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Let $a$, $b$, and $c$ be real numbers, and consider the function $f(x)$ defined by
\[f(x) =
\begin{cases}
ax+b &\quad \text{if } x > c, \\
x^2 &\quad \text{otherwise}.
\end{cases}
\]
Find all possible values of $a$ and $b$ such that there exists some $c$ for which $f(f(x))$ is a polynomial for all $x$.
For which val... | open2math |
Consider the quadratic equation
\[y^2 + y + a^2 = 0,\]
where $a$ is a positive real number.
Find all values of $a$ for which this equation has at least one integer root. | open2math |
Find all positive integers $n$ such that the greatest common divisor of $n$ and $24$ is equal to the greatest common divisor of $n$ and $30$. | open2math |
Find the positive integer $n$ such that $|(n+i)^2|=|(n-i)^3|$. | open2math |
Find the largest positive integer x less than 144 that has exactly three distinct prime factors among its divisors. | open2math |
Find the value of the positive root of the equation $x^3-6x^2-13x+42=0$ given that one of its roots is 3 less than another root. | open2math |
Find the sum of the divisors of $150$ that are also multiples of $3$. | open2math |
Determine the number of digits in the base-9 representation of the number 1458. | open2math |
A portion of the graph of $y = G(x)$ is shown in red below. The distance between grid lines is $1$ unit.
Compute $G(G(G(G(G(1)))))$.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tick... | open2math |
The product of two positive integers minus their sum is 87. The integers are relatively prime, and each is less than 25. What is the sum of the two integers? | open2math |
What is the smallest positive integer $n$ such that the sum of the digits of $n$ is a divisor of $n$, and $n$ is divisible by the sum of the digits of $n$ plus one? | open2math |
What is the smallest positive integer that can be expressed as the sum of two perfect squares in exactly three different ways? | open2math |
What is the largest positive integer $n$ such that $2^n$ is less than $100$? | open2math |
Find the largest number of terms of a geometric sequence whose product is 30 and all terms are positive integers. | open2math |
Find the maximum possible value of $x + y + z$ for positive integers $x$, $y$, and $z$ such that the sum of their reciprocals is equal to $\frac{3}{2004}$. | open2math |
Find the sum of the divisors of 120 that are also multiples of 3. | open2math |
Find the number of positive integers less than or equal to 100 that have exactly three distinct positive divisors. | open2math |
A circle is inscribed in a regular heptagon. How many distinct chords of the circle can be drawn such that they are also diagonals of the heptagon? | open2math |
Let $P(x)$ be a polynomial such that
\[P(P(x))=(P(x))^3-(P(x))^2-P(x)-1.\]Find the sum of all possible values of $P(1)$. | open2math |
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 $g(5) = 12$ and $g(8) = 18,$ find $g(25).$ | open2math |
How many real numbers $x$ satisfy the inequality $|x-3|+|x+2|>5$? | open2math |
Find the smallest positive integer $n$ such that $2^n$ is a factor of $100!$ and $2^{n+1}$ is not. | open2math |
Determine the smallest positive integer $k$ such that any $k$ distinct points in the plane with integer coordinates have a subset of 3 points that are vertices of a triangle whose area is positive. | open2math |
Let $S$ be the number of ways to color the vertices of a cube with three colors, red, blue, and green, such that each face of the cube contains exactly two red vertices and exactly one blue vertex. Additionally, no two adjacent vertices can have the same color, and all green vertices must be adjacent to at least two bl... | open2math |
The graph of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ has a tangent line at the point $(3,0)$. Find the sum of the $x$-coordinates of the points where the tangent line intersects the ellipse again. | open2math |
Let $P$ be a point outside of an ellipse $E.$ A line is drawn from $P$ such that it is tangent to ellipse $E$ at point $T.$ Meanwhile, a secant from $P$ intersects $E$ at $A$ and $B,$ such that $PA < PB.$ If the area of the triangle $PAB$ is equal to the area of the triangle $PAT$, then what is the length of the major ... | open2math |
A bookstore has 8 shelves of books. What is the greatest number of pairs of books that can be selected such that each pair consists of books from two different shelves and no two pairs are the same combination? | open2math |
Find the number of ordered pairs of integers $(a,b)$ with $2\le a<b\le 200$ such that $(-1)^a + (-1)^b$ is a positive integer. | open2math |
Let $x$ and $y$ be positive real numbers. Find the maximum possible value of $$\frac{x^2y^2+1-\sqrt{x^4y^4+2x^2y^2+1}}{xy}.$$ | open2math |
Find the value of $b$ for which the equation
$$2x^2+3x+b=0$$
has exactly one real root, and its multiplicity is 3. | open2math |
Find the largest integer value of $n$ such that $n^2-9n+18$ is negative. | open2math |
Let $m$ be the number of positive integers less than $1000$ that are a multiple of $2$ but not $4$, and let $n$ be the number of positive integers less than $1000$ that are a multiple of $6$ but not $12$. Compute $(m-n)^2$. | open2math |
Let a and b be the roots of the equation x^2+px+q=0. Given that the arithmetic mean of the squares of the roots is 6 and the geometric mean of the cubes of the roots is 5. Find p^2. | open2math |
Determine all real numbers x for which the function f(x) = 1/(1 + |x|) is increasing. | open2math |
Find the product of all values of $t$ such that $t^2 = 36$. | open2math |
In the equation
$$4x^2-kx+2=0$$
determine $k$ such that the difference of the roots be equal to unity. | open2math |
Let f(x) be a function that satisfies f(f(x)) = x for all real numbers x.
Determine the possible values of f(10) if it is given that f(f(f(10))) = 20. | open2math |
Find the number of integers n with the property that the product of the digits of n equals 1995 and n itself is divisible by 1995. | open2math |
Let $n$ be a positive number such that when $179$ is divided by $n$, the remainder is $7.$ Compute the sum of all possible two-digit values of $n$. | open2math |
The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | open2math |
Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. | open2math |
The number 3600 can be expressed as a sum of consecutive positive integers. What is the minimum number of consecutive positive integers that sum to 3600, if the sum starts with the number 12? | open2math |
When the digits in the number $2005$ are reversed we obtain the number $5002,$ and $5002 = a \cdot b \cdot c$, such that $a$, $b$ and $c$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$ and $p_3$ such that $p_1 + p_2 + p_3 = a+b+c$? | open2math |
A point (x,y) is chosen at random inside the triangle with vertices (0,0), (0,1), and (1,0). Determine the probability that $x+y<\frac{1}{2}$ | open2math |
Determine the smallest positive integer $n$ such that the decimal representation of $3^n$ has 3 as its units digit. | open2math |
For a positive integer $n$ and nonzero digits $d$, $e$, and $f$, let $D_n$ be the $n$-digit integer each of whose digits is equal to $d$; let $E_n$ be the $n$-digit integer each of whose digits is equal to $e$, and let $F_n$ be the $3n$-digit (not $n$-digit) integer each of whose digits is equal to $f$. What is the gre... | open2math |
Find the number of positive integers less than 100 that have a remainder of 2 when divided by 7. | open2math |
Determine the number of integers $n$ that satisfy the inequality $\sqrt[3]{n} < \log_{2}(n) < \sqrt[5]{n}$. | open2math |
The roots of the quadratic equation $x^2+bx+c=0$ are the reciprocals of the roots of the equation $ax^2+dx+e=0$. If $b=2d$ and $c=-\frac{1}{2}e$, find the ordered pair $(a,e)$. | open2math |
Determine the sum of all positive integer values of $a$ such that the quadratic equation $6x^2 - 7x + a = 0$ has two distinct real roots. | open2math |
The quadratic function $f(x) = x^2 - 4x - 5$ has two real roots. Find the number of integers $n$ such that $f(n) > 0$ and $f(n+1) < 0$. | open2math |
A positive integer $n$ has exactly four positive divisors. What is the smallest possible value of the sum of these four divisors? | open2math |
Find the smallest positive integer n such that the repeating decimal 0.\overline{1} can be expressed as a fraction in simplest form with denominator 2^n. | open2math |
Determine the value of the parameter $a$ such that the equation $x^4-2ax^2+x+a^2=0$ has exactly one positive solution. | open2math |
Compute the sum of all four-digit numbers that can be formed using the digits 1, 2, 3, and 4, with each digit appearing exactly once in each number. | open2math |
Let $a$ and $b$ be positive real numbers such that $a^2 + b^2 = 100.$ Find the maximum value of $ab.$ | open2math |
The graph of a function $f(x)$ is given below.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
voi... | open2math |
In a regular hexagon with side length 12, two congruent isosceles triangles are drawn such that they share one base and each has one vertex on a vertex of the hexagon. The base of each triangle is perpendicular to the side of the hexagon. What is the side length of the largest equilateral triangle that can be inscribed... | open2math |
The difference between two numbers is 9, and the sum of the squares of each number is 153. What is the value of the product of the two numbers? | open2math |
Let $a$ and $b$ be two positive real numbers such that $a + b = 50.$ Find the ordered pair $(a,b)$ for which $a^3 b^4$ is maximized. | open2math |
A circular plate with radius 20 cm has a square cut out from the center. The length of each side of the square is such that its diagonals are equal to the radius of the plate. Find the area of the remaining plate in terms of π. | open2math |
What is the greatest number of people that can be seated at 5 tables, with each table having the same number of people, given that the number of people seated must be a factor of both 40 and 60? | open2math |
The sum of the areas of two squares with sides of consecutive integers is 253. Find the side length of the larger square. | open2math |
Let $S$ be the set of all positive integers whose digits are all $1$ or $2$. Let $T$ be the set of all positive integers whose digits are all $3$ or $4$. Find the smallest integer $n$ such that there exists a one-to-one function $f: S \to T$ with the property that for any $x, y \in S$, the number of digits in the pro... | open2math |
The function $g(x)$ satisfies
\[g(xy) = g(x)y^2\]for all positive real numbers $x$ and $y.$ If $g(2) = 8,$ find $g(3).$ | open2math |
How many different convex quadrilaterals with integer side lengths are there such that the sum of the squares of their sides equals 800? | open2math |
What is the smallest whole number that must be a factor of the sum of any two consecutive positive integers? | open2math |
Given $a$ and $b$ are real numbers such that $|a+b|=2$ and $|ab|=3$, find the smallest possible value of $|a^2+b^2|$. | open2math |
In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. A circle is drawn inside the triangle so that it is tangent to all three sides. The minimum possible radius of such a circle can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | open2math |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x+y))=g(x)g(y)+x+y\]for all $x,y$. Find the sum of all possible values of $g(2)$ | open2math |
Let a and b be non-zero complex numbers such that
|a^2/b|+|b^2/a|=1.
Find the sum of all possible values of
|b^2/a|+|a^2/b|. | open2math |
Let $a$ and $b$ be positive integers. Given that $a^{2b} = 2^{15} \cdot 4^{13} \cdot 8^{11}$, find the least possible value of $ab$. | open2math |
The function $g(x)$ satisfies
\[g(x + y) = g(x) + g(y)\]for all real numbers $x$ and $y.$ Find all possible values of $g(0).$ | open2math |
Determine all real $m$ and $n$ such that for all real $x$, $x^2+mx+n=0$ has the same roots as $x^2+mx+n(x^2+mx+n)^2=0$. | open2math |
There exists a positive real number $y$ such that $ \sin (\arccot (y)) = y $. Find the value of $y^2$. | open2math |
The sum of the cubes of two consecutive integers is 693. What is the value of the product of the two integers? | open2math |
Two pairs $(x,y)$ of real numbers satisfy the equation $x+y=3xy=5$. Given that the solutions $x$ are in the form $x=\frac{a\pm b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, what is the value of $a+b+c+d$? | open2math |
Find the monic cubic polynomial, in $x$, with real coefficients, which has $2 + i$ as a root and satisfies the condition that the sum of the coefficients of the polynomial is 10. | open2math |
Find the sum of the squares of all positive integers from 1 to 25. | open2math |
Find the smallest digit $d$ such that the number $23d456$ is divisible by 11. | open2math |
Find the sum of all real solutions to the equation $x^2+9=6x+\frac{9}{x}$. | open2math |
In the first row of Pascal's Triangle, all the numbers are equal to 1. In the second row, the numbers are 1, 2, and 1, where the middle number is the sum of the two numbers just above it. In general, each number in Pascal's Triangle is the sum of the two numbers just above it and to the left and the right.
In the rows... | open2math |
Simplify the expression $\sqrt{21+12\sqrt{3}}$ into the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$. | open2math |
In the graph of $\frac{x^2+3x+2}{x^3+x^2-2x}$, let $a$ be the number of holes in the graph, $b$ be the number of vertical asympotes, $c$ be the number of horizontal asymptotes, and $d$ be the number of oblique asymptotes. Find $a+2b+3c+4d$. | open2math |
Let $A_n$ denote the set of all positive integers $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point, and let $g(n)$ be the product of all elements of $A_n$. Find the smallest positive integer $m$ such that $g(m) > 10^{10}$. | open2math |
Find all integers $a$ such that the equation $3x \equiv 2 \pmod{a}$ has a solution, and the set of solutions is equal to the set of solutions of the equation $2x \equiv 3 \pmod{a}$. | open2math |
A convex polyhedron has six faces, each being a regular hexagon. Determine the number of edges of the polyhedron if its vertices can be colored with three colors such that any face has all three colors. | open2math |
Let $m$ and $n$ be positive integers such that $2^m \cdot 5^n = 10^6$. Determine all possible values of $m+n$. | open2math |
Find the largest value of $x$ that satisfies the equation $|x-5|=12$. | open2math |
In a regular octahedron, the midpoints of the twelve edges are the vertices of a smaller octahedron. The ratio of the surface area of the smaller octahedron to that of the larger is $a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. | open2math |
A space curve is parameterized by the equations
$$\begin{cases}
x=a \cos t\\
y=b \sin t\\
z=c \sqrt{1-\sin^2t}
\end{cases}$$
where $a,$ $b,$ and $c$ are positive constants. The points $(3,0,4)$ and $(0,1,-3)$ lie on the curve. Find the point on the curve that is farthest from the origin. | open2math |
Find the sum of the odd, positive integers less than 89. | open2math |
Find the number of real values of $x$ such that the decimal representation of $\frac{x}{120}$ terminates and $x^2 + x - 6 = 0$. | open2math |
Let \(a\) and \(b\) be two positive integers such that \(a<b\). Consider a rectangle with dimensions \(a\) and \(b\). It is known that the perimeter of the rectangle is a divisor of its area, and that the area of the rectangle is a divisor of its perimeter. How many different values can the perimeter of this rectangle ... | open2math |
Find the largest integer value of $m$ that satisfies the following inequality:
$$\frac{m}{2\sqrt{m+1}} < 1$$ | open2math |
Find the minimum value of
\[x^2 - 2xy + 3y^2\]over all real numbers $x$ and $y.$ | open2math |
Find the largest three-digit number $N$ such that the value of its hundreds digit plus the absolute difference between its units and tens digits equals $P(N)$. | open2math |
Find a positive integer $x$ such that $x^2$ is a multiple of 12, and $x$ is between 12 and 20. | open2math |
Find the sum of the coefficients of the polynomial $P(x)$, given that its roots are the squares of the roots of the equation $x^3 - 2x^2 - 5x + 1 = 0$. | open2math |
Find the number of positive real numbers $x$ that satisfy the inequality
$$\frac{\sin(x-1)\sin(x-3)\sin(x-5)}{\sin(x+1)\sin(x+3)\sin(x+5)} > 0.$$ | open2math |
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