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Determine the value of the constant $b$ such that the expression $9x^2 - 24x + b$ is a perfect square trinomial. | open2math |
Let $f(x)$ be a polynomial of degree $4$ such that the value of $f(x)$ is constant and nonzero for all $x$ in the set $\{1, 2, 3, \ldots, 8\}$. Determine the largest possible number of real roots of the equation $f(x) = 0$. | open2math |
Count the number of positive integers less than 100 that have only prime digits. | open2math |
Consider the infinite geometric series $1 + x + x^2 + \cdots$. Determine $x$ such that the sum of every other term (starting from the second) is equal to twice the sum of the remaining terms. | open2math |
Determine the sum of all single-digit replacements for $z$ such that the number ${24{,}z38}$ is divisible by 6. | open2math |
A square and a circle are inscribed in the same regular hexagon. The ratio of the area of the square to the area of the circle is 1.5. Determine the side length of the hexagon. | open2math |
Given that $x$ is real and $x^3+\frac{1}{x^3}=52$, find $x+\frac{1}{x}$. | open2math |
A rectangular prism has a volume of 120 cubic units. If the dimensions of the prism are such that the product of the lengths of two of the sides is 16 units, find the length of the third side. | open2math |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xy) + f(xz) - f(x) f(yz) \ge 1\]for all real numbers $x,$ $y,$ and $z.$ | open2math |
Find the largest four-digit integer that is divisible by the product of its digits. | open2math |
Let $a$ and $b$ be positive integers such that $a>b$. Prove that the greatest common divisor of $a^b-b^a$ and $a^a-b^b$ is equal to $a-b$. | open2math |
A bookshelf contains 12 books, and each book has a unique color. The books are arranged on the shelf in a specific order. Given that the colors of the books are red, blue, green, yellow, orange, purple, pink, black, white, gray, brown, and turquoise, and that the books are arranged such that no two books of the same co... | open2math |
Let $m$ and $n$ be positive integers such that $\gcd(m,n) = 15$. Determine the smallest possible value of $\gcd(24m,30n)$ that is divisible by $3$. | open2math |
Find all complex numbers $z$ such that the number obtained by reading the digits of $z$ backwards (in the same base) is a multiple of $z$. | open2math |
Find the product of all positive integers $n$ such that $2^{n-1}<n^2<2^n$. | open2math |
Let $a$ and $b$ be the roots of the equation $2x^2-5x+3=0$. Find the sum of the squares of the numbers $a+b$, $ab$, and $1/(a^2+b^2)$. | open2math |
Find the value of $x$ for which the expression $\frac{x}{|x|}$ is equal to $\frac{2x-3}{|2x-3|}$ and which has the largest absolute value. | open2math |
If $a$ and $b$ are positive integers such that $\gcd(a,b) = 24$, then what is the smallest possible value of $\gcd(8a,12b)$? | open2math |
Find the value of $a$ that satisfies the equation $293_{a}+468_{a}=73B_{a}$, where $B_{a}=11_{10}$. | open2math |
Let
$f(x)=\ds\sqrt [3]{\log _x{(x+1)}}+\ds\sqrt [3]{\log _{x+1}{(x+2)}}+\cdots +\ds\sqrt [3]{\log _{x+n-1}{(x+n)}}$
where $n$ is a positive integer and $x>0$.
Find $n$ for which the function $f(x)$ has a relative extremum. | open2math |
Find the value of $x$ such that the equation
\[3x^2-10x+5=0\]
holds true, where the coefficients of the quadratic equation are determined by the system of equations:
\[3a^2-2b^2=1\]
\[2a^3+b^3=5\] | open2math |
Let $a$ and $b$ be two complex numbers such that $|a| = |b| = 1$. If $a^n + b^n = 1$ for all positive integers $n$, determine the maximum value of $|a + b|$. | open2math |
The sum of the first 2020 positive odd integers and the sum of their reciprocals are equal. Find the minimum value of the sum of a pair of these odd integers, one of which is a multiple of the other. | open2math |
Determine the number of different arrangements of the digits in the decimal representation of 3326 that are divisible by 2. | open2math |
What is the sum of all integer values $n$ for which $\binom{20}{n}+\binom{20}{10}=\binom{21}{11}$? | open2math |
A $\textit{palindrome}$ is a positive integer which reads the same forward and backward, like $12321$ or $4884$.
How many $5$-digit palindromes are there such that the sum of their digits is equal to $10$? | open2math |
Let $a$ and $b$ be two real numbers such that
$0 < a < b < 1$. The numbers $a$ and $b$ satisfy the following
property: when $a$ and $b$ are expressed as decimals in base 10,
with no leading zero, the digits of $a$ are a permutation of the
digits of $b$. Determine the smallest $b - a$. | open2math |
Find the number of positive integers less than or equal to 1000 that are not divisible by any of the numbers 15, 20, or 25. | open2math |
Find all values of $a$ such that the following system has a solution $(x, y)$ in real numbers:
\begin{align*}
x^y + y^x &= a^a, \\
a^{x + y} &= (xy)^a.
\end{align*} | open2math |
The function $f(x)$ takes on real values for all real $x$ except $x=1$ and $x=-1$, where it has vertical asymptotes. Its graph also has a horizontal asymptote at $y=2$. If $k$ is a randomly selected real number, what is the probability that the equation $f(x)=k$ has at least one solution that is not in the interval $(-... | open2math |
Let $P_1$, $P_2$, $P_3$, $P_4$ be four distinct points in the plane, such that the segments $\overline{P_1P_2}$, $\overline{P_2P_3}$, $\overline{P_3P_4}$, and $\overline{P_4P_1}$ form a convex quadrilateral. At each step, a particle, initially at $P_1$, moves to one of the two adjacent points with equal probability. Le... | open2math |
Find the number of ordered 12-tuples $(b_1, b_2, b_3, \dots, b_{12})$ of integers, such that the cube of any number in the 12-tuple is equal to the product of the other 11 numbers. | open2math |
Determine all possible integer values of $a$ such that there exists a pair of real numbers $(b,c)$ satisfying the equations $a+b^2 = c^2$ and $ab - bc = \frac{c^2}{2}$. | open2math |
Circle $A$ has its center at $(3, -4)$ and a radius of $8$ units. What is the sum of the $x$-coordinates of the two points on circle $A$ that are also on the $x$-axis? | open2math |
Let $x$, $y$ and $z$ be positive integers such that $xy-x$, $xy-y$, and $xy-z$ are all prime numbers. Find the largest possible value of $xy$. | open2math |
Find the sum of all real numbers $c$ such that there exists a circle centered at $C = (0,c)$ with exactly one chord $\overline{AB}$ that is tangent to the parabola $y = x^2$. | open2math |
How many four-digit numbers are there that have at least two consecutive digits that are the same? | open2math |
Find the smallest positive integer $a$ such that $x^4 + a^2$ is not prime for any integer $x.$ | open2math |
Determine the largest positive integer $m$ such that the product of the first $m$ prime numbers can be expressed as the product of $m + 2$ consecutive even positive integers. | open2math |
Compute the sum of the squares of all the divisors of 24. | open2math |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=7$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | open2math |
Let $r$ be a positive constant. Prove that if the incircle of triangle $ABC$ touches $AB$ and $AC$ at points $P$ and $Q$ respectively, then
\[\frac{1}{AP} + \frac{1}{AQ} + \frac{1}{BC}\]is constant, and find this constant in terms of $r.$ | open2math |
Find the sum of all odd positive integers that can be represented in the form $4k^2+3$, where $k$ is a positive integer, and $4k^2+3<500$. | open2math |
In the equation
$$7x^2-kx+3=0$$
determine $k$ such that the difference of the roots be equal to unity. | open2math |
Find the positive real value of $s$ that satisfies $|s-3i\sqrt{2}| |4+5i| = 35$. | open2math |
Find the only real number that can be expressed in the form \[(a + bi)^3 - 107i,\]where $i^2 = -1,$ and $a$ and $b$ are positive integers. | open2math |
The function $f(x)$ satisfies
\[f(x - y) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(x) \neq 0$ for all real numbers $x.$ Find $f(3).$ | open2math |
Find the number of positive integers x that satisfy the equation:
(1 + x)^(1 + x) = 4x^(4 - x)
Determine the values of x for which the equation is true. | open2math |
Let $n \ge 4$ be an integer and let $T = \{4,5,6,\ldots,n\}$. Find the smallest value of $n$ such that for every partition of $T$ into three subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $a+b = c$. | open2math |
Find the smallest positive integer that is both an integer power of 13 and is not a palindrome. | open2math |
Find all real values of $x$ that satisfy $\frac{1}{x+1} + \frac{3}{x+7} \ge \frac23.$ (Give your answer in interval notation.) | open2math |
Find all real numbers $a$ such that the graphs of the functions $y = e^{x + a}$ and $y = e^{1 - x}$ are symmetric with respect to the line $y = x$. | open2math |
Find the sum of all odd, positive integers less than 91. | open2math |
Find the sum of the real solutions of: $|x^2-1|=3|x-1|$. | open2math |
The solutions $p$ and $q$ of the equation
$x^2+2ax+13=0$ are such that
$\frac{1}{p}+\frac{1}{q}=3$.
Determine $a$. | open2math |
Find the positive real number(s) $y$ such that $\frac{1}{3}\left( 2y^2+5\right) = \left( y^2-30y-15\right)\left( y^2+20y+10\right)$. | open2math |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a+b+c=1.$ Find the maximum value of
\[\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}.\] | open2math |
A box contains 12 marbles, some of which are blue and the rest of which are green. When two marbles are drawn at random at the same time, the probability that both marbles are blue is $\frac{1}{20}$. How many marbles in the box are blue? | open2math |
Find the sum of all divisors of $144$ that also divide $75$, including both positive and negative divisors. | open2math |
Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $1001$, then what is the sum of the numbers written on the faces? | open2math |
Consider a polynomial function $f(x)=ax^3+bx^2+cx+d$. It is known that the function has three distinct roots $r_1$, $r_2$, and $r_3$. Moreover, it is given that the product of the roots taken two at a time is equal to the sum of the roots (i.e., $r_1r_2+r_1r_3+r_2r_3=r_1+r_2+r_3$). Find $a/b$. | open2math |
Find the values of $a$ for which the inequality
\[\frac{x^2+ax+1}{x^2+2x+2} < 1\]
holds true for all real numbers $x.$ | open2math |
A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater than the number of the green point, but les... | open2math |
Find the range of all possible values of $m$ such that the graphs of the functions $f(x)=mx-2$ and $g(x)=m^2x+m-1$ intersect on the x-axis. | open2math |
Find the value of $x$ such that $\sqrt{x+ 7} = 9$. | open2math |
Given the equation $\frac{x^2-7x-18}{x^2+3x-4}=x-5$, find the sum of the squares of its solutions. | open2math |
Determine the number of distinct triangles whose vertices are lattice points and whose sides are parallel to the coordinate axes, such that the perimeter of the triangle is equal to 30. | open2math |
Point $P$ lies on the circle $x^2+y^2=16$ and is 5 units from the point $(0,0)$. Find the sum of all possible x-coordinates that satisfy the given conditions. | open2math |
Let $f(x)$ be a polynomial of degree 2006 with real coefficients, and let its roots be $r_1,$ $r_2,$ $\dots,$ $r_{2006}.$ There are exactly 1006 distinct values among
\[|r_1|, |r_2|, \dots, |r_{2006}|.\]What is the minimum number of real roots that $f(x)$ can have? | open2math |
What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit 9?
Another problem inspired by this one:
Find the smallest positive integer $n$ such that $\frac{1}{n}$ is a repeating decimal with a repeating cycle of length 3, and $n$ contains the digit 7. | open2math |
A five-digit number $\underline{a2a4a}$ is divisible by 9. Determine the value of $a$ such that when the digits of the number are reversed, the resulting number is also divisible by 9. | open2math |
Compute the product of all the roots of the equation $(x^2+2x+1)(x^2-4x+4)-(x^2+2x+1)(x^2-6x+9)=0$. | open2math |
A square $ABCD$ is inscribed in a semicircle with diameter $d$ so that vertex $A$ lies on the diameter and vertices $B$ and $C$ lie on the semicircle. It is given that the perimeter of the square is equal to $d$. Find the ratio of the area of the square to the area of the semicircle. | open2math |
Find the sum of the first 5 terms of the geometric sequence with common ratio $r$ such that the sum of the first 3 terms is 21 and the sum of the first 6 terms is 91. | open2math |
How many positive integers less than 1000 can be formed using only the digits 3, 4, and 7, such that no digit is repeated? | open2math |
Find the smallest positive integer that has exactly five distinct prime factors. | open2math |
Let $P$ be a regular 12-sided polygon whose vertices are distinct lattice points in the coordinate plane. The area of $P$ can be expressed as $a\sqrt{b} + c\sqrt{d}$, where $a, b, c,$ and $d$ are positive integers and $b$ and $d$ are square-free. Find the smallest possible value of $a + b + c + d$. | open2math |
What is the largest positive integer that is guaranteed to divide the sum of the first n terms of any geometric sequence whose terms are positive integers and whose common ratio is greater than 1? | open2math |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | open2math |
The positive integers $m$ and $n$ are such that the decimal $0.070707\ldots$ can be represented as the fraction $\frac{8m+11}{6n+9}$. Determine the smallest possible value of the sum $m+n$. | open2math |
Find the sum of all the real solutions of $\ds x^6-4x^3+1=0$ and determine if the equation has any complex solutions. | open2math |
Given a positive integer $m$, it can be shown that every complex number of the form $p+qi$, where $p$ and $q$ are integers, can be uniquely expressed in the "base" $-m+2i$ using the integers $1,2,\ldots,m^2$ as "digits." That is, the equation\[p+qi=b_n(-m+2i)^n+b_{n-1}(-m+2i)^{n-1}+\cdots +b_1(-m+2i)+b_0\]is true for a... | open2math |
What is the smallest positive integer $n$ such that $n^{n^n}$ ends in the digits 246? | open2math |
A certain polynomial $P(x)$ has the property that whenever $x$ is a positive integer, $P(x)$ is divisible by $x$. What is the smallest positive integer $n$ such that $P(5n)$ is always divisible by 5? | open2math |
Find the number of ordered pairs $(A, B)$ of subsets of a set with $n$ elements, such that the symmetric difference of $A$ and $B$ has exactly $k$ elements, and the intersection of $A$ and $B$ has exactly $m$ elements. | open2math |
Find the sum of the coordinates of the midpoint of the segment with endpoints $(4, 8)$ and $(-2, 3)$. | open2math |
Let $n \ge 2$ be an integer and let $T = \{1, 2, 3, \ldots, n\}$. Find the smallest value of $n$ such that for every partition of $T$ into three subsets, at least one of the subsets contains integers $x$, $y$, and $z$ (not necessarily distinct) such that $x + y = z$. | open2math |
Find the value of $x$ such that $0.\overline{x234} = \frac{9}{11}$. | open2math |
A portion of a sphere is cut out by two planes, one of which passes through the center of the sphere. Given that the portion has volume 17 cubic centimeters, and that the distance between the two planes is 3 centimeters, determine the radius of the sphere. | open2math |
Let $a$ and $b$ be relatively prime positive integers such that $\frac{a}{b}$ is not equal to $\frac{c}{d}$ for any integers $c$ and $d$ such that $0<c<a$ and $0<d<b.$ Find the smallest value of $b,$ given that $|7a-19b|=1.$ | open2math |
Let $g$ be a function defined on the positive integers, such that $g(x+y) = g(x)g(y)$ for all positive integers $x$ and $y.$ Given $g(5) = 3$ and $g(7) = 2$, find $g(12)$. | open2math |
Find the number of trailing zeros in the sum of all positive integers from 1 to 1000, inclusive. | open2math |
Let $a$ be a positive integer. Define the function $f(a)$ to be the smallest positive integer $b$ such that $b^2 \geq a$. Find the value of $f(2136)$. | open2math |
Find the point on the graph of the equation $y=4x$ that is equidistant from the points $C(-4,0)$ and $D(0,6)$. | open2math |
In the complex plane, find the number of complex numbers of the form $a+bi$ where $a$ and $b$ are integers, such that the distance from $a+bi$ to $2+3i$ is equal to 5. | open2math |
Let $x,$ $y,$ $z$ be positive real numbers such that $xyz = 8.$ Find the minimum value of $x + 2y + 4z.$ | open2math |
Find the number of distinct right triangles with integer side lengths and a hypotenuse of length 25. | open2math |
Determine the number of integers $n$ such that the decimal representation of $n^2$ consists of only 2 digits. | open2math |
What is the largest possible value of $a$ such that $a^2$ is the area of a rectangle with integer side lengths, one of which is even and the other odd? | open2math |
Find the largest value of $n$ such that $3x^2 +nx + 72$ can be factored as the product of two linear factors with integer coefficients. | open2math |
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