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In a convex quadrilateral, the lengths of three sides are 20, 15, and 8 units. Find the smallest possible integral length of the fourth side such that the quadrilateral can be inscribed in a circle. | nvidia |
What is the smallest positive integer $m$ such that $\frac{m}{m+67}$ is equal to a terminating decimal? | nvidia |
What is the smallest positive integer $n$ such that the least common multiple of $n$ and $12$ is a multiple of $5$? | nvidia |
Given that $\sin 2x = 2\sin x \cos x,$ determine the value of $k$ such that the equation $\sin kx = 3\sin x \cos^2 x$ is valid for all $x.$ | nvidia |
How many three-digit numbers are there such that the product of the tens and units digits is a perfect square? | nvidia |
A certain positive integer has exactly 16 distinct factors. Find the smallest possible value of this integer. | nvidia |
A bridge is to be built between two islands, one located at (3, 5) and the other at (8, 1). However, the builder wants the bridge to be perpendicular to the line connecting the two islands. Find the equation of the line that represents the bridge, given that it intersects the line connecting the islands at its midpoint... | nvidia |
Given that the measures of angles $A$ and $B$ are both positive, integer numbers of degrees and that angle $A$ is a multiple of angle $B$, while also being complementary angles, determine the largest possible sum of the measures of angles $A$ and $B$. | nvidia |
Find the largest positive integer k such that every set of k consecutive squares contains at least one square that is divisible by 7. | nvidia |
Find all real solutions to $y^4+(3-y)^4=52$. Enter all the solutions, separated by commas. | nvidia |
Let $m \ge 3$ be an integer and let $S = \{3,4,5,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.
| nvidia |
Determine the smallest positive integer $n$ such that $5^n\equiv n^5\pmod 3$. | nvidia |
Let $A_n$ be the number of distinct positive factors of $n$. Determine all positive integers $k$ for which there exist an infinite sequence of positive integers $\{a_n\}$ such that $A_{a_n}=k$ for all $n$. | nvidia |
Consider the polynomial $x^3 + px^2 + qx + p$, where $p$ and $q$ are real numbers. Find the value of $q$ such that the sum of the absolute values of the roots of the polynomial is minimized, given that all roots are real. | nvidia |
Determine all real numbers $x$, $-\pi < x < \pi$, such that $\cos x = \cos \frac{5\pi}{7}$. | nvidia |
Given positive integers $a$ and $b$, the expression
$$\frac{a^b - 1}{a - 1}$$can be factored into the product of $b$ polynomials of degree $1$ with integer coefficients. Compute the sum of the constant terms of these polynomials. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that
\[\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}=15\qquad\mbox{and}\qquad \frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}=53.\] Find $a+b+c$. | nvidia |
Find the maximum possible value of $xy$ for positive real numbers $x$ and $y$ that satisfy the equation $x^2 + y^2 = 8$ and the inequality $x^2y^2 \geq 16$. | nvidia |
Find the number of ordered pairs $(x,y)$ of real numbers such that
\[8^{x^2 + y} + 8^{x + y^2} = 1.\] | nvidia |
Let $d$ be a positive number such that when $109$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. | nvidia |
The polynomial $x^3-5x^2+3x+9$ has a root at $x=3$. Find the value of $k$ such that the polynomial $x^3-5x^2+kx+9$ has a root at $x=2$. | nvidia |
The sum of the squares of two consecutive positive integers is 481. What is the sum of the two integers? | nvidia |
Given that $\frac{a}{b} \cdot \frac{c}{d} \cdot \frac{e}{f} = \frac{10!}{9!}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are positive integers, find the minimum value of $a+b+c+d+e+f$. | nvidia |
In a conference, the number of attendees from Europe is triple the number of attendees from Asia. There are 90 attendees from Asia, and none of the attendees from Asia are also from Europe. The total number of men attending the conference is three times the number of women. How many people are attending the conference ... | nvidia |
For a positive integer $n$, let $f(n)$ denote the remainder when $5n!$ is divided by $7$.
If $m$ is the smallest positive integer such that $f(m)=0$, determine the value of $m$. | nvidia |
A rectangle is inscribed in a right triangle with legs of lengths 6 and 8, such that one vertex of the rectangle is at the right angle of the triangle, and the opposite vertex lies on the hypotenuse. Determine the maximum possible area of the rectangle. | nvidia |
Find the quartic polynomial with real coefficients that has the complex roots $i$ and $2 + 3i$. | nvidia |
Compute the number of ways to choose 5 elements from a set of 15 elements such that no two chosen elements are consecutive in the original set. | nvidia |
Let $C$ be a circle with a center at the origin and radius $5$. Find the value of $p$ such that the two circles
$$x^2 + y^2 = 5^2$$
and
$$(x-3)^2 + (y-p)^2 = 3^2$$
are tangent externally. | nvidia |
Let $f(n)$ denote the sum of the digits of a positive integer $n$ when written in base $b$. For example, $f(12345) = 1 + 2 + 3 + 4 + 5 = 15$. Suppose that $N$ is a positive integer such that $f(N) = f(N^2)$. Find all possible values of $N$ written in base $b$. | nvidia |
Consider the polynomial $x^3+ax^2+bx+c$, where $a$, $b$, and $c$ are integers. You are told that this polynomial has a triple root, and the sum of its coefficients ($1+a+b+c$) is a perfect square. Find all possible values of the triple root. | nvidia |
Find the number of positive integers whose cubes are less than 1000 but greater than 50. | nvidia |
Find the positive integer $M$ such that
$$3^4 \times 7^6 \times M^8 = 21^{12} \ ?$$ | nvidia |
Determine the number of vertices of a polyhedron, knowing that each face is a triangle and that every edge is shared by exactly two faces, if the total number of edges is 24. | nvidia |
Let $\{a_n\}$ be a sequence such that $a_1=3$ and
\[a_{n+1}=\frac{1}{2+a_n}\]
for $n \ge 1$. Find the smallest positive integer $k$ such that
\[a_k=a_{k+8}.\] | nvidia |
Name the greatest whole number less than $100$ that has an odd number of positive factors. | nvidia |
Let a sequence of real numbers $a_1, a_2, a_3,\ldots$ satisfy the following condition:
for each positive integer $n$ the sum of the first $n$ terms of the sequence is equal to $3n^2+2$.
Determine the 100th term of the sequence. | nvidia |
Let $n$ be the greatest integer such that $\left(8^n\right)\left(10^{16}\right)$ divides the product
$\left(2^{10}\right)\left(3^{12}\right)\left(5^{15}\right)$. Find $n$.
The final answer is 4. | nvidia |
Let $g(x)$ be a function that, given a positive integer $x$, returns the smallest positive integer $y$ such that $x^y$ has at least $y$ distinct divisors. Determine the smallest positive integer $z$ such that there exist at least two distinct values of $x$ for which $g(x) = z$. | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $a^2 + b^2 = c^2$. Determine the number of possible triples $(a, b, c)$ that can be the lengths of the sides of a triangle. | nvidia |
Let $s$ be a real number, $|s| < 3,$ and let $w$ be a complex number such that
\[w + \frac{2}{w} = s.\]Find $|w|$. | nvidia |
If the base of a triangle is fixed and its altitude is a constant $k$, then find a relationship between the sum of the squares of the lengths of the legs of the triangle and the distance between the feet of these legs to a fixed point on the base. | nvidia |
Find the value of $k$ such that $7! = 2^k \cdot 3^2 \cdot 5^3$. | nvidia |
Let ABCDEFGHIJKL be a dodecagon inscribed in a circle of radius 6 so that its vertices are endpoints of the radii. What is the minimum number of colors needed to color the edges of the dodecagon so that no two edges that share a vertex have the same color? | nvidia |
Find the largest possible value of $n$ such that there exists a real number $x$ satisfying
\[\sum_{k=1}^{100} \lfloor kx \rfloor = n.\] | nvidia |
Find the smallest positive integer that is not a perfect square and has exactly seven positive divisors. | nvidia |
In a set of six consecutive integers, the sum of the two smallest integers is greater than the average of the six integers. What is the largest integer that could be in the set? | nvidia |
How many odd four-digit numbers are there such that the sum of the hundreds and units digits is 12? | nvidia |
Find the product of all positive integers $m$ such that $2.5m - 3.5 < 7.5$. | nvidia |
Find the only real number that can be expressed in the form \[(a + bi)^3 - 107i,\]where $i^2 = -1,$ and $a$ and $b$ are positive integers. | nvidia |
Three numbers are such that the sum of any two is equal to the product of the third and a given number $k$. Find the smallest positive value of $k$ for which the sum of the three numbers can be an integer. | nvidia |
Determine the sum of all the prime numbers between 1 and 20 that are not factors of 16. | nvidia |
Let $S$ be the set of all positive integers $n$ such that $n^2 + 2n + 2$ is divisible by $5$. Let $T$ be the set of all positive integers $m$ such that $m^2 - 2m + 2$ is divisible by $5$. Determine the smallest possible value of $|S \cap T|$, where $| \cdot |$ denotes the number of elements in a set. | nvidia |
Find the number of zeros at the end of the product $3^8 \times 7^5$. | nvidia |
Let $b$ and $c$ be real numbers. If the polynomial $x^2+bx+c$ has exactly one real root and $b=c+1$, find the value of the product of all possible values of $c$. | nvidia |
The product of three consecutive integers is equal to the sum of their squares. Find the middle integer. | nvidia |
What is the sum of all real numbers that are tripled when multiplied by their squares? | nvidia |
The sum of the digits of the first 100 positive integers is 900. What is the sum of the digits of the first 100 negative integers? | nvidia |
Consider the function
\[f(x) = 3x^2 + \frac{8}{x}\]
for positive real numbers $x$.
Determine the value of $x$ for which the function is smallest.
Which positive integer $n$ is closest to this value of $x$? | nvidia |
Find the product of all $x$ such that the expression $\frac{x^2+2x+1}{x^2+2x-3}$ is undefined. | nvidia |
Find the area of the region defined by the equation $|x|+|y|=k$, where $k$ is a positive constant.
Solution: The region is a square with vertices at $(k,0)$, $(0,k)$, $(-k,0)$, and $(0,-k)$. The area of the square is $4k^2$.
Note: This problem requires a different approach than the original problem, as it involves f... | nvidia |
Two dice are rolled. What is the probability that the sum of the numbers on the dice is 9 and that one of the dice shows an even number? | nvidia |
Determine the largest real value of $c$ such that the equation $x^4 - cx^2 + 8 = 0$ has no real solutions. | nvidia |
Problem:
Find the greatest value of $r$ such that there exists a regular heptagon $P_1P_2P_3P_4P_5P_6P_7$ whose vertices lie on a circle of radius $r$ centered at the origin of the coordinate plane, and the y-coordinates of its vertices are distinct elements of the set $\{-3,-2,-1,0,1,2,3\}$. Each vertex $P_i$ has a no... | nvidia |
Let $f(x)=ax^2+bx+c$ be a quadratic function such that $f(x)=f(2x-3)$ for all real $x$.
Determine the sum of all possible values of $a$, $b$, and $c$. | nvidia |
Find the largest integral value of $y$ that satisfies the inequality $\frac{2}{7}<\sqrt{\frac{y}{12}}<\frac{3}{4}$. | nvidia |
Find the sum of all 3-digit numbers that can be formed using the digits 1, 2, and 3, if each digit can be used only once in each number. | nvidia |
Find the number of positive integers $n$ such that $\sqrt{n^2 - 20n + 150}$ is a positive integer. | nvidia |
A rectangle is inscribed in the triangle with vertices $(0,0)$, $(0,4)$, and $(3,0)$, such that its sides are parallel to the coordinate axes. Find the maximum area of the rectangle. | nvidia |
Consider the 3x3 matrix B with integer entries such that B^3 = I and the sum of the elements in each row is equal to 2. Find the largest possible value of the sum of all elements in B. | nvidia |
Let $f(x)$ and $g(x)$ be two functions satisfying the equations
\begin{align*}
f(f^{-1}(x)f(y))&=g(x+y)\\
g(g^{-1}(x)g(y))&=f(x+y)
\end{align*}for all real numbers $x$ and $y$. If $f(1)=2$ and $g(1)=3$, find $f(g(2))$. | nvidia |
Find the smallest digit $d$ such that the number $43d$ is a multiple of 7. | nvidia |
Let $g(x)$ be a function that satisfies $g(x^2) + g(x^3) = x$ for all real numbers $x.$ Find $g(8).$ | nvidia |
Let $C$ be the number of three-digit numbers that are multiples of 4, and let $D$ be the number of three-digit numbers that have a remainder of 2 when divided by 4. Find the difference between $C$ and $D$. | nvidia |
For positive integers $n$, let $f(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point. How many positive integer divisors does $f(2010)$ have? | nvidia |
Let $f(x)$ be a function such that
$$f(x^2) + xf(x) = 2$$
for all real numbers $x.$
Find $f(1).$ | nvidia |
At the 2007 Math Olympics, Team Canada won $17$ out of a possible $100$ medals. Which one of the following is closest to the fraction of medals that they won? $$
\frac{1}{4} \qquad \frac{1}{5} \qquad \frac{1}{6} \qquad \frac{1}{7} \qquad \frac{1}{8}
$$ | nvidia |
Let $f(x)$ be a function such that $f(0)=1$ and for all positive integers $x$, $f(x) = x \cdot f(x-1)$. Determine the value of the following expression: $\frac{f(x)}{f(x+2)}$. | nvidia |
Determine the degree measure of an angle such that its supplement is equal to twice its complement. | nvidia |
A polynomial $p(x)$ has the property that $p(x)-x^{100}$ is divisible by $(x-1)^3$. Determine the minimum possible number of real roots of $p(x)-x^{100}$ within the interval $[-100,100]$. | nvidia |
A list of seven positive integers has the property that if any one of them is removed, the remaining six integers can be divided into two groups of three with equal sums. What is the smallest possible sum of the seven integers? | nvidia |
The graph of the function
$f(x)=\ds\frac{(x^2-1)(x^2-9)}{x^4-16}$
has four asymptotes. Determine the sum of the squares of the slopes of these asymptotes. | nvidia |
Consider a function $y=v(x)$, which is a piecewise linear function that consists of three line segments:
$y=x+1$, for $x\in[-2,0)$;
$y=-2x+3$, for $x\in[0,1)$;
$y=-2x+4$, for $x\in[1,2]$.
Find the sum of the x-coordinates of the points where the graph of the function intersects its reflection across the y-axis. | nvidia |
The figure shows a regular hexagon inscribed in a circle, with a circle inscribed in the hexagon. The sides of the hexagon are of length 6, and the radii of the two circles are distinct integer values. If the area of the larger circle is $4k\pi$, find the area of the hexagon, expressed in terms of $k$. | nvidia |
Find the smallest positive integer that is not divisible by any prime number less than 10, but is divisible by at least two distinct prime numbers greater than 10. | nvidia |
Find the height of the cone, given that the radius of the cylinder is 2/3 of the radius of the cone's base. The height of the cylinder is equal to its diameter, which is also 2/3 of the cone's height.
Note: This problem can be solved using similar triangles and proportions, rather than maximizing the volume of the cy... | nvidia |
Determine the number of positive integers $a$ less than $12$ such that the congruence $ax\equiv 1\pmod{12}$ has a solution in $x$. | nvidia |
Compute the product of all possible values of $x$ such that $\sqrt{x+1} + \sqrt{x-1} = \sqrt{4x+1}$. | nvidia |
In a right triangle, the length of one leg is given by the sum of the squares of the digits of a two-digit number, and the length of the other leg is given by the product of the digits. If the length of the hypotenuse is $\sqrt{145}$, find the two-digit number. | nvidia |
Let $a$ and $b$ be real numbers, and let $w$ be a complex number such that
\[w^2 + aw + b = 0.\]Given that $|w| = 3$, find the maximum possible value of $|a|$ in terms of $b$. | nvidia |
Find the sum of all positive integers between 1 and 100 that are not divisible by 2 or 3. | nvidia |
Evaluate the expression $\lfloor x \rfloor + \lceil 2x \rceil$, given that $x$ is a real number satisfying $3 < x < 4$. | nvidia |
Find the coordinates of the point $R$ on the line passing through the points $A(1,3)$ and $B(-4,-2)$ such that the slope of the line passing through $R$ and the origin is equal to $\frac{3}{4}$. | nvidia |
What is the smallest positive integer $n$ such that $3n$ is a perfect square and $2n$ is a perfect cube? | nvidia |
Find all positive values of $d$ such that the inequality $x^2-4x+d<0$ has real solutions for $x$. Express your answer in interval notation. | nvidia |
Let $f(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. Given that $n$ is a multiple of 15, what is the smallest value of $n$ such that $f(n) > 15$? | nvidia |
Given that $P(3,1)$ is a point on the circle with center $O(0,0)$, and the circle passes through the origin, what is the product of the coordinates of the other endpoint of the diameter that passes through $P$? | nvidia |
How many of the positive integers less than or equal to 200 can be expressed as the sum of the cubes of two nonnegative integers? | nvidia |
Find the number of digits that are greater than 5 in the base-8 representation of $185_{10}$. | nvidia |
Two cards are chosen at random from a standard deck of 52 cards. Then, without replacement, another two cards are chosen. Find the probability that the number of Aces among the four chosen cards is the same as the number of Kings, and determine the sum of the numerator and denominator of the resulting fraction. | nvidia |
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