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Suppose that $x$, $y$, and $z$ are positive integers such that $x^3_y+y^2_z+z_x=2z^3_x+3x^2_y+5y_z$. What is the largest possible value of $x+y+z$? | open2math |
Given that $f(x)$ is a function such that $f(x) > 0$ for all $x$, find all values of $a$ such that for every $b > 0$, we have $$\frac{f(a)f(b)}{f(a+b)} > \frac{f(a)}{f(a+b)}.$$ Express your answer in interval notation. | open2math |
Real numbers $a$ and $b$ are chosen with $1<a<b$ such that no triangle with positive area has side lengths $1, a,$ and $b$ or $\tfrac{1}{b}, \tfrac{1}{a},$ and $1$. What is the smallest possible value of $b$? | open2math |
Let a, b, c, d be four positive integer such that a<b<c<d.
Let p and s be two positive integer such that p=a+b+c and s=b+c+d.
Find the least value of d such that p, s and p+s=2(a+b+c+d) are all perfect squares. | open2math |
There are four complex numbers $z$ such that
\[z \overline{z}^3 + \overline{z} z^3 = 350,\]and both the real and imaginary parts of $z$ are integers. These four complex numbers are plotted in the complex plane. Find the area of the quadrilateral formed by the four complex numbers as vertices. | open2math |
The product of the numbers $\sin(\pi/3),\sin(2\pi/3),\sin(3\pi/3),\dots,\sin(100\pi/3)$ can be represented as a fraction $\frac{a}{b}$ where $a$ and $b$ are relatively prime. Find the sum of the last two digits of $a$ and $b$. | open2math |
Determine the largest prime factor of the product of all positive integers less than or equal to 30 that are relatively prime to 30. | open2math |
What is the smallest positive integer $n$ such that $3n \equiv 1356 \pmod{22}?$ | open2math |
Given that the point $(2,5)$ is on the graph of $y=g(x)$, there is one point that must be on the graph of $3y=2g(5x)-1$. What is the product of the coordinates of that point? | open2math |
Determine the value of $x$ such that the difference between the reciprocals of $x$ and $2x$ is equal to 0.25. | open2math |
Determine the number of diagonals that can be drawn for a nonagon. | open2math |
Count the number of positive integers less than 10000 that consist only of the digits 1 and 3. | open2math |
Consider the function
\[f(x) = \sum_{i=1}^{n} (x-i)^{2i},\]
where $n$ is a positive integer. Find the largest possible value of
\[\sum_{i=1}^{n} f(a_i),\]
where $a_i$ are the roots of the equation
\[f(x) = 0.\] | open2math |
The sum of the squares of the roots of the equation
$$x^3-2x^2-3x+1=0$$
is equal to 7.
Determine the product of the roots. | open2math |
Prove that for any positive integer $n$ the expression
$$(2n+1)^4-(2n-1)^4-8(2n)^2$$
is divisible by 96. | open2math |
Find the number of positive integers less than 1000 that are divisible by both 7 and 13. | open2math |
Consider the set of functions $f(n)$ that map the positive integers to the positive integers. Find the number of functions that satisfy the functional equation $f(a + b) + f(ab) = f(a) f(b) + 2$ for all positive integers $a$ and $b$. | open2math |
Find the smallest positive integer $n$ such that $|n + \sqrt{n}|$ is an integer. | open2math |
Find the radius of the smallest circle in the complex plane that contains all complex numbers $z$ such that
\[\left| z + \frac{z^2}{z-1} \right| = 1\]and the real part of $z$ is positive. | open2math |
A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $24$ cm across as the top and $8$ cm deep. What was the radius of the ball (in centimeters)?
$\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 8\sqrt{3} \qquad \textbf{(... | open2math |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x^2 + f(y^2)) = (x + y)^2\]
for all real numbers $x$ and $y$. | open2math |
Problem:
A sequence of positive integers is defined recursively as follows:
$$a_n = \left\lceil \frac{2a_{n-1}+3}{a_{n-1}+1} \right\rceil$$
Find the set of values of $a_1$ such that the sequence is monotonically increasing. | open2math |
Find the sum of all non-negative integers $a$, $b$, and $c$ such that $\binom{10}{a} \cdot \binom{20}{b} \cdot \binom{30}{c} = \binom{60}{20}$. | open2math |
Find the number of functions $f(n),$ taking the integers to the integers, such that
\[f(a \cdot b) + f(a+b) = f(a) f(b) - 1\]for all integers $a$ and $b.$ | open2math |
Find all real numbers $x$ such that
\[\frac{2^x + 3^x}{2^{x+1} + 3^{x+1}} = \frac{5}{7}.\]Enter all the solutions, separated by commas. | open2math |
Count the number of positive divisors of 1,000,000 that are not multiples of 10 and whose last two digits are not both equal to 5. | open2math |
How many integers $m \neq 0$ satisfy the inequality $\frac{1}{|m|}\geq \frac{1}{8}$?
Determine the number of integers $n$ such that $n^2-64\leq 0$ and $\frac{n}{8}+\frac{8}{n}\geq 2$. | open2math |
Let $\alpha$ be a real number such that $0<\alpha<\pi$. Compute the minimum integer value of $\ds\frac{1+\sin\alpha\cos\alpha}{\cos^2\alpha}$. | open2math |
What is the smallest positive integer $n$ such that the $n^{\text{th}}$ triangular number is a perfect square? | open2math |
Find the product of the sum and difference of two numbers $a$ and $b$, given that their sum is 12 and their difference is 4. | open2math |
Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14,$ find the smallest possible value of $|w^3+z^3|.$ | open2math |
What is the positive difference between the product of the digits of the number 214 and the sum of the digits of the number 214 squared? | open2math |
Let $n$ be a positive number such that when $123$ is divided by $n$, the remainder is $6.$ Compute the sum of all possible three-digit values of $n$. | open2math |
There are values $C$ and $D$ such that
\[\frac{Dx+15}{x^2+8x+12}=\frac{2}{x+2}+\frac{C}{x+6}.\]
Find $C+D$. | open2math |
Find the sum of the squares of the roots of the equation
$$x^3-x^2+2=0$$ | open2math |
Circle $B$ has its center at $(-6, 2)$ and a radius of $10$ units. What is the sum of the $y$-coordinates of the two points on circle $B$ that are also on the $y$-axis? | open2math |
Let $x$ be a real number. Define
$$\binom{x}{0}=1, \qquad \binom{x}{n}=\frac{x(x-1)\dots (x-n+1)}{n!}, \qquad n=1,2,\dots.$$
Prove that for any positive integer $n,$
$$\sum_{k=0}^{n} (-1)^{n-k}\binom{n}{k}\binom{x+k}{n}=\begin{cases} 0& \text{if} & x=0,1,2,\dots,n-1\\ 1&\text{if}& x=n. \end{cases}$$ | open2math |
Let $a$ be a real number such that $0 < a < 1$ and
\[\sum_{n=1}^{\infty} \frac{n^2(1-a)^n}{2^n} = \frac{6}{a}.\]Find $a.$ | open2math |
Let $f(x)=e^x$ and $g(x)=2x$. Find the sum of all $x$ that satisfy the equation $f^{-1}(g(x))=f(g^{-1}(x))$. | open2math |
If the vertices of an equilateral triangle lie on the hyperbola $xy=1$, then find the area of the triangle. | open2math |
Let $P$ be a point inside triangle $ABC$ such that triangles $ABP$, $BCP$, $CAP$ have areas $A_1$, $A_2$, $A_3$ respectively.
Assume that point $P$ is chosen inside the triangle $ABC$ so that $A_1+A_2=2A_3$.
Let $M$ be the midpoint of segment $BC$.
Find $\frac{PM}{MA}$. | open2math |
The expression $\frac{4k+8}{4}$ simplifies to an expression of the form $ak+b$ where $a$ and $b$ are integers. Find $\frac{a}{b}$ . | open2math |
There are three real numbers $x$ that are not in the domain of $$f(x) = \frac{1}{1+\frac{1}{1+\frac 1x}}.$$ What is the sum of those three numbers? | open2math |
Find the number of distinct integers that can be obtained by adding or subtracting two or more distinct terms of the sequence 2, 5, 8, 11, 14. | open2math |
Find all integer values of $a$ so that the polynomial
\[x^3 + 3x^2 + ax + 7 = 0\]has at least one integer root. Enter all possible values of $a,$ separated by commas. | open2math |
In an isosceles trapezoid that is circumscribed around a circle, the length of the shorter base is 24, and one of the base angles is equal to $\arcsin\left(\frac{3}{5}\right)$. Find the area of the trapezoid. | open2math |
Find the largest positive integer $n$ such that the least common multiple of $n$ and $72$ is less than or equal to $432$. | open2math |
Find the number of positive integers less than 100 that have exactly three divisors. | open2math |
Compute the sum of the products of all possible pairs of the numbers 99, 101, and 103. | open2math |
Determine the value of $x$ such that the inequality
$$\left(\frac{1}{2}\right)^x+\left(\frac{1}{3}\right)^x+\left(\frac{1}{5}\right)^x>1$$
holds true. | open2math |
Compute the number of ways to arrange the letters in the word "COMPUTER" such that the vowels appear together. | open2math |
Find the greatest integer $n$ such that $\sin^n(x) + \cos^n(x) \geq 1$ for all $x$. | open2math |
The units digit of the product $1! \times 2! \times 3! \times \cdots \times 2006!$ is equal to $1$. Determine the last two digits of the factorial of the smallest integer $n$ such that $n!$ is divisible by $1000!$ and $n>1000$. | open2math |
A rectangle is inscribed in a triangle whose sides are 6, 8, and 10. The length of the rectangle's side along the hypotenuse of the triangle is 4. Find the area of the rectangle. | open2math |
Find the number of distinct quadrilaterals that can be formed by connecting four different vertices of a regular octahedron, where two quadrilaterals are considered distinct if they have different sets of vertices. | open2math |
Find the greatest integer that does not exceed the number $\ds\f{n^2+2n+6}{n+1}$, if $n$ is a positive integer. | open2math |
Find the smallest possible value of $x$ such that $3x^2+30x-90=x(x+19)$. | open2math |
What is the smallest positive integer $n$ such that $3n$ is a perfect square and $2n$ is a perfect cube? | open2math |
Let
$p(z)=z^4+az^3+bz^2+cz+d$
be a polynomial of degree four such that its zeros form a geometric progression. Determine the maximum area of the quadrilateral formed by the points which represent the complex zeros of the polynomial $p(z)$. | open2math |
Find the number of 5-digit positive integers that have exactly one pair of adjacent digits that are the same, and the remaining digits are all distinct. | open2math |
Two of the roots of the equation \[ax^3+bx^2+cx+d=0\]are $3$ and $-2.$ Given that $a \neq 0,$ compute $\frac{b+c}{a}.$ | open2math |
There are values $A$ and $B$ such that
\[\frac{Bx-11}{x^2-7x+10}=\frac{A}{x-2}+\frac{3}{x-5}.\]Find $A+B$. | 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 |
The sum of the squares of two consecutive positive integers is 133. What is the sum of their cubes? | open2math |
The least value of x such that x^2 + (4 + i)^2 is a real number. | open2math |
Find the number of integers between $100$ and $200$ that are divisible by $3$ but not divisible by $4$. | open2math |
Palindromes are numbers that read the same backwards and forwards, like 5665. What is the least possible positive four-digit palindrome that is divisible by 3? | open2math |
What is the smallest positive integer $x$ such that $5^x > 3^4 \cdot 2^5$? | open2math |
Let $x$, $y$, and $z$ be distinct real numbers that sum to $0$. Compute \[ \dfrac {x^2y^2+y^2z^2+z^2x^2}{x^4+y^4+z^4}. \] | open2math |
Find the smallest positive integer $n$ such that any $n$-digit integer, when its digits are arranged in reverse order, forms a number that is a multiple of the original number. | open2math |
Let $x$ and $y$ be positive real numbers such that $x^2+y^2=1$. Find the minimum possible value of $xy+\frac{1}{xy}$. | open2math |
Count the number of positive integers $m$ such that there exists an integer $n$, for which $m$ is the smallest positive integer satisfying $2^n + m = 3^k$, where $k$ is a positive integer. | open2math |
Square $ABCD$ has center $O,\ AB=900,\ E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F, m\angle EOF =45^\circ,$ and $EF=400.$ Given that $BF=p+q\sqrt{r},$ where $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$
| open2math |
In a triangle $ABC$, the side lengths are positive integers. The length of side $AB$ is given by the sum of the lengths of sides $AC$ and $BC$. If the perimeter of the triangle is 42, find the maximum possible value of the length of side $AB$. | open2math |
Find all values of $a$ for which the product of the solutions of
\[x^2 + ax + 1 = 0\]equals the sum of the cubes of these solutions. Enter all possible values of $a,$ separated by commas. | open2math |
Let $f(n)$ be the sum of all positive integers $k$ such that $k$ and $n$ are relatively prime, where $1 \leq k \leq n$.
Find the value of
\[\frac{\sum_{d|n} \frac{f(d)}{d}}{\sum_{d|n} f(d)},\]
where the sums are taken over all positive divisors $d$ of $n$. | open2math |
Given that $f(x) = x^2 + bx + c$ is a quadratic function where $c$ can vary, determine the largest possible value of $c$ for which the graph of $f(x)$ still touches the x-axis, given that the vertex of the parabola is at $(-b/2, -b^2/4)$. | open2math |
Find all real solutions to $x^3+(x+1)^3+(x+2)^3=(x+3)^3$.
Problem:
Find all real solutions to $\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}=\sqrt{x+3}$. | open2math |
The Fibonacci sequence is defined recursively by $F(1)=1$, $F(2)=1$, and $F(n)=F(n-1)+F(n-2)$ for $n>2$. Prove that for any positive integer $k$, there exists a positive integer $n$ such that $F(n)$ is divisible by $k$. | open2math |
Let $F$ be a function from the set of all positive integers to the set of all positive integers. Suppose that for all positive integers $a$ and $b$, $F(a^2 + b^2) = F(a)^2 + F(b)^2$, and $F(1) = 1$. Find the number of positive integers $n$ such that $F(n) = n$. | open2math |
Let $m$ be a positive integer such that $m^2+2m+3$ is divisible by $m+5$. Find all possible values of $m$. | open2math |
Find the sum of the squares of two positive integers $x$ and $y$ that satisfy the equation $(x+1)(y+1) = 91$.
(Note: I tried to create a problem that requires a different approach to solve, using a similar type of equation but with a different structure, and asking for a different type of quantity, in this case, the s... | open2math |
The circle centered at $(3,2)$ is tangent to the line $x-y=a$. Find the value of $a$ such that the circle also intersects the line $y=x$ at a single point. | open2math |
Find the smallest positive integer that is both an integer power of 11 and is not a palindrome. | open2math |
An ellipse with semi-axes of lengths $a$ and $b$ is inscribed inside a square with side length $4$. The ellipse is tangent to the square at the midpoints of the sides, and the center of the ellipse is at the center of the square. Find the radius of the circle that passes through the four vertices of the square and is t... | open2math |
Given a positive integer $n$, let $T$ be the set of complex numbers of the form $a + bi,$ where $a$ and $b$ are rational numbers, such that
\[\frac{2}{3} \le a \le \frac{3}{4}.\]Find the largest positive integer $m$ such that for all positive integers $n \le m,$ there exists a complex number $z \in T$ such that $z^n \n... | open2math |
Given that $x$ and $y$ are positive integers, how many different values of $y$ satisfy the equation $x + y = 10$ for all possible values of $x$? | open2math |
Find the number of positive integers $n$ such that $\sqrt{n} + \sqrt{n+1}$ is also an integer. | open2math |
Find all positive integers n such that $n^2+ 1$ has exactly 5 positive factors. | open2math |
Let $f(x)=x^3+x^2+ax+b$ be a cubic polynomial with leading coefficient $1$ such that its roots are the reciprocals of the roots of the polynomial $x^3-6x^2+11x-6$. Find the ordered pair $(a,b)$. | open2math |
Let $S$ be the set of all subsets of the set $\{1,2,\ldots,10\}$ such that each subset in $S$ has either $4$ or $5$ elements. Find the number of subsets in $S$ that contain exactly $3$ even numbers. | open2math |
A sequence of consecutive numbers starting from $n$ is given, and the product of any three consecutive terms is a constant value $k$. Find the value of $k$ such that the sum of the first four terms of the sequence is equal to 24. | open2math |
A circle with radius $r$ is inscribed in a triangle $\triangle ABC$ such that the area of the sectors of the circle enclosed by the sides of the triangle are $3$, $6$, and $12$, respectively. Find the area of $\triangle ABC$. | open2math |
Find the absolute value of the difference between the sum of the squares of the first 10 odd numbers and the sum of the squares of the first 10 even numbers. | open2math |
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x$, $y$, and $r$ with $|x|>|y|$,
\[(x+y)^r=x^r+rx^{r-1}y^1+\frac{r(r-1)}2x^{r-2}y^2+\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\cdots\]What are the first two digits to the right of the decimal point in the decimal representation o... | open2math |
Count the number of integers between 100 and 999 that have exactly two digits equal to 3, and all other digits are prime. | open2math |
Let g(x)=3cos(2πx)+1. Determine the number of values of x on the interval −2≤x≤2 that satisfy the equation g(g(x))=−g(x). | open2math |
Find the number of integers from 1 to 1000 that are divisible by 3, but not divisible by 7. | open2math |
Determine the smallest positive integer $m$ such that all the roots of $z^6 + z^3 + 1 = 0$ are $m^{\text{th}}$ roots of unity. | open2math |
If $a$, $b$, and $c$ are integers such that $a<b<c$ and if the average of the three numbers $5a$, $5b$, and $5c$ is 15, determine the value of $b$. | open2math |
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