instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find the largest value of $n$ such that the expression $5^{2018} - 5^{2016}$ is divisible by $5^n$. | nvidia |
Let $f(x)$ be a monic cubic polynomial such that $f(1) = 1,$ $f(2) = 3,$ and $f(3) = 5.$ Find the smallest value of $n$ such that $f(n)$ is a perfect square. | nvidia |
Given that the polynomial $x^3 - 2ax^2 + bx + 24$ has only positive integer roots, find the smallest possible value of $a$ such that the sum of the roots is divisible by 3 and the product of the roots is divisible by 4. | nvidia |
Find the x-coordinate of the point on the x-axis that is equidistant from points P(0, 3) and Q(4, 1). | nvidia |
A right circular cone has a base with radius $900$ and height $300\sqrt{3}.$ A bug starts at a point on the surface of the cone whose distance from the vertex of the cone is $200$, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $450\sqrt{2}.$... | nvidia |
Let $x$ be a real number such that the expression
$$\frac{x-1}{x+1}$$
takes on positive values only. Use interval notation to represent all possible values of $x$. | nvidia |
Consider the set of all circles passing through the points $A(0,13)$, $B(13,13)$, $C(13,0)$, and $D(0,0)$. Determine the radius of the circle with the largest area that passes through these four points. | nvidia |
In the expansion of $(x+y+z)^n$, the coefficients of $x^iy^jz^k$ and $x^{n-i}y^{n-j}z^{n-k}$ are equal for all nonnegative integers $i,j,k$ such that $i+j+k=n$. Find the largest $n$ for which this property holds. | nvidia |
A $\textit{composite number}$ is a number that has two or more prime factors. The number 87 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers? | nvidia |
The sum of the squares of three positive integers is $500$, and their product is a multiple of $1000$. What is the largest possible value of the product? | nvidia |
Given that $x^3$ is a real number between $-1$ and $1$ inclusive, find the value of $x$ that satisfies the equation
$$\sqrt[3]{\cos(x^3)} + \sqrt[3]{\sin(x^3)} = \sqrt[3]{2}.$$ | nvidia |
For how many ordered pairs of positive integers $(x,y),$ with $y<x\le 200,$ do there exist positive integers $k$ and $m$ such that $x = ky + 1$ and $x + 1 = m(y + 2)$? | nvidia |
A polynomial $P(x)$ is defined for all real numbers and satisfies $P(x+1)=P(x-1)$ for all $x$. If $P(0) = 0$, determine the smallest possible degree of $P(x)$ such that it has exactly 10 distinct roots in the interval $-10\leq x \leq 10$. | nvidia |
Find the number of monic cubic polynomials of the form $x^3+ax^2+bx+c$ with integer coefficients such that $a$, $b$, and $c$ are distinct and the polynomial has exactly one real root, which is also an integer. | nvidia |
The number of men in a company is double the number of women, the number of women is three times the number of children, and the number of children is five times the number of pets. If the total number of people and pets is 550, what is the number of men in the company? | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a + b + c = 1$. Find the maximum value of
\[(a^2 + 2b^2)(b^2 + 2c^2)(c^2 + 2a^2).\] | nvidia |
In triangle $ABC$, angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D$. The lengths of the sides of $\triangle ABC$ are integers, $BD=7^4$, and $\sin B=p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | nvidia |
Suppose that $a$ and $b$ are positive integers such that $(a+bi)^2 = 3+4i$. What is $a+bi$? | nvidia |
The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $c$. (Assume that the graph has the standard unit scale.)
[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 ti... | nvidia |
Find the product of the positive integer divisors of 12 that are also multiples of 3. | nvidia |
Using only the digits 1, 2, and 3, how many positive five-digit integers can be made that are palindromes? | nvidia |
Let $x$ and $y$ be positive real numbers such that $xy=12$ and $\sqrt{x}+\sqrt{y}=13$. Find the maximum possible value of $x+y$. | nvidia |
Find the number of different positive three-digit integers that can be formed using the digits 1, 2, 3, 4, and 5, if a digit may be repeated and the sum of the digits is even. | nvidia |
A round table has 24 identical chairs, and each chair can be occupied by one person. What is the smallest number of people that can be seated around the table in such a way that each person has exactly three neighbors who are members of the same club, and each club has the same number of members? | nvidia |
Consider a square grid with side length $n$, where $n$ is a positive integer. Three vertices $A=(x_1,y_1)$, $B=(x_2,y_2)$, and $C=(x_3,y_3)$ form a triangle inside this grid, such that all three coordinates are integers and $0 \leq x_i,y_i \leq n$ for $i=1,2,3$. We define $B$ to be the reflection of $A$ across the hori... | nvidia |
Find the minimum value of $e^x - e^{2x},$ over all real numbers $x.$ | nvidia |
If $a$, $b$ are nonzero real numbers such that $a^2+b^2=8ab$, find the value of $\left|\frac{a+b}{a-b}\right|$. | nvidia |
In a certain triangle, the length of the longest side is $5x + 2$, the length of the middle side is $3x - 1$, and the length of the shortest side is $2x + 3$, where $x$ is a positive integer. Determine the number of possible values of $x$ such that the triangle is an acute triangle. | nvidia |
A set of 8 distinct positive integers has the property that no two elements of the set have the same number of factors. What is the smallest such set? | nvidia |
What is the length of the longest stick that can be placed inside a rectangular box with dimensions $5$ cm, $8$ cm, and $12$ cm, such that the stick touches all six faces of the box? | nvidia |
Find the smallest positive integer $b$ for which $x^2+bx+2008$ factors into a product of two binomials, each having integer coefficients. | nvidia |
Find the largest of four consecutive three-digit positive integers that are less than 500 and are multiples of 4. | nvidia |
The sum of the squares of two numbers is $45$ and the product of the two numbers is $12$. What is the sum of the two numbers? | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $a^2b^2c^2 = 1.$ Compute the maximum value of
\[\frac{a^2b}{c} + \frac{b^2c}{a} + \frac{c^2a}{b}.\] | nvidia |
What is the smallest positive integer n such that the product of all prime numbers less than n is greater than 1000? | nvidia |
In tetrahedron $ABCD,$ all four faces are congruent triangles with side length 1. Find the perimeter of the quadrilateral formed by connecting the midpoints of the four sides. | nvidia |
Suppose that $c$ and $d$ are nonzero real numbers, and that the equation $x^2+cx+d=0$ has solutions $x_1$ and $x_2$ such that $x_1^2+x_2^2 = c^2+d^2$. Find the relation between $c$ and $d$. | nvidia |
In a triangle, the lengths of the sides are given by $a$, $b$, and $c$, and the radius of the inscribed circle is $r$. If the tangents to the incircle at the points of tangency with the sides $a$ and $b$ intersect at a point outside the triangle, and this point is at a distance of $d$ from the vertex opposite the side ... | nvidia |
How many positive integers less than 60 have an even number of positive divisors? | nvidia |
Determine the equation of the line that passes through the midpoint of the segment joining the points $(2,2)$ and $(-1,-1)$ and is perpendicular to the line connecting these two points. | nvidia |
The sum of the squares of three consecutive positive integers is equal to 50. Determine the smallest of these integers. | nvidia |
Let $a,b,c,d,e,f$ be distinct real numbers such that the roots of the equation
$$(x-a)(x-b)(x-c)-(x-d)(x-e)(x-f)=0$$
are $a+b,b+c,c+a$. Find the value of $d+e+f$.
Good luck! | nvidia |
A sequence of equilateral triangles is constructed by starting with an equilateral triangle with side length 1 and then adding a smaller equilateral triangle on top, with side length one-third that of the previous triangle. The process is repeated indefinitely, with each new triangle having a side length one-third that... | nvidia |
Consider a sequence of positive integers defined by the recurrence relation:
$s(n) = \begin{cases}
n-1 & \text{if $n$ is composite}\\
s(n+1)+1 & \text{if $n$ is prime}
\end{cases}$
Compute $s(84)$. | nvidia |
What is the sum of the proper divisors of 256?
Another problem:
A number is called "abundant" if the sum of its proper divisors is greater than the number itself. Find the smallest abundant number that is also a perfect square. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $x+y+z=3$. What is the minimum value of \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}?\] | nvidia |
Find the smallest positive integer $n$ such that the product of its digits is divisible by 15. | nvidia |
Determine the number of ways to arrange the digits of the number 12345 such that the resulting number is divisible by 5 and has a unique digit in the units place. | nvidia |
The polynomial $f(x)$ with integer coefficients has $d$ real roots, all of which are distinct. If $a$ is the product of the integer roots and $b$ is the product of the non-integer roots, then $ab = 30$. What is the minimum possible number of integer roots $d$? | nvidia |
Given a set of consecutive integers from 1 to 20, find the sum of all the integers that are exactly 3 more than a multiple of 5. | nvidia |
Find the sum of all positive integers $a$ such that the product
\[\log_a (a+1) \cdot \log_{a+1} (a+2) \dotsm \log_{2013} 2014 \cdot \log_{2014} 2015\]
is an integer. | nvidia |
A triangle has three sides of the following lengths: $5$, $12$, and $x^3$. What are all the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order. | nvidia |
Determine the number of values of $x$ for which the expression $\frac{x^2+2x-8}{(x^2-4x-5)(x+2)}$ is undefined. | nvidia |
Find the number of 3-digit positive integers divisible by 7 that have the digit 9 in their decimal representation. | nvidia |
Find the value of $x$ that satisfies the equation:
$$\frac{x-i}{x+i} = \frac{3-2i}{3+2i}$$ | nvidia |
Three numbers are given such that the sum of the first two numbers is 7 times the third number, the sum of the last two numbers is 11 times the first number, and the sum of the first and last numbers is 13 times the second number. What is the product of the three numbers? | nvidia |
Find the number of positive integers less than or equal to 100 that are not divisible by 3 or 5. | nvidia |
What is the greatest whole number that must be a divisor of the sum of any three consecutive positive odd integers? | nvidia |
What is the largest possible value of the product of four positive integers whose sum is 24? | nvidia |
How many positive integers less than or equal to 2012 have the property that the fraction obtained by dividing the number by 2401 can be expressed as a terminating decimal? | nvidia |
What are the last two digits in the sum of factorials of the first 100 positive integers? | nvidia |
In a set of $n$ distinct points on a plane, the sum of the squares of their pairwise distances is known to be $18,000$. Additionally, the square of the distance between the point $(3,5)$ and the centroid of these points is known to be $16$.
Find the smallest positive integer $n$ for which this is possible. | nvidia |
If the point (2,5) is on the graph of y=f(x), and k(x) = 3(f(x)) for all x, then there is one point that must be on the graph of y=k(x). What is the difference of the coordinates of that point? | nvidia |
Find the least positive integer such that there exists a subset of it such that the elements of the subset have a sum that is divisible by any positive integer less than or equal to 13. | nvidia |
We are given a triangle ABC with side lengths a, b, and c. The triangle has three circles associated with it:
the incircle with radius r, the circumcircle with radius R, and the circle with center I with a radius of 2r.
The circle with center I is tangent to the circumcircle and to the two legs of the right triangle at... | nvidia |
Given that the polynomial $x^3 - 2x^2 + mx + n$ has one real and two complex conjugate roots, find the sum of all possible values of $m$ such that the real root is a rational number and the complex roots have a modulus of 1. | nvidia |
Let $h(x)$ be a cubic polynomial and $r(x)$ be a polynomial such that the degree of $h(x) \cdot r(x)$ is 5. Determine the degree of $r(x)$. | nvidia |
Compute the sum of the series:
$$\frac{1}{1 \cdot 2} + \frac{2}{2 \cdot 3} + \frac{3}{3 \cdot 4} + \dots + \frac{n}{n(n+1)}$$
where n is a positive integer. | nvidia |
What is the sum of three consecutive odd integers if the sum of the second and third integers is twice the sum of the first and second integers? | nvidia |
Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ$, find $z^{2000}+\frac 1{z^{2000}}$. | nvidia |
Let $x$, $y$, and $z$ be positive integers such that $xyz$ is a perfect cube and $x+y+z$ is a perfect square. Find the smallest possible value of $x+y+z$. | nvidia |
Consider the quadratic equation:
$$x^2 - mx + 20 = 0$$
Find the sum of all positive integer values of $m$ for which this equation has only integer solutions. | nvidia |
A complex number $z$ is said to be $n$-balanced if $z^n=1$ and $\abs{z^k-1}=\abs{z^l-1}$ for all $1 \le k,l \le n-1$ such that $k \nmid n$ and $l \nmid n.$
Let $T$ be the set of all complex numbers $x+yi$ where $x$ and $y$ are real numbers, such that $\frac{1}{3} \le x \le \frac{2}{3}.$
Find the smallest positive integ... | nvidia |
Find the maximum value of $10^x - 100^x,$ over all real numbers $x.$ | nvidia |
In the polygon shown, each side is perpendicular to its adjacent sides, and all 25 of the sides are congruent. The perimeter of the polygon is 75. Find the area of the polygon.
[asy]
unitsize(0.5 cm);
draw((2,0)--(3,0)--(3,1)--(4,1)--(4,2)--(5,2)--(5,3)--(6,3)--(6,4)--(7,4)--(7,5)--(6,5)--(6,6)--(5,6)--(5,7)--(4,7)... | nvidia |
In the Cartesian plane, there are two circles centered at points $A(-1,2)$ and $B(3,-4)$. The radius of the circle centered at $A$ is twice the length of the line segment connecting $A$ and $B$, while the radius of the circle centered at $B$ is four times the distance between the two centers. Determine the smallest pos... | nvidia |
The integers $G$ and $H$ are chosen such that
\[\frac{G}{x+5}+\frac{H}{x^2-4x}=\frac{x^2-2x+10}{x^3+x^2-20x}\]for all real values of $x$ except $-5$, $0$, and $4$. Find $H/G$. | nvidia |
Let $p$ be a prime number and $k$ be the smallest positive integer such that $p^k \equiv 1 \pmod{p^2}$. Find the remainder when $k$ is divided by $p$. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a^2+b^2=50$. Find the maximum value of $\sqrt{ab}$. | nvidia |
The graph of a function $f(x)$ passes through the point $(2,3)$ and satisfies the property that the length of any chord drawn from the point $(2,3)$ to another point on the graph is equal to the magnitude of the derivative of the function at the midpoint of the chord. Find the equation of the tangent line to the graph... | nvidia |
Let $T$ be the set of all positive integers. The function $g : T \to T$ satisfies the following two properties:
(i) First,
\[g \left( x + 1 \right) = g(x) + x + 1\]for all $x \in T.$
(ii) Second,
\[g \left( xy \right) = g(x) \times g(y)\]for all $x \in T$ and $y \in T$ such that $xy \in T.$
Let $m$ be the number of... | nvidia |
A sequence of circles is defined recursively as follows:
* The first circle has center $A$ and radius $r$.
* The $n$-th circle, for $n \ge 2$, is tangent to the $(n-1)$-th circle and has center on the line that passes through the centers of the previous two circles.
Let $P$ be a point on the first circle. Prove that t... | nvidia |
What is the smallest positive integer $n$ such that the $n^{\text{th}}$ roots of unity form a set that contains all the solutions to the equation $z^3 - 2z + 1 = 0$? | nvidia |
Given $a, b, c$ are real numbers such that
$$2a+3b+c=7$$
$$a+3b+2c=11$$
If a, b, and c are all positive integers, what is the least possible value of b? | nvidia |
Find the equation of the circle that passes through the points $(2,3)$, $(4,5)$, and $(6,7)$ and has its center on the line $x-y=1$. | nvidia |
The roots of the quadratic equation $x^2 + (2m - 3)x + m = 0$ are such that the sum of their reciprocals is $\frac{1}{4}$. Find the value of $m$. | nvidia |
Given that the quadratic equation $ax^2+bx+c=0$ has roots $2$ and $-3$, find the value of $\frac{b^2+c^2}{a^2}$. | nvidia |
Find the number of positive integers less than or equal to 1000 that have at least one digit repeated. | nvidia |
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... | nvidia |
Let $a_1, a_2, \dots, a_{100}$ be a sequence of real numbers such that for all positive integers $k$, the sum of the terms in the sequence, taken $k$ at a time, is equal to $(-1)^k$. That is,
$$\sum_{1 \le i_1 < i_2 < \dots < i_k \le 100} a_{i_1}a_{i_2}\dots a_{i_k} = (-1)^k$$
Find the value of $a_1 + a_2 + \dots + a... | nvidia |
Given a square with side length 5 units and one of its vertices at the origin (0,0), the square is reflected about the line y = 3x/4. What are the coordinates of the vertex of the reflected square that is farthest from the origin after the reflection? | nvidia |
How many positive integers less than 1000 are divisible by 9, have no repeated digits, and have no digits that are congruent modulo 9? | nvidia |
Find the smallest positive integer $m$ such that $m!$ is divisible by $7^4 \cdot 11^2$. | nvidia |
Find the smallest positive integer $x$ such that $55x$ leaves a remainder of $23$ when divided by $17$. | nvidia |
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. | nvidia |
In a set of five numbers, the median is 3.2, and the mean of the two smallest numbers is 2.5. If the sum of the three largest numbers is 20, what is the sum of all five numbers? | nvidia |
Problem:
In the figure, four circles of radius $r$ are arranged so that each circle touches two of the other circles and a line segment of length $2r$ is drawn to connect the centers of two of the circles that do not touch each other. What is the radius of the largest circle that can fit inside the rectangle formed by ... | nvidia |
Find the positive values of $x$ such that $2x = \sqrt{x^2 + x}$. | nvidia |
Let $p(n)$ be the probability that at least $n$ consecutive shots are made by a basketball player who makes each shot with a probability of $\frac{3}{5}$, starting from the first shot. Determine the largest value of $n$ such that $p(n)>\frac{1}{5}$. | nvidia |
What is the maximum number of integers that can be included in the range $[-10, 10]$ such that the sum of any two of them is not equal to 0? | nvidia |
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