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Find the smallest positive integer that is divisible by each of the first ten prime numbers. | open2math |
Let $y$ be a positive real number such that $y + \frac{1}{y} = 121.$ Find\[\sqrt[3]{y} + \frac{1}{\sqrt[3]{y}}.\] | open2math |
What is the maximum number of consecutive positive integers that can be multiplied together before the product exceeds 1000? | open2math |
Let $a\circ b=a^2+b^2$ for all real numbers $a$ and $b$. Find $x$ such that $x\circ 2=x\circ 5$. | open2math |
Let $z$ be a complex number such that $|z|=1$ and $z^3\neq 1.$ Find the real part of $\frac{z^3}{1-z^3}.$ | open2math |
Find the sum of all the positive integer divisors of a number $n$, given that the sum of its prime divisors is 20 and the number of divisors is 10. | open2math |
A circle is inscribed in a square. A smaller square is inscribed in the circle such that its vertices touch the circle and its sides are parallel to the sides of the larger square. If the area of the larger square is 16, find the area of the smaller square. | open2math |
What is the greatest whole number that must be a divisor of the sum of any three consecutive positive integers? | open2math |
Let $M$ be a set of $n$ elements and let $f:M\to M$ be a function such that
$f(x \circ y) = f(x)\circ f(y)$
holds for all $x,y\in M$, where $\circ$ is an associative binary operation.
Find the smallest $n$ such that there exists a function $f$ that is not a constant function and is not equal to the identity function. | open2math |
Consider a binary operation $\otimes$ that takes two positive real numbers $a$ and $b$ and produces a positive real number. The operation $\otimes$ satisfies the following properties for all positive real numbers $a, b, c$:
1. $(ab) \otimes b = a(b \otimes b)$
2. $(a \otimes 1) \otimes a = a \otimes 1$
Given that $1 ... | open2math |
Let $S$ be a subset of $\{1,2,3,...,50\}$ such that no pair of distinct elements in $S$ has a sum divisible by $7$. What is the maximum number of elements in $S$?
$\text{(A) } 6\quad \text{(B) } 7\quad \text{(C) } 14\quad \text{(D) } 22\quad \text{(E) } 23$
| open2math |
Let $p$ and $q$ be any two integers, with $q$ being a multiple of $p$. The smallest possible value of the expression $p^3+q^3$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 12$ | open2math |
Find the area of the triangle formed by the lines $y=2x-3$, $y=-3x+4$, and the $x$-axis. | open2math |
What is the sum of all real numbers that are tripled when subtracted from their squares? | open2math |
Find the number of positive integers less than 1000 that are not divisible by any of the prime numbers 2, 3, or 5. | open2math |
Find the sum of an infinite series whose terms are the reciprocals of the positive integers that can be written as the sum of two perfect cubes. | open2math |
Given a triangle with sides 9, 26, and 13, find the area of a triangle with sides 10, 30, and 15, if the area of the given triangle is $9 \cdot \frac{1}{13} \cdot 26.$ | open2math |
Find the smallest positive integer $x$ such that $3x^2 \equiv 7 \pmod{10}$. | open2math |
In a set of seven distinct integers, how many different triples can be formed such that the sum of the three integers is an even number? | open2math |
Given the points $A(1,0)$, $B(4,5)$, $C(7,2)$, and $D(2,6)$ in the coordinate plane, find the largest possible value of the area of the triangle formed by choosing any three of these points as its vertices. | open2math |
Given a sphere with center (0,0,0) and radius $\sqrt{65}$ and a plane defined by the equation $2x+3y-z=k$, find $k$ so that the plane intersects the sphere exactly at one point. | open2math |
Find all complex roots of the polynomial equation:
\[2x^4 + 5x^3 - 7x^2 + 2x - 3 = 0\]
Given that the sum of the real parts of the complex roots is equal to 1, find the product of all the roots. | open2math |
Find the sum of all integers $k$ such that $\binom{23}{4} + \binom{23}{5} = \binom{24}{k}$. | open2math |
Given that $a$, $b$, and $c$ are positive real numbers such that $abc = 1$ and $a+b+c=3$, find the minimum value of $a^5+b^5+c^5$. | open2math |
Find the greatest integer value of $b$ for which the expression $\frac{9x^3+4x^2+11x+7}{x^2+bx+8}$ has a domain of all real numbers. | open2math |
A sequence $a_n$ is defined for all positive integers $n$ such that:
(i) $a_1 = 1.$
(ii) For any positive integer $n,$ $a_{n+1} - a_n$ is a positive integer.
(iii) $a_{2n} = 2a_n$ for all positive integers $n.$
(iv) $a_{n+2} - a_{n+1} = a_{n+1} - a_n$ for all positive integers $n.$
Find $a_{25}.$ | open2math |
Let $p$ and $q$ be any two even numbers, with $q$ greater than $p$. The largest integer which divides all possible numbers of the form $q^2-p^2$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$ | open2math |
Determine the largest positive integer $n$ such that the number $48n$ can be expressed as the product of three distinct positive integers whose sum is minimized. | open2math |
Find the sum of all the roots of the equation $$x^3 + 7x^2 + 3x - 6 = 0$$, given that the product of the roots is -6. | open2math |
Determine the largest possible three-digit palindrome that is divisible by 11 and has all digits different from each other. | open2math |
Let $f(x)$ be a function such that $f(x) = \sqrt[3]{x}$ for $x \ge 0$ and $f(x) = -\sqrt[3]{-x}$ for $x < 0$. Compute the sum of the roots of the equation $x^2 + f(x) = 7$, given that all of the roots are real. | open2math |
Let $a$ and $b$ be two positive numbers such that $a+b=1$. Find $a^2+2ab+b^2$. | open2math |
Find the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. | open2math |
A certain infinite series is defined by the formula:
$$S = \sum_{n=0}^{\infty} \binom{n+k}{k}x^n$$
where $k$ is a positive integer and $x$ is a real number. Find the value of $k$ if it is known that the series converges to $\frac{1}{(1-x)^{k+1}}$ when $|x|<1$, and the coefficient of $x^2$ in the series is 15. | open2math |
Find the sum of the values of $x$ that satisfy the equation $x^3-4x^2+3x=16$. | open2math |
Given a square $ABCD$, the point $E$ on $BC$ is such that $DE$ is perpendicular to $BC$ and $DE=2$. The point $F$ on $AB$ is such that $BF=1$ and $EF=1$. If $\angle EFD=\theta$, determine $\cos \theta$ in the form $p+q\sqrt{r}$, where $p$, $q$, and $r$ are integers and $r$ is not divisible by the square of any prime. | open2math |
Find all functions $g(n)$ from the set of natural numbers to itself such that
\[g(\gcd(m,n))\cdot g(\text{lcm}(m,n)) = g(m)\cdot g(n)\]
for all natural numbers $m$ and $n$. | open2math |
There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number? | open2math |
What is the probability that when three fair six-sided dice are rolled, the numbers on the first two dice will be equal and the number on the third die will be greater than the number on either of the first two dice? | open2math |
In the decimal representation of positive integers, what is the smallest number of digits that can appear in a positive integer with a digit sum of 2009? | open2math |
On the $x$-axis, 5 points are selected at random in the interval $[-1, 1]$. What is the probability that there will be at least one point on the left side and at least one point on the right side of the origin? | open2math |
If $c$ is a real number and the complex number $a+bi$ is such that $\left|\ds\frac{a+bi}{a-bi}\right|=c$ for all non-zero values of $a$ and $b$, then find the possible values of $c$. | open2math |
An equilateral triangle with sides of length 10 inches is circumscribed around a circle. Prove that the fraction of the area of the circle that lies inside the region enclosed by the triangle is π/12√3. | open2math |
Find the smallest positive integer $n$ such that $n \cdot \lceil \sqrt{n} \rceil = 120$. | open2math |
Let $f(x)=|x^2+x-2|$. Find all the real values of $x$ that make $f(x)$ equal to $|x+2|$. | open2math |
Let $n$ be the smallest positive integer that satisfies the following conditions:
- $n$ is divisible by 12.
- $\frac{n}{12}$ is a perfect square.
- $\frac{n}{15}$ is a perfect cube.
Determine the smallest possible value of $n$ that satisfies all the given conditions and find the sum of its distinct prime factors. | open2math |
Find the smallest positive integer $m$ such that there exist exactly two distinct positive integers $a$ and $b$ satisfying $\frac{1}{3} < \frac{a}{m} < \frac{b}{m} < \frac{2}{5}$. | open2math |
Find the number of positive integers between 11 and 103 that are divisible by 3 but not by 5. | open2math |
Using only the digits 7, 8 and 9, how many positive seven-digit integers can be made that are palindromes? | open2math |
Suppose $x$ and $y$ are integers such that $xy+7x+2y=21$. Find the smallest possible value of $x$. | open2math |
Find the largest possible value of $b$ such that the expression $101_b \times 10_b \div 2_b$ represents a whole number in base $b$. | open2math |
Find the smallest positive integer $k$ such that $2^k \equiv 3 \pmod{11}$, given that $2^5 \equiv 10 \pmod{11}$. | open2math |
Find the sum of all positive integral values of $n$ for which $\frac{n^2+6n+1}{n}$ is a prime number. | open2math |
Given that $a$ is an odd multiple of $7767$, find the greatest common divisor of $6a^2+49a+108$ and $2a+9$.
Another problem inspired by this one:
Find the largest positive integer $n$ such that $n^2+2n+1$ is a perfect square and $n^3+3n^2+3n+1$ is a perfect cube. | open2math |
The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [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 who... | open2math |
Find the largest value of $m$ such that $2x^2 + mx + 128$ can be expressed as the product of two linear factors with integer coefficients. | open2math |
A sequence of integers is formed by writing the sum of any two consecutive numbers starting from $0$ and $1$, then writing the sum of consecutive numbers in the resulting sequence, and so on. What is the probability that the number $11$ will be included in the sequence after $5$ iterations? | open2math |
Let $S$ be the set of positive integers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(\gcd(x,y)) + 2019 \left( x+y \right)\]for all $x,$ $y > 0.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$ | open2math |
Determine the value of $a$ such that the inequality $|2x-a| < |x+1|$ has exactly one solution. | open2math |
Let $x,$ $y,$ $z$ be positive real numbers such that $x+y+z=10.$ Find the minimum value of $\frac{1}{x} + \frac{2}{y} + \frac{4}{z}.$ | open2math |
Find the sum of all positive integers less than 250 that are cubes of perfect cubes. | open2math |
Given a positive real number $m$, determine the values of $x$ and $y$ that satisfy the equation $x^2 + 2xy + my = 0$, such that the quadratic equation has exactly one real root. Express your answer in terms of $m$ and a natural number. | open2math |
Find the sum of all positive integers $n$ such that $n$ is a divisor of $6$ and $6$ is a divisor of $n+3$. | open2math |
Find the value of $x$ such that $3(x+2)-2(x-1)=5(x+1)-7$. | open2math |
Find the value of digit $B$ that will make the number $246B$ divisible by $3$. | open2math |
When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of five consecutive positive integers all of which are nonprime? | open2math |
Find the smallest positive integer \(k\) such that \(2k + 1\) is a multiple of 5. | open2math |
Let $a_n$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Find the least positive integer $m$ such that $a_m = a_{2m}$. | open2math |
Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.
Write another problem inspired by this one:
Find the sum of all positive integers $n$ such that $n^3-3n^2+2n+1$ is a perfect cube. | open2math |
A set of distinct positive integers has the following properties:
The greatest common divisor of any pair of integers in the set is $1$.
For any integer $x$ in the set, there exists another integer $y$ in the set such that the least common multiple of $x$ and $y$ is $60$.
Find all the integers in the set. | open2math |
A right triangular prism with base edges of length $5$ and $12$ and height $13$ contains a sphere of radius $r$ such that the sphere is tangent to the three faces of the prism that meet at a vertex. Find the maximum possible value of $r^3$. | open2math |
Find all values of $k$ such that the expression $\frac{x^2-k^2}{x^2-2kx-3k}$ is undefined for exactly two values of $x$. | open2math |
Determine all values of the parameter $a$ for which the equation
$$\sin x + a \sin 2x = 0$$has exactly two solutions on the interval $[0, 2\pi)$. | open2math |
Find the sum of all integers from 25 through 35, inclusive. | open2math |
Consider $n$ complex numbers $z_1, z_2, \dots, z_n$ such that $|z_i| < 1$ for $i = 1, 2, \dots, n.$ Given that the product of all $z_i$ lies on the unit circle, find the minimum value of $\left| \sum\limits_{i=1}^n z_i \right|.$ | open2math |
A polynomial with integer coefficients has two distinct positive zeros. If the product of the zeros is $60$, what is the smallest possible value of the sum of the coefficients of the polynomial? | open2math |
Larry's sister, Laura, only likes numbers that are palindromes, such as 121 or 12321. How many different numbers with exactly 3 digits are possible that Laura likes? | open2math |
Find the sum of all integer values of $n$ such that $\frac{2^n+1}{2n+1}$ is an integer. | open2math |
Find the positive value of $x$ that satisfies $p^2 = x+4$ given $|p+i|=3$. | open2math |
Find the smallest positive integer $n$ such that the equation $\frac{n}{120+2x} = \frac{3}{4}$ has a solution in positive integers for $x$. | open2math |
Find the sum of all integer divisors of $120$ that are congruent to $2$ modulo $3$. (Recall that the divisors of an integer may be positive or negative.) | open2math |
Find all values of $x$ that satisfy the equation $\frac{x+\sqrt{x+1}}{\sqrt{x^2+1}}=1$. | open2math |
Let $x$ be a positive real number. Find the minimum value of $4x^5 + 5x^{-4}.$ | open2math |
Find the value of $b$ so that the graph of the function $f(x)=x^3-bx^2+3x+1$ is symmetric about the line $x=2$. | open2math |
Find the range of all possible values of $x$ such that $\lceil{x}\rceil+\lfloor{x}\rfloor=2\lceil{x}\rceil\lfloor{x}\rfloor$, where $x>0$. Express your answer using interval notation. | open2math |
Let $S$ be a set of polynomials with real coefficients, such that $p(1)=0$ and for every polynomial $q(x)$ in $S$, there exists a nonnegative integer $n$ such that $q(x)+nx$ is in $S$. Let $m$ be the largest number of elements in a set of this kind. Determine all possible values of $m$. | 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 |
Given the equation of the circle $x^2+y^2=8x-6y-20$, find the equation of the tangent line that passes through the point $(10,2)$. | open2math |
The complex roots of a given equation with real coefficients are given to be $\{z_1, z_2, \ldots, z_{20}\}$. Let $S$ be the sum of all roots, and $T$ be the sum of all distinct products of three of the roots. Determine the real part of $\ds\sqrt{\ds\f{S^2}{T}}$ knowing that all the roots have positive imaginary parts a... | open2math |
Find the values of $a$ and $b$ such that the equation $\frac{x^2+ax+b}{x^2-2ax+b}= \frac{2x+1}{x-1}$ holds for all real values of $x$. What is the value of $a^2+b^2$? | open2math |
For positive integers $n$, let $g(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point in base 2. How many positive integer divisors does $g(2001)$ have? | open2math |
Find the greatest value of $c$ such that the expression $\frac{1}{\sqrt{x^2+4x+c}}$ has a domain of all real numbers. | open2math |
Find the maximum possible value of $a+b+c+d$ where $a$, $b$, $c$, and $d$ are nonnegative integers such that $a+b+c+d=53$ and $a+2b+3c+4d$ is divisible by 6. | open2math |
The sum of the squares of two consecutive positive integers is 85. What are these integers? | open2math |
Let $S$ be the set of positive integers $n$ such that for some positive integer $a$, the equation $x^a-x^n=1$ has an integer solution. Find the smallest positive integer $k$ such that for all integers $n>k$, we have $n\in S$. | open2math |
Find the smallest positive integer $m$ such that $2m \equiv 432 \pmod{27}.$ | open2math |
Consider a cube with a different number written on each of its eight vertices. The sum of the numbers on the vertices of each face is exactly 16. Find the sum of the numbers on the vertices along each space diagonal of the cube. | open2math |
Suppose that $a$ and $b$ are positive integers such that $(a-bi)^2 = 8-6i$. What is $a-bi$? | open2math |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x+y) = g(x) + y \cdot g \left( \frac{1}{y} \right)\]for all real numbers $x$ and $y \neq 0.$
Let $m$ be the number of possible values of $g(1),$ and let $t$ be the product of all possible values of $g(1).$ Find $m + t.$ | open2math |
Find all real values of $x$ such that the sequence $x$, $\sqrt{x}$, and $\frac{1}{\sqrt{x}}$ is an arithmetic progression. | open2math |
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