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In the triangle $ABC$, points $D$ and $E$ are the midpoints of sides $AB$ and $BC$ respectively. The perpendicular bisectors of the segments $DE$ and $AC$ intersect at point $F$. What is the area of the quadrilateral $DFEC$ if the area of the triangle $ABC$ is $12$ square units? | open2math |
Consider a regular dodecagon inscribed in a circle, with consecutive vertices labeled $A, B, C, D, E, F, G, H, I, J, K,$ and $L.$ There is a point $P$ inside the circle so that the area bounded by $\overline{PA}, \overline{PB},$ and the minor arc $\widehat{AB}$ has area $\tfrac{1}{11}$ that of the circle. Suppose also ... | open2math |
Determine the number of ways to paint the faces of a cube with 6 different colors such that no two adjacent faces have the same color. Consider two colorings to be the same if one can be obtained from the other by rotating the cube. | open2math |
From the set of integers $\{1,2,3,\dots,2019\}$, choose $m$ triples $\{x_i,y_i,z_i\}$ with $x_i<y_i<z_i$ so that no two triples have a common element. Suppose that all the sums $x_i+y_i+z_i$ are distinct and less than or equal to $2019$. Find the maximum possible value of $m$. | open2math |
Let $a$ and $b$ be real numbers such that
\[\frac{a + bi}{1 - 2i}\]is a real number. Find $\frac{a}{b}$. | open2math |
Find the sum of all integers from 1 to 100 that are not divisible by either 5 or 8. | open2math |
Determine all values of $a$ such that the inequality
\[ax^2 + 6x + 9 < 0\]
has no real solutions for $x$. | open2math |
Find all integers $n$ such that $n^2-1$ is divisible by 24 and $n^3+5$ is divisible by 27. | open2math |
Find the sum of the first ten terms of an arithmetic sequence, given that the sum of the first three terms is 15 and the sum of the last three terms is 105. | open2math |
What is the largest possible number $n$ that can be written as the sum of $k$ perfect cubes, where $k$ is a positive integer and each cube is less than or equal to $1000$? | open2math |
An integer n is prime if and only if it is not divisible by any prime p with $p^2 \le n$. Find the largest positive integer for which the product of its digits is the number of such primes less than it. | open2math |
Problem:
Ava has two eight-sided dice. One of the dice is fair, and the other die is biased so that it comes up with a 1 with probability $\frac{1}{4}$ and each of the other seven sides has a probability of $\frac{3}{28}$. Ava randomly selects one of the dice and rolls it twice. Given that the first two rolls are both ... | open2math |
Let $a$, $b$, and $c$ be positive integers such that $a$ and $b$ are relatively prime. Determine the value of $x$ that satisfies the congruence $x\equiv a\pmod{b}$ and $x\equiv c\pmod{a+b}$. | open2math |
Find the number of positive integer divisors of $n = 2^4 \cdot 3^2 \cdot 5^1$ that are also multiples of $2^2 \cdot 3^1$. | open2math |
A fair coin is flipped $10$ times, and a sequence of $10$ Heads or Tails is produced. Determine the number of sequences that can be formed, such that there are no two consecutive Tails. | open2math |
In a convex octagon, the measures of the interior angles form a decreasing arithmetic sequence. Determine the number of such sequences that are possible if the octagon is not equiangular and all of the angle degree measures are positive integers greater than 100 degrees. | open2math |
Find the smallest positive integer k, such that sin^(16k + 45) = sin 21. | open2math |
For how many three-digit positive integers is the sum of the digits equal to $5?$ | open2math |
Let $r$ and $s$ be real numbers such that $r+s=3$ and $r^2+s^2=7$. Compute $(rs)^2$. | 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 |
Determine the value of $a$ such that the sum of the first three terms of the geometric sequence with first term $a$ and common ratio $4$ is equal to the sum of the first three terms of the arithmetic sequence with first term $a$ and common difference $9$. | open2math |
Find the largest positive integer $n$ such that $n!$ is a factor of $123456789101112131415161718192021222324252627282930$. | open2math |
Determine the number of ways to color the squares of a 3x3 chessboard with black and white colors such that no two adjacent squares have the same color. | open2math |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(0) = 0$ and
\[g(x+y)+g(xy)=g(x)g(y)+1\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g \left( \frac{\pi}{2} \right),$ and let $t$ be the sum of all possible values of $g \left( \frac{\pi}{2} \right).$ Find $m \times t.$ | open2math |
Given a set of nonnegative real numbers $x_1,x_2,\ldots,x_n$, determine the minimum value of $\frac{x_1^2+x_2^2+\cdots+x_n^2}{x_1x_2+x_2x_3+\cdots+x_{n-1}x_n+x_nx_1}$ for all possible values of $n$. | open2math |
Find the smallest positive integer $n$ such that $3n+7$ leaves a remainder of 5 when divided by 11. | open2math |
Let's consider the set of all integers $n$ such that $n^2 \leq 100i < (n+1)^2$ for some $i$. For example, $T_4$ is the set ${201,202,203,...,400}$. How many of the sets $T_0, T_1, T_2,..., T_{999}$ contain at least two distinct prime numbers? | open2math |
Let $m$ and $n$ be any two even numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2 - n^2$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$ | open2math |
Find the number of positive integers less than 2000 that are neither 11-nice nor 12-nice. | open2math |
Given the equation of a circle $(x-2)^2 + (y-3)^2 = 4$, find the sum of the $x$-coordinates of the points where the circle intersects the $x$-axis and the sum of the $y$-coordinates of the points where the circle intersects the $y$-axis. | open2math |
Find the number of positive integers less than or equal to $1000$ that can be written as the product of two prime numbers. | open2math |
Given $x,y,z$ are non-negative real numbers such that $xyz = 25$, let $M = \max\{x,y,z\}$. Find the smallest possible value of $M$ such that $x^2 + y^2 + z^2 = 50$. | open2math |
Find the number of sequences of real numbers $a_1, a_2, a_3,...$ such that for any positive integer $n$, the product of the first $n$ terms of the sequence is equal to the sum of the first $n$ terms of the sequence. | open2math |
Determine the number of positive integers whose logarithm is an integer.
In other words, find how many positive integers $n$ have the property that $\log_{10} n$ is an integer.
(Remember, this problem should require a different approach to solve than the original one.) | open2math |
Find the number of different real numbers $x$ with the properties that $0<x<\pi$ and $\sin(x^{12}) - \sin(x^6)$ is a rational number. | open2math |
What is the largest positive integer $n$ such that $5^{2n+1}$ is a divisor of $5^{35} + 5^{28}$? | open2math |
How many positive whole numbers have square roots that are between $5$ and $15$, inclusive? | open2math |
What is the greatest two-digit positive integer that has exactly three positive divisors? | open2math |
Find the smallest positive integer that has exactly 5 distinct prime factors. | open2math |
Find the value of n so that the area of the triangle formed by the points (0, 0), (1, 1), and (n, 2) is half the perimeter of the triangle. | open2math |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) + y) = g(x) + g(g(y) + g(-x)) - x^2\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(0),$ and let $t$ be the sum of all possible values of $g(0).$ Find $m \times t.$ | open2math |
Let $S_n$ be the sum of all the terms of the form $\frac{1}{n^2}$ where $n$ is not a multiple of 4, and let $T_n$ be the sum of all the terms of the form $\frac{1}{n^2}$ where $n$ is a multiple of 4.
Determine $\frac{S_n}{T_n}$. | open2math |
Given that $y$ is real and $y^4+\frac{1}{y^4}=83$, find $y^2+\frac{1}{y^2}$. | open2math |
Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than $30$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$. | open2math |
Find $w$, such that $5^65^w=25$. | open2math |
Given the circles $\omega_1$, $\omega_2$, and $\omega_3$ that are mutually tangent to each other and have radii of $2$, $3$, and $6$ units, respectively, find the area of the triangle formed by connecting the centers of these circles. Express the answer in the form $\frac{1}{2}m \sqrt{n}$, where $m$ and $n$ are positiv... | open2math |
A triangle is formed by the points $A(2,0)$, $B(0,4)$, and $C(0,0)$. A line with a slope of 3 intersects the positive $x$-axis at $D$ and the positive $y$-axis at $E$. The lines intersect at $F(1,3)$. What is the area of the shaded triangle $AFC$? | open2math |
Let $x$ and $y$ be positive real numbers such that $\ds \frac{x}{y}+\frac{y}{x}=17$ and $x+y=14$. Find the value of $xy$. | open2math |
In the six-digit integer $4A7,\!255$, what is the smallest digit $A$ so that the sum of the squares of its digits is divisible by 5? | open2math |
The sides of a triangle are in arithmetic progression. The area of the triangle is equal to the perimeter. Determine the number of such triangles with integer side lengths. | 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 largest value of $x$ that satisfies the equation $\sqrt{2x}=4x$? Express your answer in simplest fractional form.
New Problem:
Determine the smallest positive integer $n$ such that the equation $\sqrt{n+x}=\frac{1}{3}x^2$ has a real solution. | open2math |
Find the smallest positive integer $n$ such that $\frac{1}{n}$ is a repeating decimal and $n$ contains the digit 7. | open2math |
Find the sum of all integer solutions to the inequality $|x| < |x^2-1| < 8$. | open2math |
Find all $x$ such that $x^2+5x<6$. Express your answer in interval notation. | open2math |
Let $f(x)$ be a function defined on the set of positive integers, such that $f(1) = 1$, $f(2) = 2$, and for $n \geq 3$, $f(n)$ is the sum of the products of all possible pairs of positive integers that add up to $n$. For how many functions $g(x)$ does there exist a function $h(x)$ such that $f(g(x)) = f(x) \cdot h(x)$ ... | open2math |
Find the sum of all positive integers less than 20 that have exactly two distinct prime factors. | open2math |
Find all values of $x$ that satisfy the equation $|x-2|=3x-1$. Express your answers in simplest fractional form. | open2math |
What is the sum of all the numbers in the sequence
$$1-2+3-4+5-6+7-8+\dots+199-200$$
and the sum of the squares of these numbers?
Can you determine the final result? | open2math |
A set of five numbers has both a mean and median equal to 10. If the largest number in the set is 20, and the sum of the two smallest numbers is 10, what is the sum of the remaining two numbers? | open2math |
Find the smallest positive integer $n$ such that the graph of the quadratic function $f(x) = x^2 + 4x + n$ does not intersect the x-axis. | open2math |
Find the maximum value of
\[\frac{|a+b|+|b+c|+|c+d|+|d+a|}{|a|+|b|+|c|+|d|}\]
where $a,b,c,d$ are non-zero real numbers and $a+b+c+d=0$. | open2math |
Given the points $(-5,-4),(-3,-2),(0,1)$ and $(6,3)$ are on the graph of the parabola $y=ax^2+bx+c$.
Determine the $x$-coordinate of the vertex of this parabola. | open2math |
Let $f(n)$ denote the number of zeros at the end of the decimal representation of $n!$. Find the largest $n$ such that $f(n)=f(2n)$. | open2math |
The sum of the cubes of two positive integers is 945. The difference of the two integers is 3. What is the sum of the two integers? | open2math |
Find the sum of all possible values of $c$ for which the polynomial $x^3+cx^2+2008x+c$ can be factored into the product of three linear factors with integer coefficients. | open2math |
Find the number of positive integers $n$ such that $2^n+1$ is a perfect square and $2^n-1$ is a prime number. | open2math |
Let's say the roots of the quadratic equation $ax^2+bx+c=0$ are $x_1$ and $x_2$.
Determine $a$ such that the product of $x_1$ and $x_2$ is equal to the sum of the coefficients of the quadratic equation. | open2math |
Prove that for any angle $x$, the following inequality holds:
\[\frac{\sin^2 x}{\cos x} + \frac{\cos^2 x}{\sin x} \geq 1.\] | open2math |
Find the sum of all positive divisors of 36 that are also divisors of 24. | open2math |
Find a positive integer that is divisible by 25 and whose square root is a number between 10 and 10.2. | open2math |
In a triangle $ABC$, the angles satisfy the condition that the sum of any two angles is $k$ times the third angle. Find the smallest possible value of $k$, given that $k$ is an integer. | open2math |
Let $g$ be a function that maps the set of all positive integers to the set of all positive integers such that $g(ab)+g(a+b)=a(g(b)+1)+b(g(a)+1)$ for all positive integers $a$ and $b$. Find the least possible value of $g(10)$ and the greatest possible value of $g(11)$. | open2math |
Find all values of $n$ for which $\binom{12}{4}+\binom{12}{n}=\binom{13}{5}$ and $\binom{12}{5}+\binom{12}{n}=\binom{13}{6}$ simultaneously. | open2math |
Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| open2math |
If $p$ and $q$ are positive integers such that $p$ has exactly $3$ divisors, and $q$ has exactly $p$ divisors, then what is the greatest possible value of $\gcd(p,q)$? | open2math |
Find the sum of all solutions of the equation $\ds x^{\log_3x} = \frac{1}{3}$. | open2math |
Let $a$ and $b$ be two distinct positive integers, and let $n$ be a positive integer such that
$$(a-b)^n = a^n - a^{n-1}b + a^{n-2}b^2 - \cdots + ab^{n-1} - b^n.$$Determine the value of $a+b$. | open2math |
In a right-angled triangle, the center of the inscribed circle is at (2,3). One vertex of the triangle is at (0,0), while the other two vertices lie on the line y=3. Find the length of the hypotenuse. | open2math |
A square and a regular hexagon are inscribed in the same circle. The side length of the square is $12$ cm. Find the area of the hexagon. | open2math |
Find the smallest positive integer $n$ such that $n$, $n+1$, and $n+2$ are each divisible by a distinct prime number. | open2math |
Find the sum of all the terms of the form $\frac{1}{2^n}$, where $n$ is a positive integer and $n$ is not a multiple of 3. Express the answer as a common fraction. | open2math |
Determine the value of $b$ such that the quadratic expression $2x^2+bx+8$ can be factored into the product of two binomials with integer coefficients. | 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 |
Find the positive integer $k$ such that the $k$th decimal digit after the decimal point in the decimal representation of $\frac{7}{11}$ coincides with the $k$th digit after the decimal point in the decimal representation of $\frac{13}{17}$. | open2math |
Determine the greatest integer value of $x$ such that $|x-1|+2|x+3|<13$. | open2math |
If the point $(4,8)$ is on the graph of $y=f(x)$, and $k(x)=(f(x))^3$ for all $x$, then there is one point that must be on the graph of $y=k(x)$. What is the sum of the coordinates of that point? | open2math |
Let $S_n$ be the set of all positive integers that can be written as the sum of exactly $n$ distinct perfect squares. Let $f(n)$ be the minimum value that the greatest common divisor of any two elements of $S_n$ can take. What is the maximum possible value that $f(n)$ can take on? | open2math |
Find the number of positive integers $x$ such that $x^3-x^2+2x-1$ is a perfect square.
[Finish] | open2math |
Find the smallest positive integer $n$ such that the sum of $n^3$ and $n^2$ is a perfect square. | open2math |
Let $a$, $b$ and $c$ be three different prime numbers. Find the number of positive divisors of the number $a^2b^3c^4$ that are not divisible by the product $abc$. | open2math |
Determine the largest positive integer $n$ such that the inequality
\[|\sin x|^n + |\cos x|^n \ge \frac{2}{n}\]holds for all real numbers $x$. | open2math |
If $x$, $y$ are nonzero real numbers such that $x^2+y^2=6xy$, find the value of $\left|\frac{x+y}{x-y}\right|$. | open2math |
Consider a sequence of real numbers $\{b_k\}$ such that $b_1=1$ and $b_{m+n}=b_m \cdot b_n + 2^{m+n},$ for all positive integers $m$ and $n.$ Find $b_{10}.$ | open2math |
In triangle $ABC$, $D$ and $E$ are points on $\overline{BC}$ such that $BD:DE:EC=1:2:3$. Find the ratio of the area of triangle $ADE$ to the area of triangle $ABC$. | open2math |
Find the number of three-digit positive integers with exactly two 7s as digits. | open2math |
Let $a$ and $b$ be complex numbers such that $|a+bi|=3\sqrt{2}$ and $|a-bi|=5\sqrt{2}$. Find the value of $|a^2-b^2i|$. | open2math |
Find the area of a triangle inscribed in a circle of radius 5, where two of the sides of the triangle are tangent to the circle and the third side is a diameter of the circle. | open2math |
Find the sum of all positive integral values of $n$ for which $\frac{n+3}{n}$ is an integer. | open2math |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=10$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | open2math |
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