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Find all values of $x$ such that $\log_3(\sqrt{x})=\frac{1}{2}(\log_3(x)+\log_3(5))$. | open2math |
Let $f(x)$ be a function such that $f(2x) = f(x) + 1$ for all positive integers $x$. If $f(1) = 3$, find the value of $f(32)$. | open2math |
Let $p$ and $q$ be real numbers such that $p^2 + q^2 = 4.$ Determine the set of all possible values of $pq(p + q).$ | open2math |
Find the sum of all possible values of $x$ that satisfy the equation $|x-2|+|x-5|=3$. | open2math |
Find the smallest four-digit integer whose digits are distinct and form an arithmetic sequence, with the additional constraint that the sum of the digits is a perfect square. | open2math |
Let $f(n)$ be the sum of the proper positive integer divisors of $n$. For how many values of $n$ is $f(n)$ a perfect square, given that $2 \le n \le 100$? | open2math |
The region $R$ is defined by the set of all points $(x,y,z)$ such that $|x+y| \le 2,$ $|x+z| \le 3,$ and $|y+z| \le 4.$ Find the volume of $R.$ | open2math |
How many of the positive integers can be expressed as the sum of two squares of integers? | open2math |
A deck of 52 cards is shuffled and 5 cards are drawn. It is given that there is at least one pair of cards of the same rank, but no three cards of the same rank. If the cards that are not part of the pair are replaced with new cards from the remaining deck, what is the probability that at least three of the five cards ... | open2math |
Suppose the parabola defined by the equation $y=x^2+bx+c$ has vertex $(h,k)$ where $h$ and $k$ are integers. Find the number of possible values of $b$ for which there are three points on the parabola where both coordinates are integers. | open2math |
Find the smallest positive integer n for which there exists a polynomial f(x) with integer coefficients and degree 3 such that f(n) = n and f(x) ≠ x for any integer x ≠ n. | open2math |
Count the number of permutations of the numbers 1 through 5 such that the sum of any two consecutive numbers is odd. | open2math |
Compute the product of all possible values of $x$ such that $\sqrt{x+1} + \sqrt{x-1} = \sqrt{4x+1}$. | open2math |
The sum of the cubes of two numbers is 126, and the product of the two numbers is 10. What is the value of the sum of the two numbers? | open2math |
What is the positive integer $N$ for which $$ 22^2 \times 55^2 = 10^2 \times N^2 \ ? $$ | open2math |
How many positive integers less than 60 have an even number of positive divisors? | open2math |
Let $x$ be a positive real number. Find the maximum value of $x^3 + \sin \left( \frac{\pi}{x^3} \right)$. | open2math |
Determine the smallest positive integer $m$ such that when the decimal representation of $m$ is reversed, the resulting number is a multiple of 24. Compute the ratio of $m$ to its reverse. | open2math |
Find the sum of all positive integers n such that 109 is a divisor of 53^n + 1. | open2math |
Determine $x$ such that $\sqrt[4]{(\sqrt{x})^3}=18$. | open2math |
Let \( y \) be a positive real number. Find the largest possible value of \( n \) such that for any positive integer \( k \leq n \), the following inequality holds:
\[ k \cdot \lceil y \rceil - \lceil k \cdot y \rceil \geq 2.\] | open2math |
Given that $a$ is an integer such that $a^3 \equiv 17 \pmod{101}$, find $a^6 \pmod{103}$. | open2math |
Find the number of ways to color the vertices of an icosahedron with three colors so that every face contains exactly one vertex of each color. | open2math |
In the base $16$ representation of the number $\sqrt {n}$, where $n$ is an integer such that $256 < n < 512$, the rightmost digit is $3$.
Determine $n$. | open2math |
What is the smallest whole number that must be a divisor of the sum of any three consecutive positive integers? | open2math |
Find the product of all real values of $r$ for which $\frac{1}{2x}=\frac{r-x}{7}$ has exactly one real solution. | open2math |
Let $a_n$ be the product of all positive integer divisors of $n$. Determine all values of $n$ for which $a_n = n + 1$. | open2math |
Let $h$ be a tenuous function defined on the set of positive integers, and let $h(n)$ be the minimum value that satisfies the inequality $h(x) + h(y) > y^2$ for all positive integers $x$ and $y.$ Suppose $s$ is a positive integer greater than 2, and let $H$ be the sum of $h(1) + h(2) + \dots + h(s).$ Prove that $H$ i... | open2math |
What is the largest positive integer $n$ such that $2^n$ is a factor of $100!$? | open2math |
Find the number of positive integers $n$ less than or equal to $2018$ such that $6n + 1$ is not relatively prime to $42$. | open2math |
In the Cartesian plane, a circle has a radius of 4 units and its center at (2, 1). The sum of the y-coordinates of the intersection points of the circle and the line y = x^2 is 1.5. Find the sum of the x-coordinates of the intersection points. | open2math |
Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\sin x + \cos x = \sqrt{2}.$ Enter all the solutions, separated by commas. | open2math |
Find the sum of all even numbers between 11 and 109, inclusive, that are not divisible by 3. | open2math |
The product of two positive integers plus their sum is 103. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | open2math |
Find the sum of the coordinates of the midpoint of the segment with endpoints $(4, 8)$ and $(-2, 3)$. | open2math |
Consider the function $g(x)=\{x^3\}$. Find the largest positive integer $m$ such that the equation $m^2g(g(g(x)))=x^2$ has exactly 5 distinct real solutions.
Note: The function $g(x)$ takes the fractional part of $x^3$, not $x$. | open2math |
Let \[x=\sqrt[3]{\sqrt{\frac{1}{256}}}\]. Find the smallest positive integer $n$ such that $x^{6n}$ is a perfect square. | open2math |
Two different fair dice are rolled simultaneously. What is the probability that the sum of the numbers on the two dice is 7, given that one of the dice shows an even number? | open2math |
Given the function $f(x)=3x^3+2$, find the value of $x$ so that $f^{-1}(x)=4$. | open2math |
Six congruent circles are drawn such that each circle is tangent to two sides of a regular hexagon. The circles have radius 1 and the area of their union is $m/n\pi,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ | open2math |
Find the smallest positive integer $n$ such that the equation
$$\sin^2 \left( \frac{k\pi}{n} \right) = \frac{1}{2}$$
has infinitely many solutions for $k$. | open2math |
Find the number of three-digit numbers that are divisible by 5 and have a tens digit that is twice the units digit. | open2math |
Find the positive integer $n$ such that
\[\tan \left( \frac{\pi}{2n} \right) = \sqrt{n} - 1.\] | open2math |
A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls is a multiple of 3, given that at least one of the rolls is a 6? | open2math |
In a certain triangle, the length of the hypotenuse is equal to the sum of the lengths of the other two sides, and the perimeter is 60. If one of the sides is half as long as the other, find the lengths of the sides of the triangle. | open2math |
Compute the sum of the roots of unity $e^{\frac{2\pi ki}{n}}$ for $k=0,1,2,\ldots,n-1$, where $n$ is a positive integer greater than 1. | open2math |
Consider a triangle with sides $a$, $b$, and $c$, where $a\ge b\ge c$. Let $\alpha$ be the maximum possible value of $\frac {a}{c}$ for which the triangle has a cevian (a line segment connecting a vertex of a triangle with the opposite side) of length $d$, such that $d^2$ is equal to the sum of the squares of the dista... | open2math |
In the equation $x^2 - kx + 1 = 0$, determine $k$ such that the product of the roots is equal to the sum of the roots. | open2math |
Twelve books are arranged on a shelf. Find the number of subsets of these books that contain at least two adjacent books. | open2math |
Find the number of non-negative integer solutions to the equation $x+y+z=11$ such that $x+y$, $y+z$, and $z+x$ are all even. | open2math |
Let $a$ and $b$ be positive integers such that $a+b=91$. Determine the maximum value of $\gcd(a,b)$. | open2math |
Given a rectangular prism with dimensions $a$, $b$, and $c$, where $a
Find the point $P$ on the surface of the prism that minimizes the sum of the distances from $P$ to the four vertices of the base of the prism. | open2math |
Find the smallest, positive five-digit multiple of 24 that has exactly two distinct digits. | open2math |
Find the largest integer value of $n$ for which the decimal representation of $\frac{1}{n}$ repeats with a cycle length of exactly 4. | open2math |
Let $n$ be a positive number such that when $237$ is divided by $n$, the remainder is $7$. Compute the sum of all possible two-digit values of $n$. | open2math |
Find the number of positive 4-digit numbers whose digits form a strictly increasing sequence. | open2math |
The sum of the lengths of all the edges of a cube is 60 cm. Find the number of cubic centimeters in the volume of the cube. | open2math |
Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-6x+5|$ and $y=\frac{29}{4}-x$. | open2math |
Let $F$ be the matrix defined by $$F=\begin{pmatrix}0&1&1\\ 1&1&0\\ 1&0&1\end{pmatrix}.$$ Prove that for all positive integers $n$, the matrix $F^n$ has all its entries positive. | open2math |
Given that
\begin{eqnarray*}&(1)& x\text{ and }y\text{ are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& y\text{ is the number formed by reversing the digits of }x\text{; and}\\ &(3)& z=|x-y|. \end{eqnarray*}
How many distinct values of $z$ are possible?
| open2math |
A fair six-sided die has faces labeled with distinct positive integers. Exactly four of the six faces have prime numbers. If the remaining two faces are filled with the positive integers $a$ and $b$, what is the probability that the product of the numbers on the top face and bottom face of the die is composite? | open2math |
The sum of the first three terms of an arithmetic sequence is 15, and the sum of the squares of the same terms is 93. Find the sum of the squares of the next three terms of this sequence. | open2math |
Three numbers are independently chosen from the set of positive integers less than or equal to 7. What is the probability that the sum of the three numbers is less than their product? Express your answer as a common fraction. | open2math |
Find the number of positive integers $n$ less than 100 such that the roots of the equation $x^2-nx+n^2=0$ are both non-real and satisfy the condition $|x_1-x_2|=\sqrt{2}$. | open2math |
Consider two continuous functions $h(x)$ and $p(x)$ defined on the interval $[c,d]$ with $c<d$. If the minimum value of $h$ occurs at $x=d$, the maximum value of $p$ occurs at $x=c$, and $h(c)-p(d)=6$, find the largest possible value of the maximum of the function $h(x)-p(x)$. | 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 |
What is the smallest positive integer $a$ such that for any positive integer $n$, $\frac{n^2 + a}{n + 1}$ is always an integer? | open2math |
Find the number of 5-digit palindromes with exactly two non-zero digits, where the non-zero digits must be different. | open2math |
Compute $\sqrt{(31)(30)(29)(28)+1}.$
Another problem inspired by this one:
A polynomial $P(x)$ has the property that $P(x)P(x+1) = x^4 + 1$ for all real numbers $x$. Find $P(2)$. | open2math |
Find the number of positive integers less than or equal to $n$ that are not divisible by any of the numbers $a$, $b$, and $c$, where $n=840$, $a=7$, $b=15$, and $c=20$. | open2math |
Determine all real numbers x that satisfy the equation
$${{\left( \frac{1-x}{1+x} \right)}^{2017}}+{{\left( \frac{1+x}{1-x} \right)}^{2017}}=2.$$ | open2math |
The positive integer $N$ has the property that $N^3$ has exactly 20 divisors and $N^2$ has exactly 15 divisors. How many positive integers are divisors of $N$ but are not divisors of $N^3$? | open2math |
In a regular pentagon, the length of each side is $x$ units. A segment of length $y$ units is drawn from one vertex of the pentagon to the midpoint of the opposite side. Find the sum of the lengths of the two diagonals that can be drawn from the endpoint of the segment to the other vertices of the pentagon. | open2math |
A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that for each complex number $z$, $f(z)$ is equidistant from both $z$ and the origin. Given that $|a+bi|=8$, find $b^2.$ | open2math |
Let $g(x)$ be a function that, given a positive real number $x$, returns the largest possible real number $p$ such that the polynomial $p(x)$ has exactly three distinct real roots, where $p(x) = x^3 + ax^2 + bx + c$ for some constants $a, b, c$. Given that $x$ is a root of the polynomial $q(x) = x^4 - 3x^2 + 2$, what i... | open2math |
The product of two positive integers is 50, and the sum of their reciprocals is 1/5. What is the sum of the two integers? | open2math |
Let $a$, $b$ and $c$ be three positive integers such that $a^b=b^c$ and $a$ and $b$ are relatively prime.
Show that $a=b^x$ and $b=a^y$ for some integers $x$ and $y$.
Determine all possible pairs $(a,b)$ satisfying the above equation. | open2math |
Given that $f(x)=\dfrac{x^2+1}{x-4}$, find the value of $x$ such that $f(x)=3$. | open2math |
Find the value of $x$ that satisfies the equation:
$$\sqrt[3]{x^2+2x+1} \cdot \sqrt[4]{x^3+3x^2+3x+1} = x^2+2x+1$$ | open2math |
In a triangle, $\angle A = 30^\circ$ and $\angle B = 60^\circ$. Find the measure of $\angle C$ if it is known that the sum of the measures of the three angles of the triangle is $180^\circ$. | open2math |
How many positive three-digit integers less than 500 have at least two digits that are the same? | open2math |
The sum of the positive divisors of a positive integer of the form $2^i3^j$ is equal to $600$. What is $i + j$? | open2math |
Let $m$ be a positive integer such that $3m^2-5m+1$ is divisible by a perfect square $k^2$, where $k>1$. Find the largest possible value of $k$. | open2math |
Let $P(x)$ be a polynomial such that $P(x^2+1)=x^4+4x^2$. Find $P(12)$. | open2math |
Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$ Find the maximum value of $xy.$ | open2math |
What is the positive integer $N$ for which $$ 22^2 \times 55^2 = 10^2 \times N^2 \? $$
Another problem inspired by this one:
Find the integer $N$ such that $$ 22^2 \times 55^2 = 10^2 \times N^2 \ \text{and} \ 11^2 \times 44^2 = 5^2 \times N^2.$$ | open2math |
Let $f(x)$ be a polynomial with rational coefficients such that $f(1) = 1$ and $f(2) = 4.$ Find the largest possible number of rational roots of the equation $f(x) = x^2.$ | open2math |
Find the area of the region inside the graph of the equation $x^2 + y^2 = 1$ that is outside the region of the isosceles triangle with vertices $(0,0)$, $(1,0)$, and $\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. Express your answer in terms of $\pi$. | open2math |
Determine the value of $a$ such that the circle with equation $(x-a)^2+(y+1)^2=1$ passes through the point $(3,-2)$. | open2math |
Find the value of $x$ in the equation $\sqrt{2x+5}=\frac{1}{\sqrt{x-1}}$, and then find the value of $2^x$. | open2math |
Consider a quadrilateral $ABCD$ with angles $A, B, C,$ and $D$ of measures $a, b, c,$ and $d$ respectively. Assume that angles $PAB, PBC,$ and $PCA$ are all congruent, and angle $PDC$ is twice the measure of angle $PAB$. Find the minimum perimeter of $ABCD$ that allows such configuration, given that the side lengths a... | open2math |
Let $g(n)$ be the product of the digits of $\frac{1}{7^{{}^n}}$ that are to the left of the decimal point. What is the smallest positive integer $n$ such that $g(n)$ is a perfect square? | open2math |
Let $a$ and $b$ be two positive real numbers such that the expression $\frac{\sqrt[3]{a}+\sqrt[3]{b}}{\sqrt[3]{a\cdot b}}$ is as large as possible. Find the product $a\cdot b$ if $\frac{a}{b}=\frac{27}{64}$. | open2math |
Count the number of positive integers with four digits that can be expressed in two different ways as the sum of two perfect squares, using distinct squares. | open2math |
What is the maximum number of consecutive positive integers that can be added together before the sum exceeds 800? | open2math |
Let $A$ be a $2 \times 2$ matrix such that
\[\det (A^n) = \det(A)^n = (-2)^n,\]for every positive integer $n.$ If the sum of the squares of the eigenvalues of $A$ is equal to $3,$ compute $\det(A).$ | open2math |
Consider two positive integers $a$ and $b$ such that the sum of their squares is 50 and the sum of their cubes is 200. How many different values could be the product of $a$ and $b$? | open2math |
Find the number of divisors of $555$ that are not divisible by $5$. | open2math |
Find the number of three-digit positive integers that have exactly four distinct positive divisors. | open2math |
Determine all real values of $x$ such that $x^3 + 2x^2 + x + 1$ is divisible by $x^2 + 3x + 2$. | open2math |
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