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Let $m$ and $n$ be positive integers with $m < n.$ For how many positive integers $d$ does there exist a pair of positive integers $m$ and $n$ such that $d$ is a divisor of both $m^2 + n^2$ and $m^2 - n^2,$ where $m$ and $n$ are relatively prime? | open2math |
What is the value of $a$ such that the equation $x^3 - 6x^2 + ax + 2 = 0$ has exactly one real root and the sum of the real parts of all three roots is equal to 4? | open2math |
The vectors $\mathbf{a}$ and $\mathbf{b}$ are such that
the area of the parallelogram generated by $\mathbf{a}$ and $\mathbf{b}$ is equal to 12, and the area of the parallelogram generated by $\mathbf{b}$ and $3\mathbf{a}-\mathbf{b}$ is equal to 35.
Find the area of the parallelogram generated by $\mathbf{a}$ and $5\ma... | open2math |
Find the sum of all positive integers $n$ such that $n^3$ is less than or equal to the sum of the cubes of its digits. | open2math |
Find the number of positive integers between 11 and 103 that have a units digit of 3. | open2math |
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \neq 0$$ | open2math |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}^n = \mathbf{I}.\] | open2math |
What is the number of 4-digit numbers with distinct digits such that the sum of the first two digits is equal to the sum of the last two digits? | open2math |
Let $S_n$ denote the $n$-th hexagonal number, which is the number of dots contained in a hexagon with $n$ dots on a side.
Find the number of solutions to $S_x+S_y=S_z$, where $x$, $y$, and $z$ are positive integers such that $1\le x\le y\le z\le 10$.
[asy]
draw((1,0)--(2,0)--(3,0)--(3,1)--(3,2)--(2,2)--(1,2)--cycle);
... | open2math |
Find the smallest positive integer that is both an integer power of 12 and has a digit sum that is a perfect square. | open2math |
Find the sum of the first seven prime numbers that have a units digit of 7. | open2math |
Find the sum of all the even numbers between -15 and 25, inclusive, that are not divisible by 4. | open2math |
Suppose that $a$ is a positive integer such that $a$ is a divisor of both $15$ and $20$. What is the number of possible values of $a$? | open2math |
Consider a set of functions $f_1(x), f_2(x), \dots, f_n(x)$ defined on the set of real numbers. Each function takes a real number to a real number, and for any two distinct functions $f_i(x)$ and $f_j(x)$ in the set, the equation $f_i(f_j(x)) = x$ holds for all real numbers $x$. How many functions are in the set if $f_... | open2math |
What is the smallest positive integer that is a multiple of the sum of its digits and has exactly three distinct prime factors? | open2math |
In the equation
$$2x^2-kx+3=0$$
determine $k$ such that the sum of the roots be equal to $2$. | open2math |
The numbers $x_1,$ $x_2,$ $x_3,$ $x_4$ are positive integers such that $x_1+x_2+x_3+x_4=12$. Find the maximum possible value of
\[x_1x_2x_3x_4(x_1+x_2+x_3+x_4).\] | open2math |
Let $a^2=\frac{16}{44}$ and $b^2=\frac{(2+\sqrt{5})^2}{11}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a+b)^3$ can be expressed in the simplified form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$? | open2math |
Let $f(x)$ be a quadratic function such that $f(7)=0$, $f(8)=4$ and $f(10)=3$. Determine all possible values of $f(12)$. | open2math |
A circle is inscribed in a square such that the center of the circle lies on the line $x = 2$ and the circle passes through the point $(0,0)$. What is the perimeter of the square? | open2math |
The square of an integer is 182 greater than the integer itself. What is the sum of all integers for which this is true? | open2math |
Find the sum of all the different possible areas of a rhombus, where the number of square units in its area is equal to three times the number of units in its perimeter, and the lengths of its diagonals are integers. | open2math |
Find the smallest positive integer $n$ such that there exists a function $f:\{1,2,\dots,n\}\to \mathbb{R}$ satisfying the following properties:
1. For all $x,y\in \{1,2,\dots,n\}$, $f(x)f(y)-f(xy)=0$.
2. For all $x,y\in \{1,2,\dots,n\}$ with $x<y$, $f(x)<f(y)$.
3. For all $x,y,z\in \{1,2,\dots,n\}$ with $x<y<z$, $f(x)... | open2math |
Find the value of $x$ such that
$\left(1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\cdots\right)\left(1-\frac{1}{x}+\frac{1}{x^2}-\frac{1}{x^3}+\cdots\right)=\left(\frac{1}{2}\right)^x+\left(\frac{1}{2}\right)^{x^2}+\left(\frac{1}{2}\right)^{x^3}+\cdots$. | open2math |
The lengths of the sides of a triangle with positive area are $\log_{10}12$, $\log_{10}75$, and $\log_{10}n$, where $n$ is a positive integer. Find the number of possible values for $n$. | open2math |
Let $a$, $b$ and $c$ be positive real numbers forming a geometric progression. Determine the maximum possible value of $ab+ac+bc$ if $a+b+c=20$. | open2math |
Find the area of the triangle with vertices $(0,7,10),$ $(-1,6,6),$ and $(-4,9,6).$ | open2math |
Let $S$ be the set of all integers $a$ for which the polynomial $x^2 + ax + a^3$ has two distinct integer roots. Compute the largest possible value of $a$ in $S$. | open2math |
Find the smallest integer $n$ such that $2^n$ is a perfect square and is divisible by 8, and lies between 60 and 100. | open2math |
Find the positive integer $n$ such that
\[\sin \left( \frac{\pi}{2n} \right) + \cos \left (\frac{\pi}{2n} \right) = \frac{\sqrt{n}}{2}.\] | open2math |
Find the product of all the positive divisors of 315 that are one more than a multiple of 3. | open2math |
Given that $\left|a+bi\right|=5$, where $a$ and $b$ are positive integers, determine the smallest possible value of $\left|a-b\right|$. | open2math |
Find all perfect cubes between 100 and 200 that can be expressed as the sum of two perfect squares. | open2math |
Let $f(x) = ax^2+bx+c$ be a quadratic function with integer coefficients such that the expression $\sum\limits_{i=1}^n f(x_i)$ has the maximum possible value of 2023, where $x_1, x_2, \dots, x_n$ is a subset of $\{1, 2, \dots, 100\}$. Determine the number of ordered triples $(a, b, c)$ for which this maximum is attaine... | open2math |
What is the smallest positive integer $n$ such that the roots of the quadratic equation $x^2+nx-6=0$ differ by more than 5? | open2math |
Find the smallest positive integer $a$ such that the equation $x^2 \equiv a \pmod{20}$ has a solution, and $a$ is a multiple of $3$. | open2math |
Find the largest value of $k$ such that the equation $f(x) = 2x^3 - 4x^2 + k$ has three real roots. | open2math |
Given that $x$, $y$, and $z$ are positive real numbers such that $x+y+z=1$, which of the following inequalities must hold?
A) $x^2+y^2<z^2$
B) $x^2+z^2<y^2$
C) $x^2+y^2<2z^2$
D) $2x^2+2y^2<z^2$
E) $x^2+y^2+z^2<1$
Enter your answer as a list of those options that are always true. | open2math |
Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). The area of the pentagon can be expressed as the sum of the areas of five triangles formed by the sides of the pentagon and the radii of the inscribe... | open2math |
Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and
\[f(x - y) = \sqrt{f(xy) + 2}\]for all $x > y > 0.$ Determine $f(2009).$ | open2math |
Find the number of all four-digit positive integers such that the sum of their digits is 10. | open2math |
Find the number of distinct positive integer divisors of 36 that are also multiples of 3. | open2math |
Find the smallest positive integer such that the sum of its digits is equal to 7 and the sum of the squares of its digits is a perfect square. | open2math |
Prove that $\ds\sqrt{2+2\cos \ds\frac{\pi}{10}}+\sqrt{2-2\cos \ds\frac{\pi}{10}}=2$. | open2math |
Find the number of positive integers less than 100 that are both a perfect square and a perfect power of 5. | open2math |
What is the smallest positive integer $n$ such that the sum of the squares of the digits of $n^2$ is equal to the sum of the squares of the digits of $n$? | open2math |
Determine the smallest positive integer $k$ such that the sum of the remainders when divided by 25 of all the integers from 1 to 2007 is divisible by $k$. | open2math |
Consider the triangular array of numbers with 0, 1, 2, 3, $\dots$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.
\[
\begin{array}{ccccccccccc}
& & & & & 0 & & & & & \\
& & & & 1 & & 1 & & & & \\
& & & 2 & & 2 & & 2 & & & \\
& & 3 & & 4... | open2math |
Consider a standard 5-digit postal code, where each digit can be 0-9. Given that each 5-digit arrangement is equally likely, the probability that such a postal code will contain at least one pair of consecutive digits that are the same (e.g., 11, 22, 33, etc.) is $\dfrac{a}{b}$, where $a$ and $b$ are relatively prime p... | open2math |
The largest integer $n$ such that $n!$ contains exactly $200$ zeros at its end.
Determine $n$. | open2math |
A sequence of numbers is formed as follows: starting with the number $n$ in the middle, the number above it and to the left is $(n-1)/2$, the number above it and to the right is $(n+1)/2$, and this process continues with each subsequent number being the average of the two numbers below it. What is the sum of the number... | open2math |
Regular decagon $B_1B_2B_3B_4B_5B_6B_7B_8B_9B_{10}$ is inscribed in a circle of area $2.$ Point $Q$ lies inside the circle so that the region bounded by $\overline{QB_1},\overline{QB_2},$ and the minor arc $\widehat{B_1B_2}$ of the circle has area $\tfrac{1}{6},$ while the region bounded by $\overline{QB_4},\overline{Q... | open2math |
Given a positive integer $M$, find a positive integer $N$ with six or fewer digits such that $N$ is the multiplicative inverse of $M$ modulo 1,000,000. Additionally, the sum of the digits of $N$ is equal to the sum of the digits of $M$. | open2math |
Let $x$ and $y$ be positive real numbers such that $[x]+[y]=3$.
Evaluate $\left[\frac{x+y}{2}\right]$. | open2math |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[15,m]=45$ and $\mathop{\text{lcm}}[m,20]=60$, what is $m$? | open2math |
What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions? | open2math |
The sum of the reciprocals of the roots of the quadratic equation $ax^2+bx+c=0$ is $-7/3$, and the product of the roots is $5/3$. If $a$, $b$, and $c$ are positive integers, find the value of $a+b+c$. | open2math |
Simplify the expression $\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}$ assuming that $x$ is a real number between $-1$ and $1$. | open2math |
There are four distinct positive integers $a,b,c,d$ less than $8$ which are invertible modulo $8$. Find the remainder when $(abc+abd+acd+bcd)(abcd)^{-1}$ is divided by $8$.
Write another problem inspired by this one:
Let $n$ be a positive integer. Find the smallest possible value of $n$ such that there exist distinct... | open2math |
Given the real function $f(x)=\frac{x^3}{1+x^2}$,
how many integer values of $x$ exist such that
$0.95 < f(x) < 1.05$ ? | open2math |
Find the smallest three-digit palindrome number that is divisible by the sum of its digits. | open2math |
Find the product of all positive real numbers $x$ such that $x+\frac{x^2-4x+4}{x^2+x+1}=7$. | open2math |
Find the greatest value of $c$ such that $|x-c|\ge 2$ for all $x$ satisfying the inequality $x^2-6x+8\le 0$. | open2math |
Find all values of $a$ such that the equation $x^3+(x+a)^3=(2x)^3$ has exactly one real solution. | open2math |
The positive integers $a$ and $b$ have the same number of positive integer divisors, and for every prime number $p$ that divides $a$, there is exactly one prime number $q$ that divides $b$ such that $p \equiv q \pmod{2}$. What is the minimum possible value of $|a - b|$? | open2math |
In an equilateral triangle $ABC$, a point $P$ is chosen inside the triangle so that the area of the region bounded by $\overline{PA}$, $\overline{PB}$, and the side $\overline{AB}$ is $\tfrac{1}{5}$ of the area of the entire triangle, and the area of the region bounded by $\overline{PC}$, $\overline{PA}$, and the side ... | open2math |
Given the equation
$$5x^2-kx+1=0$$
determine $k$ such that the difference of the roots be equal to unity.
Write another problem inspired by this one:
Given the equation
$$x^2-3ax+a^2=0$$
find $a$ such that the sum of the roots is equal to $1.75$.
(Note: I've changed the problem by keeping the quadratic equation but ... | open2math |
Given the polynomial $x^3+bx^2+cx+d$, find the value of $k$ such that the polynomial can be written in the form $(x+k)^3+m$, where $m$ is a constant. Express $k$ in terms of $b$, $c$, and $d$. | open2math |
Find all real numbers $b$ such that the equation
\[2x^3 - bx^2 - 3bx + b^2 + 2 = 0\]
has exactly one real solution in $x$. | open2math |
Find the number of ordered 10-tuples $(b_1, b_2, b_3, \dots, b_{10})$ of positive integers, such that the product of any two numbers in the 10-tuple is equal to the sum of the remaining 8 numbers. | open2math |
Let $f(n)$ denote the sum of the cubes of the digits of a positive integer $n$. For example, $f(145) = 1^3 + 4^3 + 5^3 = 190$. Find the largest three-digit number $n$ such that $n$ is divisible by $f(n)$. | open2math |
Consider two complex numbers $z_1$ and $z_2$ that are roots of the polynomial $f(z)$. Determine the possible values of the sum of the squares of the coefficients of $f(z)$ when the product of $z_1$ and $z_2$ is equal to the sum of $z_1$ and $z_2$. | open2math |
Find the sum of all integers between 1 and 100 that are not divisible by 3 or 5. | open2math |
Let $a,$ $b,$ $c$ be non-zero real numbers such that $a + b + c = 0.$ Find all possible values of
\[\frac{a^3 + b^3 + c^3}{abc}.\]Enter all the possible values, separated by commas. | open2math |
Let $M$ be a set of $n$ elements, and let $\mathcal{T}$ be a family of subsets of size $5$ of $M$, such that no two members of $\mathcal{T}$ have exactly the same $5$ elements. For each subset $K\subseteq M$, let $t(K)$ denote the number of members of $\mathcal{T}$ that contain $K$ as a subset. Find all values of $n$, ... | open2math |
What is the sum of all positive integers less than 200 that are cubes of perfect cubes? | open2math |
The point with coordinates $(3, 5)$ is the midpoint of the segment with one endpoint at $(0, -2)$. Find the sum of the coordinates of the other endpoint. | open2math |
Find the sum of all even numbers between 1 and 200, excluding those that are divisible by 4. | open2math |
How many three digit numbers are left if we exclude all three digit numbers in which there are exactly two digits the same, but these two digits are not adjacent? | open2math |
What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers? | open2math |
In a right triangle $ABC$ with hypotenuse $AB = 24$, a sphere is inscribed such that it touches the triangle at points $D$, $E$, and $F$. The center of the sphere is denoted by $O$. If the radius of the sphere is $6$, find the sum of the lengths of segments $OD$, $OE$, and $OF$. | open2math |
A circle of radius $2\sqrt{3}$ is circumscribed around an equilateral triangle $ABC$. The area of the triangle is inversely proportional to the square of the distance from the center of the circle to the midpoint $M$ of side $BC$. If the distance from the center of the circle to $M$ is $2$ when the area of the triangle... | open2math |
Find the range of values for $x$ such that the expression $\arcsin(\arctan(\arccos(x)))$ is defined. | open2math |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) - y) = g(x^2 + y) + 2g(x) y^2\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the sum of all possible values of $g(2).$ Find $m \times t.$ | open2math |
Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{9}{10}.\] | open2math |
Let $\alpha$, $\beta$, and $\gamma$ be real numbers such that
$\alpha+\beta=5$, $\alpha+\gamma=3$, $\beta+\gamma=4$, and
$$\left(\frac{4}{5}\right)^{\alpha}\left(\frac{2}{3}\right)^{\beta}\left(\frac{1}{2}\right)^{\gamma}=\frac{1}{\sqrt{15}}.$$
Determine $10^{\alpha}$. | open2math |
Let $S$ be the set of all positive integers that have exactly 4 positive divisors and are less than or equal to 100. What is the sum of the elements of $S$? | open2math |
The product of the cubes of three consecutive positive integers is 2500 times their sum. What is the sum of the squares of the three original integers? | open2math |
Find the smallest value of $x$ that satisfies the equation $|3x+7|=26$. | open2math |
Find the least positive integer $n$ such that $40304n$ is divisible by $30203$. | open2math |
Find the number of divisors of $2^{2018}$ that are not multiples of $8$. | open2math |
Let \{$p_n$\} be a sequence of numbers such that $$p_n=\sqrt[n]{17^n}+\sqrt[n]{289^n}+\sqrt[n]{4913^n},$$ where $n$ is a positive integer greater than or equal to $2$. How many terms of the sequence are integers? | open2math |
Find the sum of the coordinates of the midpoint of the segment with endpoints $(4, 9)$ and $(2, -3)$. | open2math |
How many integers $N$ less than $1000$ can be written as the sum of $j$ consecutive positive odd integers from exactly 5 values of $j\ge 1$?
| open2math |
Let $a$, $b$, $c$ be distinct real numbers such that the roots of the equation
$$x^2+ax+b=0$$
are equal in absolute value, but have opposite signs.
Prove that $c$ is the only real root of the equation
$$|x-a|+|x-b|=|x-c|.$$ | open2math |
Let $a$ be a number such that the units digit of $a^2$ is 5. How many distinct three-digit positive integers $b$ are such that the units digit of $ab$ is 5? | open2math |
Let $a$ be a positive integer with $a \equiv 1 \pmod 6$. For which values of $a$ is it possible to have
$a^{n+5} \equiv a^n \pmod{162}$ for every positive integer $n$? | open2math |
Let $\mathcal{S}$ be the set of all positive integers that can be represented as $x+\lceil{x}\rceil$ for some real number $x$. Determine the largest integer $k$ such that for every positive integer $n$, there exists an integer $m$ in $\mathcal{S}$ with $|m-n|\leq k$. | open2math |
Let $A$ be the set of all real numbers except $0$, $1$ and $-1$. Find the value of $n \in A$ that maximizes the sum
\[S_n = \sum_{k = 0}^{n} \frac{1}{\binom{n}{k}} + \sum_{k = 0}^{n} \frac{k}{\binom{n}{k}}.\] | open2math |
Find the sum of the reciprocals of all the positive divisors of 48 that are not perfect squares. | open2math |
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