia |
Find the maximum value of $x^2y^2(4-x^2-y^2),$ where $x$ and $y$ are real numbers satisfying $x^2 + y^2 \leq 4.$ | nvidia |
Determine the largest positive integer $n$ such that there exist positive integers $x, y, z$ so that \[
n^2 = x^2+y^2+z^2+2xy+2yz+2zx+3x+3y+3z-6
\] | nvidia |
Consider the sequence of positive integers whose digits are all 1, starting with 1, 11, 111, and so on. Let $T_n$ be the sum of the reciprocals of the digits of the integers from 1 to $10^n$ inclusive, excluding the integers in the sequence of all 1s. Find the smallest positive integer $n$ for which $T_n$ is an integer... | nvidia |
Find a nonzero $q$ such that $qx^2+15x+9=0$ has only one solution. | nvidia |
Find the smallest positive integer $n$ such that the product of its digits is equal to the sum of the cubes of its digits, and $n$ is divisible by the sum of its digits. | nvidia |
A quadrilateral has three angles whose measures are $x^\circ$, $2x^\circ$ and $3x^\circ$. Determine the measure of the fourth angle, given that it is twice as large as the sum of the other three angles. | nvidia |
A rhombus $ABCD$ is circumscribed around a circle of radius 1 with center $O$ such that the circle is tangent to the midpoints of the sides of the rhombus. Find the area of the rhombus. | nvidia |
Find the number of lattice points on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ and $b$ are relatively prime positive integers. | nvidia |
If the four solutions of $x^4=-16$ are written in the form $a+bi$, where $a$ and $b$ are real, find the sum of those solutions with $b<0$. | nvidia |
Find the value of $x$ that satisfies $\sqrt{x^2+\sqrt[3]{x^4}}=2$. | nvidia |
Suppose $x$ and $y$ are integers such that $xy+3x+7y=14$. Find the greatest possible value of $x$. | nvidia |
A semicircle is inscribed in a square in such a way that the diameter of the semicircle coincides with one of the sides of the square, and the arc of the semicircle is inside the square. What is the length of the radius of the semicircle, in inches, if the product of the circumference of the semicircle, in inches, and ... | nvidia |
Find the minimum positive difference between two composite numbers whose product is 120. | nvidia |
What is the smallest positive integer that has exactly 10 distinct positive factors? | nvidia |
In the complex plane, point $P$ is represented by the complex number $2 + i$, and point $R$ is represented by the complex number $12 + 15i$. The line segment $\overline{PR}$ is rotated $90^\circ$ counterclockwise about the origin. Find the sum of the real and imaginary parts of the complex number representing the image... | nvidia |
Determine $a$ such that the equation $\ds\frac{2x^2+x-3}{ax^2+3x-2}=2x$ has solutions whose sum is equal to $-\frac{5}{2}$. | nvidia |
Find the smallest positive integer $b$ such that $x^6 + bx^3 + 1$ is always composite for all integers $x$. | nvidia |
The function $f(x)$ satisfies
\[f(x + y) = f(x) + f(y)\]for all real numbers $x$ and $y.$ Find all possible values of $f(0).$
Enter all the possible values, separated by commas. | nvidia |
The sum of two positive integers is 78 and their difference is 18. What is the value of the positive difference of the cubes of the integers? | nvidia |
Let a closed loop of string have a length of 13 inches. If the string is placed on a flat surface, a circle may be formed with a circumference of 13 inches. Suppose then that the string is placed to form the perimeter of a region that has the shape of a regular polygon with four sides, then its area will be some number... | nvidia |
Let $a$ and $b$ be real numbers such that the roots $x_1$ and $x_2$ of the quadratic equation
$$ax^2+bx+a=0$$
satisfy $x_1^2+x_2^2=16$. Find the product of the roots of the equation
$$x^2+\frac{b}{a}x+1=0$$ | nvidia |
Evaluate the expression \[ \frac{\sqrt{a+b}}{\sqrt{a-b}} \cdot \frac{\sqrt{b+c}}{\sqrt{b-c}} \cdot \frac{\sqrt{c+a}}{\sqrt{c-a}}, \] given that $a$, $b$, and $c$ are the sides of a triangle, and none of the expressions inside the square roots are negative. | nvidia |
What is the greatest integer that is less than the sum of the squares of the reciprocals of the first $n$ positive integers, where $n$ is a positive integer greater than or equal to 3? | nvidia |
Let $g(n)$ be the product of all the proper divisors of a positive integer $n$. If $g(g(n)) = n^2-1$, then call $n$ hyperdeficient. How many hyperdeficient positive integers are there? | nvidia |
Given that $x+\sin y=2008$ and $x+2008 \cos y=2007$, where $0 \leq y \leq \frac{\pi}{2}$, find the value of $x+y$.
Find $x$ and $y$ if $x+\sin y=2010$ and $x+\cos(2010+y)=2009$. | nvidia |
How many positive integers less than 1000 can be expressed as the product of two positive perfect squares? | nvidia |
Let $m$ be the product of all positive integers less than $4!$ which are invertible modulo $4!$. Find the remainder when $m$ is divided by $4!$.
(Here $n!$ denotes $1\times\cdots\times n$ for each positive integer $n$.) | nvidia |
Given that $\mathbf{u}$ and $\mathbf{v}$ are two vectors in $\mathbb{R}^3$ such that $\|\mathbf{u}\| = 2$ and $\|\mathbf{v}\| = 3$, and the angle between them is $\frac{\pi}{4}$, find the area of the parallelogram generated by $\mathbf{u}$ and $\mathbf{v}$.
(Note: $\|\mathbf{u}\|$ denotes the magnitude of the vector $... | nvidia |
If $x$ is a real number and $x \geq 3$, find the minimum value of $x+\lceil x^2 \rceil$. | nvidia |
Let $A$, $B$, $C$, $D$, and $O$ be points in a plane such that $OA=OB=OC=OD=20$ and $ABCD$ is a convex quadrilateral. Suppose that the diagonals $AC$ and $BD$ intersect at a point $P$. Given that $CP^2+DP^2=154$, determine the area of triangle $BPC$. | nvidia |
Let $a$ and $b$ be positive real numbers such that each of the equations $x^2 + ax + 2b = 0$ and $x^2 + 2bx + a = 0$ has real roots. Find the smallest possible value of $a + b.$ | nvidia |
The functions $g(x)$ and $h(x)$ satisfy the equations
$g(h(x))=x$ and $h(g(x))=x^2$ for all real numbers $x.$
Find $g(1).$ | nvidia |
Milton spilled some ink on his homework paper. He can't read the coefficient of $x$, but he knows that the equation has two distinct negative, integer solutions. What is the sum of all of the distinct possible integers that could be under the ink stain?
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)-... | nvidia |
What is the largest positive integer that cannot be expressed as the sum of distinct powers of 5? | nvidia |
A quadrilateral that has consecutive sides of lengths $70,90,130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length $x$ and $y$. Find $|x-y|$.
$\text{(A) } 12\quad \text{(B) } 13\qua... | nvidia |
Find the smallest positive integer that is not the product of a positive integral multiple of 3 and a positive odd integer. | nvidia |
Determine the smallest positive integer $k$ such that $5k+2$ is congruent to $0$ modulo $13$. | nvidia |
In a quadrilateral, the square of the length of one diagonal is equal to the sum of the squares of the lengths of the other diagonal and two adjacent sides. If the length of the first diagonal is 25 cm, find the sum of the squares of the lengths of the other diagonal and the remaining two sides. | nvidia |
Determine all values of $a$ such that the greatest prime divisor of $a^2+2a+2$ is a factor of $a+1$. | nvidia |
A positive integer $n$ is said to be $\textit{square-free}$ if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free? | nvidia |
Find the greatest $a$ such that $\frac{7\sqrt{(2a)^2+(1)^2}-4a^2-1}{\sqrt{1+4a^2}+3}=2$. | nvidia |
Find the positive integer $n$ such that
$$\arctan\frac {1}{3} + \arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.$$ | 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 |
Determine all real numbers x that satisfy the equation:
$\sqrt{x^2 + \sqrt{x + 1}} = x + 1$ | nvidia |
The product of two positive integers is 1001. How many different pairs of positive integers have a product of 1001? | nvidia |
Find the number of positive integers less than 1000 that are divisible by 7 and have a sum of digits equal to 10. | nvidia |
Find a cubic polynomial, with real coefficients, that has $2 + 3i$ as a root, and where the sum of the coefficients is equal to 5. | nvidia |
Given that $x$ is real and $x^3+\frac{1}{x^3}=52$, find $x+\frac{1}{x}$. | nvidia |
How many positive integers $k$ less than 50 have a corresponding integer $p$ such that the roots of $x^2-px+k^2=0$ have a sum that is a perfect square? | nvidia |
Find the value of $x$ such that $\sqrt[3]{2x + 5} = 3$. | nvidia |
What is the smallest three-digit number such that the sum of its digits is a prime number and the product of its digits is a perfect square? | nvidia |
A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order. | nvidia |
An integer between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?
| nvidia |
Determine the value of a such that $x^3+ax^2+7x-9$ is exactly divisible by $x-1$. | nvidia |
Let $n$ be a positive integer such that $4^n \cdot 5^n$ is a perfect square. Determine the smallest possible value of $n$. | nvidia |
Find the number of positive integer values of $x$ such that the sum of the areas of the three pairs of opposite faces of a rectangular prism box with length $x^2$, width $x+1$, and height $x+2$ is less than 200 square units. | nvidia |
Find the smallest positive integer $n$ such that the sum of its prime factors is equal to the number of its positive integral divisors. | nvidia |
In $\triangle DEF$, the sides have integer lengths and $DE=DF$. Circle $\Omega$ has its center at the circumcenter of $\triangle DEF$. An incircle of $\triangle DEF$ is a circle in the interior of $\triangle DEF$ that is tangent to all three sides of the triangle. Suppose that the incircle is internally tangent to $\Om... | nvidia |
How many positive three-digit integers less than 500 have at least two digits that are the same? | nvidia |
What is the smallest positive integer that has exactly six distinct positive factors? | nvidia |
Let p_(0), p_(1), ..., p_(n) be positive integers such that p_(i+1)=p^(2)_(i)+1 for i=0,1,...,n-1. Find the least positive integer d such that there exists n≥1 for which p_(0)p_(1)...p_(n) is divisible by 4^(d). | nvidia |
Let $g(z)$ be a polynomial of degree $n$ with complex coefficients, and let its roots be $z_1,$ $z_2,$ $\dots,$ $z_n.$ The polynomial $g(z)$ is such that the sum of the imaginary parts of the roots is equal to $2013.$ Determine the minimum number of roots that lie in the upper half-plane. | nvidia |
Determine the value of $a$ such that the area of the shaded region in the right triangle with legs $3a$ and $4a$ is equal to $\frac{48}{5}$ square units.
[asy]
fill((0,0)--(0,4a)--(3a,0)--cycle,gray(0.7));
draw((0,0)--(0,4a)--(3a,0)--cycle);
[/asy] | nvidia |
Find the number of positive integer solutions to the equation
\[\cos \left( \frac{\pi}{n} \right) = \left( \frac{1}{\sqrt{2}} \right)^n\]for $1 \leq n \leq 100$. | nvidia |
Find the value of $k$ such that the solution to the system of equations \begin{align*}
x^2+y^2&=k,\\
4x-3y&=2,
\end{align*}is unique. | nvidia |
Let $f(x)$ be a polynomial function such that $f(x)=f(2x)$ and $f(3x)=2x^2+5$ for all real numbers $x$. Determine the smallest possible degree of $f(x)$. | nvidia |
In a circle, two chords are drawn such that the ratio of their lengths is 3:5. If the length of the shorter chord is 12 cm, find the length of the longer chord. | nvidia |
Find the number of positive integers $n$ such that $n^2+2n+1$ is a perfect cube and $100 < n < 900$. | nvidia |
Prove that for any positive integer $n$, we have
$$\sin^2 \frac{\pi}{n} + \sin^2 \frac{2\pi}{n} + \dots + \sin^2 \frac{(n-1)\pi}{n} = \frac{n-1}{2}.$$ | nvidia |
A large library has a vast collection of books, and it wants to organize them on shelves. Each shelf can hold at most 20 books, and the library has 15 shelves. The books are to be arranged such that no two books of the same author are placed on the same shelf. What is the maximum number of authors that can be represent... | nvidia |
The sum of the cubes of two positive integers is 306. The difference of the two integers is 3. What is the product of the two integers? | nvidia |
How many real numbers $x$ satisfy the inequality $\frac{x^2+1}{|x|} \leq 10$? | nvidia |
Find the number of positive integers $n$ that satisfy the inequality $2^n < 500 < 2^{n+1}$. | nvidia |
Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$
| nvidia |
A jar contains $n$ balls, each with a different number from 1 to $n$ written on it. A ball is chosen at random from the jar and replaced, and then another ball is chosen at random. The probability that the first ball has a number that is a divisor of the second ball's number is $\frac{1}{2}$. Find the largest possible ... | nvidia |
Determine all positive integers less than or equal to $30$ such that when divided by $5$ the remainder is $3$, and when divided by $7$ the remainder is $4$. Find the product of these numbers. | nvidia |
Find the smallest positive integer that is both an integer power of a prime number and has a sum of its digits that is divisible by 3. | nvidia |
Find all two-digit numbers $N = \underline{a}\,\underline{b}$ such that the three-digit numbers $\underline{a}\,\underline{b}\,\underline{a}$ and $\underline{b}\,\underline{a}\,\underline{b}$ are both divisible by 7. | nvidia |
Let $f(n)$ be the sum of all positive divisors of $n$. What is the largest integer $n$ with $f(f(f(n)))=f(n)$? | nvidia |
Find the digit represented by $\star$ that satisfies the following addition problem:
$$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & 4 & 3 & 2 & \star_7\\ & & & \star & 5 & 1_7\\ &+ & & & \star & 3_7\\ \cline{2-6} & & 5 & 3 & \star & 0_7\\ \end{array} $$ | nvidia |
Suppose that $x$ and $y$ are positive integers such that $x^2 - y^2 = 12$ and $\text{lcm}(x, y) = 20$. Find the largest possible value of $x$. | nvidia |
Find the largest possible value of $n$ such that $2^n$ is a factor of $21!$, where $n$ is a positive integer. | nvidia |
Find the equation of the line that passes through the focus of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and is perpendicular to the line that passes through the foci of the ellipse. | nvidia |
The function $f(z)=\ds\frac{3z+5}{z+3}$ maps the complex number $z=2+i$ to a point $w$ on the unit circle. Find the complex number $a$ such that the function $f(z)+a$ maps $z=2+i$ to the origin. | nvidia |
Determine the maximum value of the expression $$\frac{bc+ca+ab}{(b-c)^2+(c-a)^2+(a-b)^2},$$ where $a$, $b$, and $c$ are positive real numbers and $a \neq b$. | nvidia |
Find the number of positive four-digit integers that can be formed using only the digits 3 and 7. | nvidia |
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