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The equation of the line passing through the points $(1,1)$ and $(9,5)$ is $y=x/2+1/2$. Find $b$ so that the line $y=mx+b$ is perpendicular to this line and passes through the point with coordinates $(5,3)$. | nvidia |
Find the number of positive integers less than 1000 that have a remainder of 7 when divided by 11. | nvidia |
Find the smallest positive integer $k$ such that $4k$ is a perfect fifth power and $3k$ is a perfect seventh power. | nvidia |
The sum of the squares of three consecutive positive integers is 7805. What is the sum of the cubes of the three original integers? | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $\gcd(a,b)=1$ and $ab=c^n$ for some positive integer $n$. If $a+b$ has 14 digits, then what is the maximum number of digits that $\gcd(a,c)$ can have? | nvidia |
Find the number of positive integers $n,$ $1 \le n \le 1000,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients, where the coefficients of the linear factors are also positive integers. | nvidia |
Let $a$ and $b$ be nonnegative real numbers such that
\[\cos (ax + b) = \cos 23x\]for all integers $x.$ Find the smallest possible value of $a.$ | nvidia |
The last two digits of a number n are 42. Determine the number of possible values of n mod 1000. | nvidia |
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum? | nvidia |
Let $p(x) = x^3 + ax + b$ be a polynomial such that all of its roots are integers. Find the smallest possible value of $b$ such that $p(x)$ has exactly two distinct roots.
The value of $b$ should be such that it is the smallest possible value that will make $p(x)$ have exactly two distinct roots. | nvidia |
What is the sum of all positive integers less than 100 that are squares of perfect squares? | nvidia |
In a triangle, the lengths of the sides are in the ratio 3:4:5. Find the length of the longest side if the perimeter of the triangle is 36 cm. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(0) = 0$ and
\[g(x^2 + y^2) = (x + y) (g(x) - g(y))\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(1),$ and let $t$ be the sum of all possible values of $g(1).$ Find $m \times t.$ | nvidia |
If $\alpha$ and $\beta$ are acute angles such that
$\sin{\alpha}=\cos{\beta},$
$\cos{\alpha}=1-\sin{\beta},$
and $\sin{\alpha}<\cos{\alpha},$ find the value of $\tan{\frac{\alpha}{2}}.$ | nvidia |
What is the smallest positive integer $n$ such that the sum of the digits of $5^n$ is divisible by 7? | nvidia |
Find the largest integer $n$ such that the equation $\lceil x \rceil^n - \lfloor x \rfloor^n = 2019$ has at least one solution. | nvidia |
In triangle $ABC$, a circle of radius $r$ with center $O$ is drawn so that it is tangent to $\overline{AB}$ and $\overline{BC}$. The angle $BAC$ is twice the angle $BCA$. Given that $r$ is a positive integer, find the maximum value of $r$ such that the triangle $ABC$ has integer side lengths. | nvidia |
Let $f(n)$ be the sum of all positive divisors of $n$. Determine the smallest integer $N$ such that there exist three distinct positive integers $a$, $b$, and $c$ satisfying $f(a) + f(b) + f(c) = N$. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a^2 + b^2 = 100.$ Find the maximum value of $a^2b^2.$ | nvidia |
Let $f(n)$ be the number of ways to arrange the numbers $1,2,\dots,n$ in a sequence such that each number is either larger or smaller than all the numbers to its left. If the average value of $f(n)$ over all positive integers $n$ is equal to $\frac{1}{5}$, find the value of $\lim_{n \to \infty} \frac{f(n)}{n}$. | nvidia |
A right pyramid with an octagonal base is placed on a coordinate plane so that its vertices are located at $(0,0)$ and $(4\sqrt{2},4\sqrt{2})$. A point $P$ is located on the edge of the pyramid so that it is 8 units from each of the two mentioned vertices. What is the length of the line segment from $P$ to the center o... | nvidia |
What is the largest positive integer $n$ such that $2^n$ divides $24!$? | nvidia |
The sum of the squares of three consecutive positive integers is 365. What is the middle integer of the three? | nvidia |
The length of a rectangle is given by (x+1), and its width is given by (x-2). A triangle has a base of (x+3) and a height of (x-1). If the area of the rectangle is three times the area of the triangle, find the sum of the possible values of x. | nvidia |
Compute
\[\sum_{1 \le a < b < c} \frac{1}{2^a 3^b 5^c}.\](The sum is taken over all triples $(a,b,c)$ of positive integers such that $1 \le a < b < c.$) | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $ab + bc + ca = 1$. Find the minimum value of
\[\frac{a^2 + b^2 + c^2}{abc}.\] | nvidia |
Find a nonzero $q$ such that $qx^2+20x+5=0$ has only one solution. | nvidia |
Find the sum of all values of $z$ that satisfy the equation $\frac{4}{2z^2+5z-2}+\frac{5}{z^2+6z+9}=1$. | nvidia |
Find all integer values of $a$ so that the polynomial
\[x^3 + 3x^2 + ax + 7 = 0\]has at least one integer root. Enter all possible values of $a,$ separated by commas. | nvidia |
Let $x,y,z$ be positive real numbers such that $x+y+z=12$ and $xy+yz+zx=36$. Let $M=\max\{x,y,z\}$. Find the smallest possible value of $M$.
Note: This problem requires a different approach than the original problem, as it asks for the smallest possible value of the maximum of the three numbers, rather than the larg... | nvidia |
Find all positive real solutions x to the equation \[x\cos{x}=\cos{(2x^2)}+\frac{1}{x}\]and calculate their product. | nvidia |
Let $a$ and $b$ be two positive real numbers such that $a^2+b^2=1$ and $a^3+b^3=\frac{1}{2}$. Find the value of $\frac{b}{a}$. | nvidia |
Let $N$ denote the number obtained by concatenating the decimal expansion of $\sqrt{2}$ and $\pi$. That is, $N=1.4142135623730951...15926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679...$. Determine the smallest positive integer $n$ such that $N^n$ contains exactly $33$ di... | nvidia |
Factor the expression $(x-1)(x^2+2x+1)+(x+1)(x^2-2x+1)$ and find the sum of the coefficients of the resulting polynomial. | nvidia |
A sequence of positive integers starts with 1 and has the property that every term after the first is the arithmetic mean of the previous term and a positive integer that is not necessarily part of the sequence. For example, if the sequence starts 1, 3, 6, ... then we can get the next term in the sequence by taking the... | nvidia |
How many positive three-digit integers less than 500 have at least two digits that are the same? | nvidia |
Consider the function $f(n)$ that counts the number of factors of a positive integer $n$. Compute the sum $f(1) + f(2) + f(3) + \cdots + f(20)$. | nvidia |
Find the smallest value of $x$ that satisfies the inequality $|2x-3| \leq 7$. | nvidia |
If $x$ and $y$ are positive integers such that $\gcd(x,y) = 15$, then what is the smallest possible value of $\gcd(12x,20y)$? | nvidia |
Count the number of positive integers smaller than $1{,}000{,}000$ that are powers of $3$ but not powers of $9$. You may find it useful to consider that $3^6=729$. | nvidia |
Find the value of $x$ that satisfies the equation:
$$\frac{2}{7}+\frac{8}{10}=\frac{x}{7}+\frac{2x}{10}$$
Good luck! | nvidia |
Find the number of ways to arrange the digits of the number 12345 into a 5-digit number such that no two consecutive digits are the same. | nvidia |
$n$ coins are simultaneously flipped. The probability that at most one of them shows tails is $\frac{3}{16}$. Find $n$. | nvidia |
The integers divisible by a natural number n are written on cards, one integer per card. When these cards are arranged in order, it is noticed that there is no card with the number divisible by 21 on it. How many cards are there? | nvidia |
Find the area of the triangle with vertices $(3,-5),$ $(-2,0),$ and $(1,-6).$ | nvidia |
Let $P(x)$ be a monic polynomial with integer coefficients such that $P(x)P(-x)=x^6-7x^4-5x^2+1$. Find the sum of all possible values of $P(1)$. | nvidia |
Determine the largest number that can be expressed as the sum of two distinct prime numbers such that their difference is less than 10. | nvidia |
Find the number of three-digit integers that are perfect squares and have only even digits. | nvidia |
Given an n-sided polygon with n>3, determine the maximum number of obtuse interior angles that can exist within the polygon. | nvidia |
In a trapezoid, the lengths of the two parallel sides are $12$ and $30$, and the distance between them is $10$. Find the length of the segment that connects the midpoints of the two non-parallel sides. | nvidia |
Find the sum of the series
$\sum_{k=1}^\infty \frac{1}{k^2}\left(1-\frac{1}{2^k}\right)$,
where $k$ ranges over all positive integers. | nvidia |
Consider the quadratic equation $x^2 + bx + c = 0$, where $b$ and $c$ are integers. If the equation has two distinct real roots and the product of the roots is $5$ times the sum of the roots, find the ordered pair $(b, c)$. | nvidia |
Let $g(x)$ be a function such that $g(x) + g(y) = g(xy)$ for all real numbers $x$ and $y.$ If $g(3) = 5,$ find $g(9).$ | nvidia |
In the triangle $ABC$, the points $D$ and $E$ are the feet of the altitudes from the vertices $A$ and $B$, respectively. It is known that $AD=BE$ and $BD:DC=1:2$. Find $\angle ACB$. | nvidia |
Find the largest possible value of $a$ such that the equation $ax^2+4x-7=0$ has two distinct roots, one of which is equal to the product of the roots of the quadratic equation $x^2+x-6=0$. | nvidia |
Determine the sum of all the perfect squares between 1 and 30 that are also palindromes. | nvidia |
Problem:
In a triangle $ABC$, the altitude from vertex $A$ to side $BC$ intersects the circumcircle of the triangle at a point $D$ other than $A$. Prove that the segment $AD$ is tangent to the circle that passes through the midpoints of the sides of the triangle $ABC$. Compute the radius of this circle.
Given: $AB = ... | nvidia |
Let $P(x)$ be a polynomial with integer coefficients, and let $r$ be a positive integer. If $P(x)$ has exactly $r$ real roots, and all of them are distinct integers, determine the largest possible number of integer solutions to the equation $P(x) \equiv 0 \pmod{r}$. | nvidia |
Let $w$ and $z$ be two complex numbers such that $|wz|=4$, $|w^3+z^3|=32$. Find the smallest possible value of $|w+z|$. | nvidia |
A certain polynomial has the following properties: the product of all its coefficients is $10$ and the sum of the products of every possible pair of its coefficients is $32$. If the product of the first and the last coefficients is $40$, find the sum of all the coefficients of the polynomial. | nvidia |
Determine the number of positive integers less than or equal to 100 that have exactly 5 factors. | nvidia |
A parabola passes through the points $(1,a)$, $(b,1)$, $(c,c)$ and $(d,d^2)$, where $a,b,c,$ and $d$ are real numbers, not necessarily in that order. Find all possible values of $d$, separated by commas. | nvidia |
Let $\mathbf{B}$ be a $3 \times 3$ matrix, with real entries, such that $\mathbf{B}^2 = \mathbf{I}$, where $\mathbf{I}$ is the $3 \times 3$ identity matrix. Find the number of different possible values that $\det(\mathbf{B})$ can be. | nvidia |
Find the sum of all positive integers such that their expression in base $8$ digits is the reverse of their expression in base $10$ digits, with the additional constraint that the sum of the digits in base $8$ is equal to the sum of the digits in base $10$. Express your answer in base $12$. | nvidia |
Find the minimum value of $b$ such that the inequality $(b-3)(b-7)>5$ is satisfied. | nvidia |
Let $P$ be the set of all points $(x,y)$ such that
$x=\ds\cos^2\alpha-2\sin\alpha\cos\alpha+\sin^2\alpha$ and
$y=\ds\cos^2\alpha+2\sin\alpha\cos\alpha+\sin^2\alpha$ for some real number $\alpha$.
Prove that all the points in $P$ lie on a line and determine the line's slope. | nvidia |
Let $f(n)$ be the sum of all the positive divisors of $n$, except $n$ itself. Compute $f(f(f(15)))$. | nvidia |
The three roots of the cubic $ 30 x^3 - 50x^2 + 22x - 1$ are distinct real numbers strictly between $ 0$ and $ 1$. If the roots are $p$, $q$, and $r$, what is the sum
\[ \frac{1}{1-p} + \frac{1}{1-q} +\frac{1}{1-r} ?\] | nvidia |
In the equation $x^2 + kx + 2 = 0$, determine the value of $k$ such that the product of the roots is equal to the sum of the squares of the roots. | nvidia |
If $x$ is a real number such that $3^{2x}+3^{-2x}=10$, find the integer part of $\log_{10}{x}$. | 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 |
Find the smallest positive integer $m$ such that $m^2$ is a multiple of $7^2$ and $m^3$ is a multiple of $5^3$. | nvidia |
Determine the real value of $a$ such that the function $f(x) = \frac{1}{x^2+a}$ has only one discontinuity point on the real axis. | 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 |
Problem:
The sequence $T=\{t_1,t_2,t_3,\ldots\}$ is defined by $t_1=3$ and $t_n=t_{n-1}+2^{t_{n-1}}$ for each integer $n>1$. Find the smallest positive integer $n$ such that $t_n$ is divisible by 11. | nvidia |
Problem:
Determine the value of $a$ such that $\log_a5 = 3\log_a2$. | nvidia |
Determine the number of positive integers n less than 100 such that \frac{n^2+1}{2n} is an integer when expressed as a decimal. | nvidia |
Find the number of distinct units digits of numbers of the form $3^n$, where $n$ is a positive integer. | nvidia |
The sequences of positive integers $1,a_2, a_3, \dots$ and $1,b_2, b_3, \dots$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$.
Write another problem inspired by this one:
Let $a... | nvidia |
Find the length of the altitude to the hypotenuse of a right triangle with legs of length $5$ and $12$, given that the area of the triangle is $30$ square units. | nvidia |
What is the value of $x$ for which $(2008+x)^2=x^2$?
Another problem inspired by this one:
The sum of the squares of two positive integers is 2008. The sum of the two integers is $x$. Find the value of $x$.
This problem requires a different approach, as it involves finding two integers whose sum of squares is 2008, ... | nvidia |
Find a positive integer that is divisible by 18 and whose cube root is a number between 8 and 8.1.
New Problem:
Find a positive integer that is divisible by 18 and has a square root that is a whole number. | nvidia |
The sum of the cubes of two consecutive integers is 91. What are the integers? | nvidia |
Determine the number of different combinations of letters that can be made from 5 letters, with the condition that exactly 2 of the letters must be vowels and the rest must be consonants, if there are 10 vowels and 12 consonants available for selection. | nvidia |
What is the sum of all positive integers less than 100 that are squares of perfect squares? | nvidia |
Find the number of divisors of 180 that are perfect cubes. | nvidia |
A person is standing at a point (x, y) on the edge of a circular lake, where x and y are positive integers. The person wants to reach a point (a, b) on the other side of the lake, where a and b are also positive integers. The person can only move along the edge of the lake, and the distance between the two points is th... | 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 |
Find all three-digit numbers that are divisible by 8 and whose digits form a geometric progression with common ratio greater than 1. | nvidia |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/29 of the original integer. | nvidia |
Let $a, b, c$ be complex numbers such that $a+b+c=ab+bc+ca=3$ and $abc=-1$. Find $a^2+b^2+c^2$. | nvidia |
Let $f(x)$ be a function such that
\[\sin^3 x = f(x)\cos 3x + 2 \cos x\]for all real numbers $x.$ Determine the value of $f(0).$ | nvidia |
Given the function $g(x) = x^2 - x + b$ for all real numbers $x$, where $b$ is some real number, find the value of $b$ such that the graph of the function $g(g(x))$ intersects the graph of $y=g(x)$ at exactly $2$ distinct points. | 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 |
Given that $2^2+3^2+5^2=38$ and $2^3+3^3+5^3=160$, find the value of $2^4+3^4+5^4$. | nvidia |
Find the smallest value of $k$ such that the polynomial $x^3 + kx^2 + 12x + 9$ can be expressed as the product of a quadratic polynomial with integer coefficients and a linear polynomial with a coefficient of $x$ equal to $1$. | nvidia |
Given a number $x$ that is equal to the product of two prime numbers $p$ and $q$, $x = p^m \cdot q^n$, what is the smallest positive integer $y$ such that the product $xy$ is a perfect square and also divisible by 5? | nvidia |
Let $p(x)$ be a polynomial of degree $n$ such that $p(0) = 0$ and $p(1) = 1$. Compute the sum of the coefficients of the polynomial $p(x)$. | nvidia |
Find the sum of all even integers between 20 and 50, inclusive. | nvidia |
The vertices of a regular polygon with 12 sides are colored in two colors: red and blue. Each side of the polygon is painted with a color that differs from the colors of its endpoints. Find the number of possible colorings if two colorings are considered equivalent if one can be obtained from the other by a rotation or... | nvidia |
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