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A triangle with sides $a$, $b$, and $c$ is inscribed in a circle. The angles opposite sides $a$ and $b$ are bisected, and the points where the bisectors intersect the circle are connected to form a smaller triangle. If the area of the larger triangle is 40, find the area of the smaller triangle. | open2math |
Find the smallest positive integer $n$ such that $n^2$ is a palindrome and contains exactly three digits. | open2math |
Find the sum of all real solutions to the equation \[\frac{x-2}{x^2+4x+1} = \frac{x-5}{x^2-10x}.\] | open2math |
Let $a$ be a real number such that $a > 1$ and
\[\sum_{n=1}^{\infty} \frac{n^2}{a^n} = \frac{18}{5}.\]Find $a$. | open2math |
Evaluate the product \[ (a-10) \cdot (a-9) \cdot \dotsm \cdot (a-1) \cdot a, \] where $a=2$. | open2math |
A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Determine the possible values for the other two zeros.
Answer: The other two zeros must be complex conjugates, and their product must be a rational number. This is because the polynomial has ... | open2math |
Let $n$ be the least common multiple of the positive integers from 1 to 15. Find the largest number $m$ such that $m^2$ divides $n$. | open2math |
Let $T$ be the set of all positive real numbers. Let $g : T \to T$ be a function such that
\[g(x) + g(y) = g(x^y + y^x)\]for all $x,$ $y \in T$. Find the largest possible value of $g(3)$. | open2math |
Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \[\begin{aligned} xy+x+y&=71 \\ x^2y+xy^2 &= 880.\end{aligned}\] | open2math |
The function $g(n)$ is defined on the positive integers such that $g(g(n)) = 2n + 1$ and $g(2n + 2) = 4n + 1$ for all positive integers $n.$ Find $g(250).$ | open2math |
Find the sum of all numbers $x$ satisfying $x+25/x = 10.$ | open2math |
Find the smallest possible value of $n$ for which the sum of $n$ consecutive positive integers is a perfect square, and the largest term in the sum is less than $2n$. | open2math |
For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 50$.
(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? | open2math |
Find the number of integer solutions to the inequality $|2x-5| \geq |x+1|$ that satisfy the condition $-10 \leq x \leq 10$. | open2math |
Find the number of ordered pairs of non-negative integers $(a,b)$ such that $a^3 + b^2 < 100$ and $a < b$. | 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 |
Determine the value of $k$ such that the polynomial $x^3-kx^2-3x+8$ is divisible by $x+2$. | open2math |
A sequence is defined by $b_n = 2^n + 3^n$. Find the smallest positive integer $n$ such that $b_n$ is divisible by $13$. | open2math |
Find the maximum value of
\[\frac{x^2 + 11}{\sqrt{x^2 + 5}}\]over all real numbers $x.$ | open2math |
Find the maximum value of $|(z^2-4)^2(z^2+4)|$, given that $z$ is a complex number with $|z|=\sqrt{3}$. | open2math |
Let $P(x)$ be a polynomial of degree 6 such that
\[P(x) = P(2016-x).\]
Find the maximum number of real roots that $P(x) - 1$ can have. | open2math |
How many different $4\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?
Problem:
How many different $4\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each... | open2math |
Let $a,$ $b,$ $c$ be positive real numbers such that
\[a^2b^2 + b^2c^2 + c^2a^2 = a^2b^2c^2.\]
Find the maximum value of $a + b + c.$ | open2math |
Find the number of ordered pairs of positive integers $(a, b)$ such that the equation $ax^3 + bx^2 + 1 = 0$ has at least one rational root. | open2math |
Let $a$ and $b$ be positive real numbers such that $a + b = 2.$ Find set of all possible values of $\frac{1}{a} + \frac{1}{b}.$ | open2math |
Let $a$ be a real number such that
\[a^3 + |a|^3 = 28.\]Find $a$. | open2math |
Find the number of positive integers $n \le 500$ such that $24n$ is a perfect cube. | open2math |
Find the largest positive integer $n$ such that $n^2$ is divisible by 72, $n^3$ is divisible by 168, and $n^4$ is divisible by 400. | open2math |
The roots $x_1$ and $x_2$ of the equation
$$x^2+2x+a^2=0$$
are such that
$x_1^2+x_2^2=12$.
Determine $a$. | open2math |
Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2=1$. What is the minimum value of the expression \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+a+b+c?\] | open2math |
A quadratic equation in the form $x^2+bx+c=0$ has one root that is twice the other root. Find the sum of all possible values of $b$. | open2math |
In the equation $x^3 + ax^2 + bx + c = 0$, the sum of the roots is $-2$, the sum of the pairwise products of the roots is $5$, and the product of the roots is $-6$. Find the sum of the cubes of the roots. | open2math |
Find the largest positive real number $c$ such that for all real numbers $a$ and $b$,
$$|a^3 + b^3| + |a^2b + ab^2| + |ab(a + b)| \ge c|a + b|^3.$$ | open2math |
Determine the number of integers between 10 and 100 that have at least one digit equal to 5. | open2math |
Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}.$ Find the number of possible values of $S.$
| open2math |
Find the volume of the tetrahedron whose vertices are $(0,0,0)$, $(1,0,0)$, $(0,2,0)$ and $(x,y,z)$, where $x,y,z$ are positive real numbers and the point $(x,y,z)$ lies on the surface defined by the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. The numbers $a,b,c$ are positive and are given by $a=1, b=... | open2math |
Determine the real values of $m$ for which the cubic equation $$x^3-mx^2-mx+m^2=0$$ has only one positive root. | open2math |
Let $x$ be a positive real number such that $\lfloor x\rfloor+\lceil 1/x\rceil=11$. How many possible values of $x$ are there? | open2math |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xy) + f(xz) - f(x) f(yz) \ge 1\]for all real numbers $x,$ $y,$ and $z.$ | open2math |
Let $f(x)$ be a function defined on the set of positive integers, such that $f(x)$ is the sum of the digits of $x$ when $x$ is a perfect square, and $f(x) = f(x - 1) + 2$ otherwise. Find the maximum value of $f(x)$ for $1 \le x \le 100$. | open2math |
Find all values of x such that x^2 + 3x + 2 = 0 and the sum of the roots is equal to -3. | open2math |
Three players roll a standard dice in succession. Find the probability that at least one of the players will roll a number that is greater than the number obtained by the previous player. | open2math |
In the five-digit numeral $ABCBA$, $A$ is a prime number and the sum of all digits is a multiple of 4. Determine the largest possible value of $A$ that makes the number $ABCBA$ a multiple of 5. | open2math |
What is the smallest positive integer n such that the sum of the elements in the nth row of Pascal's triangle is a perfect square? | open2math |
Let $a_n$ be the smallest positive integer such that $a_n!$ is divisible by $2^n$.
Find the largest value of $n$ for which $a_n<1992$. | open2math |
Which of the following quantities is the largest? (Write $A$, $B$, or $C$.) \[ A.\ \ \frac{2006}{2005}+\frac{2006}{2007} \qquad B.\ \ \frac{2006}{2007}+\frac{2008}{2007} \qquad C.\ \ \frac{2007}{2006}+\frac{2007}{2008} \] | open2math |
Find $x$, such that $4^{\log_7x}=16$. | open2math |
Let $x$, $y$, and $z$ be positive real numbers such that
\[x^2+y^2+z^2=3\sqrt[3]{xyz}.\]
If $T=x+y+z$, determine the maximum value of $T^3$.
[Your response should end here] | open2math |
Find the smallest positive integer that is a perfect cube and has exactly 4 distinct prime factors. | open2math |
Determine the smallest positive integer $m$ such that
\[\tan^m x + \cot^m x \le 2\]
for all $x$ in the interval $(\pi/4, \pi/2)$. | open2math |
Find the minimum value of the expression $x^2+y^2-8x-6y+22$ for real $x$ and $y$. | open2math |
Let $a$, $b$, and $c$ be real numbers such that $\sqrt{a}+\sqrt{b}+\sqrt{c} = 5$. Determine the minimum value of $a+b+c$ such that all square roots involved are defined. | open2math |
Find the sum of all integral values of $d$ with $d\ge -20$ for which the equation $z=x^3-5x-d$ has exactly one real root. | open2math |
Let $a$ and $b$ be real numbers such that
$$ab=\frac{1}{3} \mbox{ and } a^2+b^2=6.$$
Compute the value of
\[(a+b)^2\left(1+\frac{1}{(a+b)^2}\right).\] | open2math |
Find the difference between the sum of the squares of the first 10 prime numbers and the sum of the cubes of the first 7 prime numbers. | open2math |
Find the positive real number x such that [ x ] ⋅ x = 30 . Express x as a decimal. (Here [ x ] denotes the greatest integer less than or equal to x .) | open2math |
How many of the 200 smallest positive integers are such that the sum of their digits is a multiple of 3? | open2math |
Quadrilateral $ABCD$ is a parallelogram with center $O$. Let the feet of the perpendiculars from $O$ to $\overline{AB}$ and $\overline{BC}$ be $E$ and $F$, respectively. Given that $AB = 121$ and $AD = 40$, and that the length of one of the diagonals of the parallelogram is $71$, find the area of the parallelogram. | open2math |
Determine the number of positive four-digit integers of the form $\_\_99$ that have a remainder of 3 when divided by 7 and have a sum of digits that is divisible by 9. | open2math |
The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the the smallest possible value of $m+n.$ | open2math |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$$\frac{1}{x_1}+\frac{1}{x_2}=\frac{1}{2}.$$
Determine $a$. | open2math |
Find the sum of all positive integers $k$ that have a unique expansion of the form
\[k = (a_3a_2a_1a_0)_i\]
where $a_3 \neq 0$ and the base $i$ satisfies $i^2 = -1$. | open2math |
The vertices of a convex pentagon lie on a circle. What is the largest possible value of the sum of the interior angles of the pentagon that are formed by three consecutive vertices? | open2math |
The real function $g$ has the property that, whenever $a,$ $b,$ $n$ are positive integers such that $a \times b = 2^n,$ the equation
\[g(a) \times g(b) = n^3\]holds. What is $g(8)$? | open2math |
Compute the sum of the infinite series $1 + 2x + 3x^2 + 4x^3 +...$, where $x$ is a real number such that $|x| < 1$. | open2math |
Find the maximum number of consecutive integers whose sum is equal to 2022 and for which the sum of their squares is less than or equal to 2022 times 999. | open2math |
Compute the number of distinct subsets of size $3$ that can be formed from a set containing $10$ red balls and $15$ blue balls. | open2math |
Find the equation of the circle that passes through the points $(0,0)$, $(3,4)$, and $(6,8)$. | open2math |
What is the smallest positive integer $n$ such that $\frac{n}{n+101}$ is equal to a terminating decimal? | open2math |
Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$. | open2math |
Find all values of $x$ so that $\arccos x > \arcsin x.$ | open2math |
A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$? | open2math |
The sum of the squares of two consecutive integers is 17 times the square of the larger integer. Determine the value of the smaller integer. | open2math |
A certain sequence of integers begins with 0 and ends with 1000, and contains exactly one instance of every positive integer with exactly three digits. The sequence is formed by writing the integers in a specific order, such that the last two digits of each integer are the first two digits of the next integer in the se... | open2math |
If $x$ and $y$ are real numbers such that $x^2 + y^2 = 25$ and $xy = 12$, what is the sum of all possible values of $x$? | open2math |
Find the smallest positive $a$ such that $a$ is a multiple of $4$ and $a$ is a multiple of $14.$
Another problem inspired by this one:
Find the smallest positive integer $n$ such that $n$ is a multiple of $6$ and $n+3$ is a multiple of $9$. | open2math |
Find the real roots of the equation: $$\sqrt[3]{x+\frac{1}{x}}=\sqrt{x}-1$$ | open2math |
Determine the greatest number of distinct non-zero digits that can occur in the decimal representation of the product of two 3-digit numbers. | open2math |
Find the product of the possible values of $y$ that satisfy the equation $\frac{y-1}{y+1} \cdot \frac{y+2}{y-2} = 3$. | open2math |
Find the area of the triangle with vertices $(-1,4),$ $(7,0),$ and $(11,5).$ | open2math |
Let $x$ be a positive integer such that its cube $x^3$ ends with four zeros.
What is the least possible number of zeros that the integer $x$ can have? | open2math |
The sum of the squares of two positive integers is 117. The positive difference of their squares is 45. What is the product of the two integers? | open2math |
Let $A$ be the set of all positive integers less than $20$ that are either a multiple of $4$ or a multiple of $5$. Let $B$ be the set of all positive integers less than $20$ that are a multiple of $4$ and a multiple of $5$. Compute $(|A| - |B|)^2$. | open2math |
Let $Q$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $Q$ and the point $\left(\frac{3}{4}, \frac{1}{4} \right)$ is less than or equal to $-\frac{1}{3}$ can be written as $\frac{p}{q}... | open2math |
Find the number of all integer powers of 2 that lie between $\sqrt{200}$ and $\sqrt{3500}$. | open2math |
Let $n$ be the smallest positive integer that is a multiple of $75$ and has exactly $75$ positive integral divisors, including $1$ and itself. Find $\frac{n}{75}$.
| open2math |
Find the number of positive divisors of 8100 that are not multiples of 5. | open2math |
What is the largest possible value of the sum of the coefficients of a polynomial with integer coefficients if the polynomial has degree 5, the leading coefficient is positive, and the sum of the roots is -3? | open2math |
In triangle $ABC$, $AC=BC$, and $D$ is a point on $\overline{AB}$ such that $AD=CD$. If $\angle CAD=12^\circ$, find the measure of $\angle ACD$. | open2math |
Find the number of positive integers $n$ such that $n^3+1000$ is a perfect square. | open2math |
The sum of the squares of two positive integers is 200. Find the greatest common divisor of these two integers. | open2math |
What is the least positive integer that must divide the sum of the first fifteen terms of any arithmetic sequence whose terms are positive integers? | open2math |
An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free? | open2math |
Find the smallest prime number that can be expressed as the difference of two consecutive cubes. | open2math |
A circle with center $(3,2)$ passes through the point $(5,4)$. Find the equation of the line that is tangent to the circle at the point $(5,4)$. | open2math |
Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that
the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear,
each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint,
all of t... | open2math |
Let $S$ be the set of all positive real numbers $x$ such that $\lfloor x \rfloor \geq \lfloor 2x \rfloor$. Find the number of positive integers $N$ such that the equation $2^x = N$ has a solution for $x$ in $S$. | open2math |
A positive real number x is randomly selected such that x < 1. What is the probability that the integer nearest to x is odd? | open2math |
The sum of the squares of two consecutive positive integers can be represented as $(n+1)^2+n^2=2n^2+2n+1$. Show that this expression can also be written as $N^2+(N+1)^2$ for some integer $N$. What is the smallest value of $N$? | open2math |
What is the smallest positive integer n such that the product of all positive integers less than or equal to n is divisible by 2007? | open2math |
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