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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 $a+b$. | nvidia |
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. | nvidia |
Find the greatest number of non-overlapping unit squares that can be placed inside a circle with radius 6 units, assuming each square can be oriented in any direction. | nvidia |
The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$?
| nvidia |
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. | nvidia |
The measures of the interior angles of a convex octagon form a sequence of consecutive Fibonacci numbers. How many such sequences are possible if the octagon is not equiangular and all of the angle degree measures are positive integers less than $170$ degrees? | nvidia |
Find the smallest positive integer $n$ such that $n$, $n+1$, and $n+2$ are each divisible by a distinct prime number. | nvidia |
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. | nvidia |
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. | nvidia |
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? | nvidia |
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 $\frac{13}{17}$. | nvidia |
Determine the greatest integer value of $x$ such that $|x-1|+2|x+3|<13$. | nvidia |
How many different 4-digit numbers can be formed using the digits 0, 1, 2, 3, 4, and 5, such that exactly two of the digits are odd, and exactly one digit is zero, and no digit is repeated? | nvidia |
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? | nvidia |
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? | nvidia |
Find the number of positive integers $x$ such that $x^3-x^2+2x-1$ is a perfect square.
[Finish] | nvidia |
Find the smallest positive integer $n$ such that the sum of $n^3$ and $n^2$ is a perfect square. | nvidia |
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 $abc$. | nvidia |
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 $x$. | nvidia |
If $x$, $y$ are nonzero real numbers such that $x^2+y^2=6xy$, find the value of $\left|\frac{x+y}{x-y}\right|$. | nvidia |
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 $b_{10}.$ | nvidia |
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 $ABC$. | nvidia |
Find the number of three-digit positive integers with exactly two 7s as digits. | nvidia |
Let $a$ and $b$ be complex numbers such that $|a+bi|=3\sqrt{2}$ and $|a-bi|=5\sqrt{2}$. Find the value of $|a^2-b^2i|$. | nvidia |
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. | nvidia |
Find the sum of all positive integral values of $n$ for which $\frac{n+3}{n}$ is an integer. | nvidia |
A point $P$ is chosen at random on a straight line segment of length $x$. If the point $P$ is at distance $y$ from the midpoint of the line segment, find the expected value of $y^2$. | nvidia |
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 $xyz$? | nvidia |
Let $a$, $b$, and $c$ be real numbers, and consider the function $f(x)$ defined by
\[f(x) =
\begin{cases}
ax+b &\quad \text{if } x > c, \\
x^2 &\quad \text{otherwise}.
\end{cases}
\]
Find all possible values of $a$ and $b$ such that there exists some $c$ for which $f(f(x))$ is a polynomial for all $x$.
For which val... | nvidia |
What is the least positive integer that is divisible by four distinct primes? | nvidia |
A book consists of 100 pages, numbered from 1 to 100. Each page has exactly 200 words printed on it, and no word appears more than once throughout the book. For each page, the product of the page number and the total number of vowels on that page is equal to the sum of the squares of the number of occurrences of each v... | nvidia |
Consider the quadratic equation
\[y^2 + y + a^2 = 0,\]
where $a$ is a positive real number.
Find all values of $a$ for which this equation has at least one integer root. | nvidia |
Find all positive integers $n$ such that the greatest common divisor of $n$ and $24$ is equal to the greatest common divisor of $n$ and $30$. | nvidia |
Find the positive integer $n$ such that $|(n+i)^2|=|(n-i)^3|$. | nvidia |
Find the largest positive integer x less than 144 that has exactly three distinct prime factors among its divisors. | nvidia |
Find all real values of x that satisfy \(\frac{\sin^2(x)}{1 + \cos(x)} \le \frac{2}{3}.\) (Give your answer in interval notation.) | nvidia |
Find the value of the positive root of the equation $x^3-6x^2-13x+42=0$ given that one of its roots is 3 less than another root. | nvidia |
Find the sum of the divisors of $150$ that are also multiples of $3$. | nvidia |
Determine the number of digits in the base-9 representation of the number 1458. | nvidia |
A portion of the graph of $y = G(x)$ is shown in red below. The distance between grid lines is $1$ unit.
Compute $G(G(G(G(G(1)))))$.
[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 tick... | nvidia |
The product of two positive integers minus their sum is 87. The integers are relatively prime, and each is less than 25. What is the sum of the two integers? | nvidia |
What is the smallest positive integer $n$ such that the sum of the digits of $n$ is a divisor of $n$, and $n$ is divisible by the sum of the digits of $n$ plus one? | nvidia |
Find the angle between the plane of the equilateral triangle $ABC$ with side length $4$ and the plane of the isosceles triangle $DEF$ with base $DE$ of length $6$ and congruent legs $DF$ and $EF$ of length $5$, where $A$ and $D$ are the same point and $B$, $E$, $C$, and $F$ are distinct points in space. | nvidia |
What is the smallest positive integer that can be expressed as the sum of two perfect squares in exactly three different ways? | nvidia |
What is the largest positive integer $n$ such that $2^n$ is less than $100$? | nvidia |
Find the largest number of terms of a geometric sequence whose product is 30 and all terms are positive integers. | nvidia |
Find the maximum possible value of $x + y + z$ for positive integers $x$, $y$, and $z$ such that the sum of their reciprocals is equal to $\frac{3}{2004}$. | nvidia |
Find the sum of the divisors of 120 that are also multiples of 3. | nvidia |
Find the number of positive integers less than or equal to 100 that have exactly three distinct positive divisors. | nvidia |
Consider a circle centered at the origin with radius 4, and a point $P$ on the circle with coordinates $(x, y)$. Let $A$ be the point on the circle diametrically opposite to $P$, and let $B$ be the point on the x-axis that lies on the line passing through $A$ and $P$. Find the vector $\begin{pmatrix}v_1 \\ v_2 \end{pma... | nvidia |
A circle is inscribed in a regular heptagon. How many distinct chords of the circle can be drawn such that they are also diagonals of the heptagon? | nvidia |
Let $P(x)$ be a polynomial such that
\[P(P(x))=(P(x))^3-(P(x))^2-P(x)-1.\]Find the sum of all possible values of $P(1)$. | nvidia |
Let $g$ be a function defined on the positive integers, such that
\[g(x+y) = g(x) \cdot g(y)\]for all positive integers $x$ and $y.$ Given $g(5) = 12$ and $g(8) = 18,$ find $g(25).$ | nvidia |
How many real numbers $x$ satisfy the inequality $|x-3|+|x+2|>5$? | nvidia |
Find the smallest positive integer $n$ such that $2^n$ is a factor of $100!$ and $2^{n+1}$ is not. | nvidia |
Determine the smallest positive integer $k$ such that any $k$ distinct points in the plane with integer coordinates have a subset of 3 points that are vertices of a triangle whose area is positive. | nvidia |
Let $S$ be the number of ways to color the vertices of a cube with three colors, red, blue, and green, such that each face of the cube contains exactly two red vertices and exactly one blue vertex. Additionally, no two adjacent vertices can have the same color, and all green vertices must be adjacent to at least two bl... | nvidia |
The graph of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ has a tangent line at the point $(3,0)$. Find the sum of the $x$-coordinates of the points where the tangent line intersects the ellipse again. | nvidia |
Let $P$ be a point outside of an ellipse $E.$ A line is drawn from $P$ such that it is tangent to ellipse $E$ at point $T.$ Meanwhile, a secant from $P$ intersects $E$ at $A$ and $B,$ such that $PA < PB.$ If the area of the triangle $PAB$ is equal to the area of the triangle $PAT$, then what is the length of the major ... | nvidia |
A bookstore has 8 shelves of books. What is the greatest number of pairs of books that can be selected such that each pair consists of books from two different shelves and no two pairs are the same combination? | nvidia |
Find the number of ordered pairs of integers $(a,b)$ with $2\le a<b\le 200$ such that $(-1)^a + (-1)^b$ is a positive integer. | nvidia |
Let $x$ and $y$ be positive real numbers. Find the maximum possible value of $$\frac{x^2y^2+1-\sqrt{x^4y^4+2x^2y^2+1}}{xy}.$$ | nvidia |
Find the value of $b$ for which the equation
$$2x^2+3x+b=0$$
has exactly one real root, and its multiplicity is 3. | nvidia |
Find the largest integer value of $n$ such that $n^2-9n+18$ is negative. | nvidia |
Let $m$ be the number of positive integers less than $1000$ that are a multiple of $2$ but not $4$, and let $n$ be the number of positive integers less than $1000$ that are a multiple of $6$ but not $12$. Compute $(m-n)^2$. | nvidia |
Let a and b be the roots of the equation x^2+px+q=0. Given that the arithmetic mean of the squares of the roots is 6 and the geometric mean of the cubes of the roots is 5. Find p^2. | nvidia |
Determine all real numbers x for which the function f(x) = 1/(1 + |x|) is increasing. | nvidia |
Find the product of all values of $t$ such that $t^2 = 36$. | nvidia |
In the equation
$$4x^2-kx+2=0$$
determine $k$ such that the difference of the roots be equal to unity. | nvidia |
Let f(x) be a function that satisfies f(f(x)) = x for all real numbers x.
Determine the possible values of f(10) if it is given that f(f(f(10))) = 20. | nvidia |
Find the number of integers n with the property that the product of the digits of n equals 1995 and n itself is divisible by 1995. | nvidia |
Let $n$ be a positive number such that when $179$ is divided by $n$, the remainder is $7.$ Compute the sum of all possible two-digit values of $n$. | nvidia |
Determine the value of $a+b$ where the value of $\log_{10}{52641}$ falls between two consecutive integers $a$ and $b$. | nvidia |
The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | nvidia |
Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. | nvidia |
The number 3600 can be expressed as a sum of consecutive positive integers. What is the minimum number of consecutive positive integers that sum to 3600, if the sum starts with the number 12? | nvidia |
When the digits in the number $2005$ are reversed we obtain the number $5002,$ and $5002 = a \cdot b \cdot c$, such that $a$, $b$ and $c$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$ and $p_3$ such that $p_1 + p_2 + p_3 = a+b+c$? | nvidia |
A point (x,y) is chosen at random inside the triangle with vertices (0,0), (0,1), and (1,0). Determine the probability that $x+y<\frac{1}{2}$ | nvidia |
Determine the smallest positive integer $n$ such that the decimal representation of $3^n$ has 3 as its units digit. | nvidia |
For a positive integer $n$ and nonzero digits $d$, $e$, and $f$, let $D_n$ be the $n$-digit integer each of whose digits is equal to $d$; let $E_n$ be the $n$-digit integer each of whose digits is equal to $e$, and let $F_n$ be the $3n$-digit (not $n$-digit) integer each of whose digits is equal to $f$. What is the gre... | nvidia |
Find the number of positive integers less than 100 that have a remainder of 2 when divided by 7. | nvidia |
Determine the number of integers $n$ that satisfy the inequality $\sqrt[3]{n} < \log_{2}(n) < \sqrt[5]{n}$. | nvidia |
Find the sum of the divisors of 72. | nvidia |
The roots of the quadratic equation $x^2+bx+c=0$ are the reciprocals of the roots of the equation $ax^2+dx+e=0$. If $b=2d$ and $c=-\frac{1}{2}e$, find the ordered pair $(a,e)$. | nvidia |
Determine the sum of all positive integer values of $a$ such that the quadratic equation $6x^2 - 7x + a = 0$ has two distinct real roots. | nvidia |
The quadratic function $f(x) = x^2 - 4x - 5$ has two real roots. Find the number of integers $n$ such that $f(n) > 0$ and $f(n+1) < 0$. | nvidia |
The roots of the equation $x^4 + 4x^2 + 4 = 0$ can be expressed in the form $a + bi$, where $a$ and $b$ are real. Find the sum of the squares of the roots with $b > 0$. | nvidia |
A positive integer $n$ has exactly four positive divisors. What is the smallest possible value of the sum of these four divisors? | nvidia |
Find the smallest positive integer n such that the repeating decimal 0.\overline{1} can be expressed as a fraction in simplest form with denominator 2^n. | nvidia |
Determine the value of the parameter $a$ such that the equation $x^4-2ax^2+x+a^2=0$ has exactly one positive solution. | nvidia |
Compute the sum of all four-digit numbers that can be formed using the digits 1, 2, 3, and 4, with each digit appearing exactly once in each number. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a^2 + b^2 = 100.$ Find the maximum value of $ab.$ | nvidia |
The graph of a function $f(x)$ is given below.
[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 wholetickdown=tickdown;
voi... | nvidia |
In a regular hexagon with side length 12, two congruent isosceles triangles are drawn such that they share one base and each has one vertex on a vertex of the hexagon. The base of each triangle is perpendicular to the side of the hexagon. What is the side length of the largest equilateral triangle that can be inscribed... | nvidia |
The three vertices of a regular tetrahedron lie on the vertices of a unit cube with vertices $(0,0,0)$, $(0,0,1)$, $(0,1,0)$, $(0,1,1)$, $(1,0,0)$, $(1,0,1)$, $(1,1,0)$, and $(1,1,1)$. Find the coordinates of the fourth vertex, given that the fourth vertex also lies on one of the vertices of the cube. | nvidia |
The difference between two numbers is 9, and the sum of the squares of each number is 153. What is the value of the product of the two numbers? | nvidia |
Let $a$ and $b$ be two positive real numbers such that $a + b = 50.$ Find the ordered pair $(a,b)$ for which $a^3 b^4$ is maximized. | nvidia |
A circular plate with radius 20 cm has a square cut out from the center. The length of each side of the square is such that its diagonals are equal to the radius of the plate. Find the area of the remaining plate in terms of π. | nvidia |
What is the greatest number of people that can be seated at 5 tables, with each table having the same number of people, given that the number of people seated must be a factor of both 40 and 60? | nvidia |
The sum of the interior angles of a convex polygon with an increasing number of sides forms a geometric progression. If the sum of the interior angles of a triangle is 180 degrees, and the sum of the interior angles of a square is 360 degrees, find the sum of the interior angles of a polygon with 12 sides. | nvidia |
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