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When two fair 8-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that the two digit number $ab$ (where $a$ and $b$ are digits) and $a$ and $b$ are all divisible by 4? | nvidia |
Find the sum of all integer divisors of $42$ that also divide $18$. (Recall that the divisors of an integer may be positive or negative.) | nvidia |
Let $S$ be the set of all positive integers $n$ such that $n^2$ has exactly 3 digits when written in base 5. Find the remainder when the product of all the elements in $S$ is divided by 25. | nvidia |
Let $n$ be a positive integer such that $n^2+1$ is divisible by the sum of the first $n$ positive integers. Determine $n$. | nvidia |
Let $P$ be a point on the circle with equation $x^2+y^2=2$. The tangents from $P$ to the parabola $y=x^2$ intersect the $x$-axis at points $A$ and $B$. Find the sum of the squares of the distances from $P$ to the points $A$ and $B$. | nvidia |
Find the largest integer $n$ such that the increasing sequence $\left\lfloor \frac{2^{k_1}+2^{k_2}+\ldots+2^{k_n}}{n}\right\rfloor$, where $k_i\geq 1$ are distinct integers for all $i$, contains all positive integers. | nvidia |
Find the sum of $a$ and $b$ in the expression $\frac{\sqrt{a}}{b}$, which represents the positive difference between the roots of the quadratic equation $2x^2 + 5x - 6 = 0$, where $a$ is an integer not divisible by the square of any prime number and $b$ is an integer. | nvidia |
In a regular polygon, the sum of the interior angles is equal to the sum of the exterior angles. Determine the number of sides of this polygon. | nvidia |
Let $A$ and $B$ be real numbers such that $\frac{A}{x-5}+B(x+1)=\frac{-3x^2+12x+22}{x-5}$. What is $A+B$? | nvidia |
What is the smallest positive integer $n$ such that the sum of the first $n$ odd integers is divisible by $1234$? | nvidia |
Let $S$ be the sum of all positive integers $n$ such that $n^2+12n-2007$ is a perfect square. Find the remainder when $S$ is divided by $1000.$
Inspired Problem:
Find the sum of all positive integers $n$ such that $n^2+12n-2007$ is a perfect cube. Determine the remainder when this sum is divided by $1000$. | nvidia |
Find all values of $a$ such that the inequality $\frac{2a}{x+2}+3>1$ has real solutions for $x$ in the interval $[-5, 0)$. Express your answer in set notation. | nvidia |
Find the positive integer $n$ such that
$$\arctan\frac {1}{3} + \arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.$$
Write another problem inspired by this one:
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
Find the largest value of $y$ that satisfies the inequality $|2y-5|\leq 11$. | nvidia |
The equation $x^3-ax^2-bx+c=0$ has three distinct integer roots, and the sum of the squares of the roots is $a^2$. Find the sum of all possible values of $a$. | nvidia |
What is the minimum number of sides a polygon can have such that the sum of its interior angles is greater than 180 degrees? | nvidia |
Suppose that $a$ and $b$ are positive integers such that $(a+bi)^2 = 3+4i$. What is $a+bi$? | nvidia |
Let $A$ be a $4 \times 4$ matrix whose entries are $\pm 1$. Suppose that the sum of the entries in each row, column, and diagonal is $0$. Find the largest possible value of the sum of the entries of $A^{-1}$. | nvidia |
Problem:
Determine the value of $a$ such that the product of the roots of the cubic equation $x^3-2ax^2+(a-1)x+2=0$ is equal to half the sum of the roots. | nvidia |
Find the sum of all the positive integers from 1 to 99 that are not divisible by 3. | nvidia |
Find the number of integer values of $x$ such that $\frac{1}{x^2} < \frac{1}{100}$.
Can you find a solution to this new problem? | nvidia |
Determine the number of positive integers n such that the expression $n^2+3$ is between 5 and 50, inclusive. | nvidia |
Find all integers $x$ such that $x^2 + 5x + 7$ and $x^2 + 5x + 9$ are both prime numbers. | nvidia |
A number $x$ is equal to $7\cdot24\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? | nvidia |
Determine the sum of the digits of the smallest repeating block in the decimal expansion of $\frac{13}{23}$. | nvidia |
Find all real numbers $a$ such that the roots of the polynomial
$$x^3 - 6x^2 + 21x + a$$form an arithmetic progression and are not all real. | nvidia |
Find the sum of all numbers $x$ such that $x^2 + 25x^{-2} = 18.$ | nvidia |
Find the positive integer $m$ such that
$$\arctan\frac{1}{2} + \arctan\frac{1}{5} + \arctan\frac{1}{7} + \arctan\frac{1}{m} = \frac{\pi}{3}.$$ | nvidia |
Given that $x$ is real and $x^3+\frac{1}{x^3}=52$, find $x+\frac{1}{x}$. | nvidia |
Consider a bookshelf with 5 shelves, and 6 books of different sizes that can be arranged on these shelves. How many ways can the books be arranged such that each shelf contains at most one book? | nvidia |
Find the smallest positive integer that is a multiple of 27 and has exactly 5 prime factors. | nvidia |
A set of five positive integers has a median of 12 and a mode of 15. If the sum of the integers is 70, what is the smallest possible value of the largest integer in the set? | nvidia |
Find the remainder when the sum of all positive integers from 1 to 99 is divided by 100. | nvidia |
Let $a$ and $b$ be positive integers such that $b \le 100.$
Find the number of positive integers $n$ such that
$\ds\sqrt {\ds\f{a^2}{n^2}+b^2}$ is an integer. | nvidia |
A point is randomly chosen on a line segment of length 12. The probability that the lengths of the two segments formed are in the ratio 4:64 is k/7, where k is an integer. Find k. | nvidia |
Given a positive integer $n$, find the largest constant $c_n$ such that for any real numbers $x_1, x_2, \ldots, x_n$ satisfying $x_i > 0$ and $\sum_{i=1}^{n} x_i = 1$, the inequality $$\sum_{i=1}^{n} \frac{x_i}{\sqrt{1 - x_i}} > c_n$$ holds. | nvidia |
In a regular hexagon $ABCDEF$, a segment $AD$ is drawn, connecting two non-adjacent vertices. A point $P$ is chosen on segment $AD$ such that the area of triangle $ABP$ is equal to the area of triangle $FDP$. If the side length of the hexagon is 10 cm, find the length of segment $AP$. | nvidia |
What is the sum of all real numbers that are tripled when added to their squares? | nvidia |
Find the units digit of the product $(7^{6n+1}+1)(13^{2n+1}+5)$ for any positive integer $n$. | nvidia |
Given the polynomial $P(x) = x^4 + ax^3 + bx^2 + cx + d,$ where $a,b,c,d$ are positive integers. The equation $P(x) = 0$ has four distinct real roots, and the product of the roots taken three at a time is a constant. Find the smallest possible value of the constant. | nvidia |
Find the sequence of functions $\{f_n(x)\}$, with $f_n(x)$ being a polynomial of degree $n$, such that $f_n(x) + f_n(y) = f_n(x + y)$ for all real numbers $x$ and $y$, and the following conditions hold:
$f_0(x) = 1$
$f_1(x) = x$
$f_n(x)f_m(x) = f_{n+m}(x)$ for all non-negative integers $n$ and $m$
Determine the exp... | nvidia |
Find all real numbers x such that the inequality
$$\frac{x^2-2x+1}{|x-1|}\geq 1$$
holds true. | nvidia |
Find the largest integer $m$ such that for any $n > m$, there exists a prime number $p$ with $n < p < 2n$. | nvidia |
The largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$.
Find the largest value of $c$ such that the range of $f(x)=x^2+3x+c$ contains the interval $(-2, 1)$. | nvidia |
Find the value of $x,$ $0 < x < \pi,$ such that $\sin x = \sin \frac{13\pi}{12}.$ | nvidia |
Given that the point $(3,5)$ is on the graph of $y=g(x)$, there is one point that must be on the graph of $3y=\frac{g(3x)}3+1$. What is the product of the coordinates of that point? | nvidia |
Find a number consisting of three digits such that the sum of the cubes of its digits is 216, and each digit is larger than the one on its left.
Determine the sum of the digits of the largest integer with these properties. | nvidia |
Find the largest three-digit number that is divisible by each of the first three prime numbers. | nvidia |
For how many positive integers $x$ is $100 \leq x^2 \leq 200$? | nvidia |
What is the sum of all real numbers that are tripled when added to twice their reciprocals? | nvidia |
When three numbers are added two at a time the sums are 29, 46, and 53. What is the sum of all three numbers? | nvidia |
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? | nvidia |
Find the integer $n$, $10 \le n \le 15$, such that
$n \equiv 2311 \pmod{7}$. | nvidia |
Problem:
The equation $(x+y)^2 - (x-y)^2 = 5xy$ has two positive real solutions, where $x$ and $y$ are integers. If $(x+y)^4$ can be expressed in the simplified form $a^2 + b^2$ where $a$ and $b$ are positive integers, what is the value of the sum $a+b$? | nvidia |
Find the number of real solutions of the equation
\[\frac{x}{10} = \cos x.\] | nvidia |
Let $a$ and $b$ be two real numbers such that
$a^3+b^3 = a + b = 1$. Find $ab$. | nvidia |
Compute the sum of all the integers from 1 to 99 that are not divisible by 3. | nvidia |
How many positive integers $p$ less than 200 have a corresponding integer $q$ divisible by 4 such that the roots of $x^2-px+q=0$ are consecutive even positive integers? | nvidia |
In the equation
$$4x^3-5x^2-x+1=0$$
determine $k$ such that the product of the roots be equal to unity. | nvidia |
Find $y,$ given that $y$ is a nonzero integer and the numbers $\frac{1}{y},$ $y,$ and $y^2$ form a geometric sequence in that order. | nvidia |
The positive integer $x$ is an increasing function of the value of the digit $a$, where
$$x = \frac{a^3+19}{a}$$
and $a$ can take values from 1 to 9.
How many different integer values can the positive integer $x$ take? | nvidia |
Let $s(n)$ be the sum of the digits of a positive integer $n$ when written in base $b$. Find the smallest value of $b>2$ such that there exists a positive integer $m$ for which $s(m)=s(m^2)$. | nvidia |
Find all values of $c$ such that the range of
\[g(x) = \frac{cx^2 + 2x - 3}{x^2 + 2x + c}\]is the set of all real numbers. | nvidia |
Find the largest possible value of $x$ such that $3^x$ divides evenly into $13^{12}$. | nvidia |
Find $x$ such that $\sin(\frac{\pi}{6}) = \log_{10}(x)$ | nvidia |
Let $x$ be a number such that $5\%$ of $20\%$ of $x$ is equal to $10\%$ of $25\%$ of $x$. Find $x$. | nvidia |
Find all real numbers $x$ which satisfy \[\frac{x-1}{x-3} \ge 2.\](Give your answer in interval notation.) | nvidia |
What is the least positive integer that has exactly six positive factors and is divisible by 12? | nvidia |
Let $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ be a polynomial of degree $n$ with distinct zeroes $r_1,\ldots,r_n$. Define a polynomial $g(x)=x^n+b_{n-1}x^{n-1}+\cdots+b_0$ such that
\[g\left(r_j+r_j^2\right)=0\]for $j=1,\ldots,n$. Given that $a_{n-1}+b_{n-1}=100$ and $f(1)=g(1)=101$, determine the value of $b_0$. | nvidia |
Find the number of positive integers less than 1000 whose cube is a perfect square. | 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 smallest positive integer $m$ such that \[m \equiv 4321 \pmod{11}.\] | nvidia |
Find the sum of all solutions to the equation $(x-6)^2=25$. | nvidia |
Find the largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$. | nvidia |
Determine the number of ways to arrange the letters of the word ABCDEFGH such that no letter is in its original position. | nvidia |
Six 6-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal? | nvidia |
The number $2^{29}$ can be expressed as a 9-digit number $\overline{a_8a_7a_6a_5a_4a_3a_2a_1a_0}$ in base 10. Find the sum of the squares of the digits of $2^{29}$. | nvidia |
A circle of radius $10$ is tangent to two parallel lines $12$ units apart. Two points $A$ and $B$ on the circle are $15$ units apart and the line segment $\overline{AB}$ intersects the two parallel lines at points $C$ and $D$. Find the sum of the squares of the distances from the center of the circle to $C$ and $D$. | nvidia |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 312^\circ.$ | nvidia |
The squares of a chessboard are labelled with numbers from 1 to 64 in a specific pattern. The pattern is such that the sum of the numbers in each row, column, and diagonal is the same. How many different ways can the chessboard be labelled in this manner? | nvidia |
Let $x$, $y$, and $z$ be real numbers such that $\sin(x+y)=\cos(z+x)$ and $\cos(y+z)=\sin(x+y)$. Find the value of
$\frac{\sin(x+y+z)}{\cos(x+y+z)}$. | nvidia |
Find the sum of all the real solutions of $x^3 - 2x^2 - 5x + 10 = 0$, given that two of the solutions are integers. | 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 |
A certain polynomial $P(x)$ has a degree of $n$, where $n$ is a positive integer less than or equal to 10. The coefficients of the polynomial are all positive integers less than or equal to 5. What is the largest possible number of distinct positive roots that $P(x)$ can have? | nvidia |
Given that the graphs of $y=f(x)$ and $y=g(x)$ intersect at $(1,1),$ $(3,3),$ $(5,5),$ and $(7,7),$ there is one point where the graphs of $y=f(3x)$ and $y=3g(x)$ must intersect. What is the sum of the coordinates of that point? | nvidia |
A segment of length 1 is divided into two parts by a point chosen at random. A circle with center at this point and radius equal to one of the parts is drawn. What is the probability that the area of the circle is greater than 1/4 but less than 1/2 of the area of the square with side 1? | nvidia |
Consider the set of all positive integers whose base-10 representation consists only of 1's and 2's. What is the smallest positive integer that is not in this set? | nvidia |
Let $S$ be the set of all points in the plane that can be reached by a sequence of rotations (about any point) and translations (by any distance in any direction), starting from a given point $P$.
Determine the subset of points $T$ in the plane that cannot be reached from $P$ by this process if there are two points $A... | nvidia |
Let $f(x)$ be a continuous function such that $f(0) = 0$ and for any positive real numbers $x$ and $y$ we have
\[ |f(x)-f(y)| = |f(x-y)|+|f(x)-f(x-y)|. \]What is the number of possible values of $f(1)$? | nvidia |
Let $g(n)$ be the number of positive divisors of $n$ that are not multiples of 3. Find the sum of $g(20), g(21),$ and $g(22)$. | nvidia |
Given that $3=x^3+\frac{1}{x^3}$, determine the sum of all possible values of $x+\frac{1}{x}$. | nvidia |
A triangular number is a positive integer which can be represented as the sum of all positive integers up to some number. For example, the number 10 is a triangular number because $10=1+2+3+4$. How many triangular numbers are there between 100 and 200? | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c=\pi/2$. Prove that
$$\sin a\sin b\sin c\leq\frac{1}{8}.$$ | nvidia |
Find the number of solutions to the equation $2x + 3y = 7$, where $x$ and $y$ are integers with $0 < x, y < 10$. | nvidia |
There exists a positive real number $x$ such that $ \cos (\arctan (x)) = x $. Find the value of $x^2$. | nvidia |
Determine the number of integer values for $n$ in the range $1 \leq n \leq 1000$ such that the expression $\frac{n(n+1)}{2}$ results in a perfect cube. | nvidia |
Let $\{b_k\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_mb_n+m+n$ for all positive integers $m$ and $n.$ Find the smallest value of $k$ such that $b_k$ is a multiple of 10. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(0) = 0$ and
\[f(x^2 + f(y)) = xf(y) + f(x)\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$ | nvidia |
Given the coordinates of $A=(1,0)$, $B=(0,1)$, $C=(-1,0)$, and $D=(0,-1)$ on the unit circle centered at the origin, calculate the sum of the dot products of the vectors $\overrightarrow{OA}$, $\overrightarrow{OB}$, $\overrightarrow{OC}$, and $\overrightarrow{OD}$ with the vector $\overrightarrow{OE}=(1,\sqrt{3})$. | nvidia |
How many four-digit numbers can be formed using only the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, such that the sum of the digits in each number is exactly 15, and the number does not contain any consecutive digits that are the same? | nvidia |
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