instruction_seed stringlengths 30 7.99k | source stringclasses 1
value |
|---|---|
Let $A, B, C, D$ be a four-digit positive integer with different digits such that the sum of the digits is $6.$ If $AB$ is divisible by $3,$ and $CD$ is divisible by $2,$ determine how many such numbers exist. | nvidia |
Given that $a$, $b$, and $c$ are positive real numbers such that $a+b+c=\pi$, $\sin a = \frac{1}{3}$, and $\cos b = \frac{1}{2}$, find the value of $\sin c$. | nvidia |
Let $x$ and $y$ be real numbers such that $x^2+y^2=3x-2y=10$. Given that the solutions $x$ are in the form $x = \frac{a \pm b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, what is the value of $a + b + c + d$? | nvidia |
A palindrome number is a number that remains the same when its digits are reversed. A four-digit palindrome number is in the form of $ABBA$, where $A$ and $B$ are digits. Find a four-digit palindrome number $ABBA$ such that $A^B$ is divisible by $B^A$. | nvidia |
Let $g(x)$ be a function defined for all real numbers $x,$ such that $g(1) = 1,$ and
\[g(a + b) = g(a)g(b) + 2ag(b) + 2bg(a)\]for all real numbers $a$ and $b.$ Compute $g(x)$ for all $x.$ | nvidia |
Find the largest number that can be expressed as a sum of two different members of the set {2, 4, 6, 8, 10}. | nvidia |
Find the value of digit $B$ that will make the number $246B$ divisible by $3$. | nvidia |
A rectangular prism has dimensions $a$, $b$, and $c$, where $a$ and $b$ are integers and $c$ is a positive real number. The surface area of the prism is $120$ square units, and the volume is $180$ cubic units. What is the positive difference between the greatest and least possible values of $a+b$? | nvidia |
Let $A$ and $B$ be points on a circle with center $O$, such that the arc $AB$ is $\frac{1}{5}$ of the circumference of the circle.
For how many points $C$ on the circle is the arc $AC$ longer than the arc $BC$? | nvidia |
Find the number of positive integers $n \le 1000$ that can be expressed in the form
\[\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor = n\]for some real number $x.$ | nvidia |
In a rectangular garden, the length is three times the width, and the perimeter is 120 meters. What is the sum of the area of the largest square that can be inscribed in the garden and the length of the diagonal of the garden? | nvidia |
The graph of $y=f(x)$ is shown below, with $1$ unit between grid lines. Assume $f(x)$ is defined only on the domain shown.
What is the sum of all integers $c$ for which the equation $f(x)=c$ has exactly $4$ solutions?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14... | nvidia |
Given an altitude $\overline{AE}$ in triangle $ABC$, determine the value of $x$ such that quadrilateral $FEDC$ is cyclic, where $F$ is a point on side $AB$ such that $\overline{AF}: \overline{FB}=x: 2x$, and $D$ is a point on side $AC$ such that $\overline{AD}: \overline{DC}=x: x^2$. | nvidia |
Find the smallest whole number $x$ that will satisfy the inequality:
$$\frac{2x-1}{x+1} > \frac{x-2}{x+3}$$ | nvidia |
The product of two positive numbers is 1, and the sum of their fourth roots is 4. Find the sum of the numbers. | nvidia |
The sum of the squares of two consecutive even numbers is 100. Find the difference between the numbers. | nvidia |
How many different positive integer values of $y$ will make this statement true: there are exactly $3$ positive divisors of $y$ that are also prime numbers? | nvidia |
Find the sum of the reciprocals of the divisors of $120$. | nvidia |
For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?
Inspired problem:
Let $p_1,p_2,...,p_k$ be the first $k$ prime numbers, and let $n$ be a positive integer. For which value of ... | nvidia |
Find all real numbers $x$ for which the expression $\ds\sqrt{x^2+\sqrt{x^4+4}}$ is equal to $\ds\frac{x^2+1}{\sqrt{x^2+1}}$ | nvidia |
Consider a sequence of fractions where each term has the form $\frac{n}{n+1}-\frac{n-1}{n}$ for positive integers $n$ starting from 2. Determine the sum of the first 200 terms of this sequence. | nvidia |
Evaluate the product \[ (n-1) \cdot n \cdot (n+1) \cdot (n+2) \cdot (n+3), \] where $n=2$. | nvidia |
Determine the number of positive integers that are not perfect squares and have a square root that lies between 7 and 15. | nvidia |
Find the value of $k$ such that
$(4^{1003}+2^{1003})^2-(4^{1003}-2^{1003})^2$
is equal to
$k\cdot8^{1003}$. | nvidia |
Find all the three-digit numbers that have three digits with a difference of 2 when read from left to right. What is the sum of all the possible numbers you have found? | nvidia |
Find the number of distinct equilateral triangles that can be formed using the vertices of a regular octagon. | nvidia |
Let $a_1, a_2, \ldots, a_{20}$ be the distinct numbers in the set $\{1,2,3,\ldots,20\}$ such that the sum $a_1+a_2+\ldots+a_{20}$ is divisible by $10$.
Find the largest possible value of the product $a_1 a_2 \ldots a_{20}$. | nvidia |
A sequence $a_n$ of positive integers is defined by the formula $a_n = (n^2 + n + 1)(n^2 + n)$. Determine $n$ if the sum of the cubes of three consecutive terms of this sequence is equal to $27\cdot a_{n+1}$. | nvidia |
Find the sum of all the roots of the equation $z^{12} - 1 = 0$, where $z$ is a complex number. | 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 |
Let $n$ be a natural number such that when $n$ is written in base prime, the exponents form an arithmetic sequence, i.e., $a, a+d, a+2d, \ldots, a+(k-1)d$, where $a$ is the exponent of the smallest prime and $d$ is the common difference. Find all possible values of $n$ such that the sum of its digits in base prime is e... | nvidia |
Let's create a new problem inspired by the given one.
Problem:
Determine the smallest positive integer $n$ such that the product of $n$ and any integer greater than $n$ is divisible by $24$ and $90$. | nvidia |
Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\overline{CD}$ and vertices $G$ and $H$ on the circle. If the area of square $ABCD$ is $1$, then the area of square $EFGH$ can be expressed as $\frac {m}{n}$ where $m$ and $n$ are relatively prime positive integers and $m < n$. Find $10n... | nvidia |
Consider the set $T$ of all positive real numbers. The function $g : T \to T$ satisfies the following two properties:
(i) First,
\[g \left( \frac{x}{2} \right) = \frac{1}{2} g(x)\]for all $x \in T.$
(ii) Second,
\[g \left( \frac{x}{2} \right) + g \left( \frac{y}{2} \right) = 1 + g \left( \frac{x + y}{2} \right)\]fo... | nvidia |
If the sum of two numbers is $-\sqrt{2}$ and the sum of their squares is 6, find the product of the two numbers. | nvidia |
Find the number of ways to arrange the letters of the word "ABCDDDD" in a row such that no two identical letters are adjacent. | nvidia |
Find the minimum value of $a$ for which the equation
$$\left(\frac{x^2+1}{x+2}\right)^2-\left(\frac{x^2+1}{x+2}\right)-a=0$$
has exactly one solution. | nvidia |
Given that $A=2+3i$, $B=-1-i$, and $C=3+2i$, find the complex number $X$ such that $(A+B+C)X=A(B+C)$. | nvidia |
Find the number of positive integral values of $b$ for which $y = 1$ is the only positive integer solution of the system of inequalities
$$
\begin{cases}
3y > 2y + 5 \\
2y - b > -3
\end{cases}
$$ | nvidia |
Find the sum of the values of $y$ which satisfy $y^2 +2023y = 2024$. | nvidia |
Determine $a$, given that the product of the roots of the equation $x^2-10x+a=0$ is 7 greater than the sum of the roots. | nvidia |
Find the median of the set of positive divisors of 72. | nvidia |
Let $f(x)=3x-1$ and $g(x)$ be a function such that $$f(g(x)) = \frac{x-2}{x+2}.$$ Find all possible functions $g(x)$. | nvidia |
Find the value of $a$ such that the expression
$\left(\sqrt{\left(\sqrt{a}\right)^{\sqrt{a}}}\right)^{\sqrt{a}}$
equals $a$. | nvidia |
Find all $a$, $0 < a < 2\pi$, such that $\sin a$, $\sin 2a$, and $\sin 3a$ form a geometric sequence, in that order. | nvidia |
Let $a$, $b$ and $c$ be positive integers such that \[\begin{aligned} a+b+c&=10 \\ ab+bc+ca&= 37 \\ abc&=84.\end{aligned}\]
Find the value of $a^3+b^3+c^3$. | nvidia |
If $x$ and $y$ are positive real numbers such that $5x^2 + 10xy = x^3 + 2x^2 y,$ what is the value of $x$? | nvidia |
What is the probability that a randomly chosen positive integer from the set {1, 2, 3,..., 100} is a divisor of at least one of the numbers 36, 48, or 60? | nvidia |
Let $f(x)$ be a real-valued function such that
\[f(x) f(y) - f(xy) = x + y\]for all real numbers $x$ and $y.$ Find $f(2).$ | nvidia |
Let $g(x)$ be a function satisfying $g(x+y) = g(x)+g(y)$ for all real numbers $x$ and $y$. Let $n$ be a positive integer. Given that $g(x)=x^{n+1} g(1)$ for all positive integers $x$, find $g(5)$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the maximum value of $x^2y + y^2z + z^2x$. | nvidia |
A portion of the graph of $y = g(x)$ is shown in blue below, where $g(x)$ is a quadratic function. The distance between grid lines is $1$ unit.
What is the product of all distinct numbers $x$ such that $g(g(g(x)))=4$?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0... | nvidia |
Find the minimum value of
\[x^2 + xy + y^2\]over all real numbers $x$ and $y.$ | nvidia |
Find the sum of the squares of the coefficients in the expansion of $(x+y)^{12}$. | nvidia |
Let $f(n)$ be the sum of all the divisors of a positive integer $n$. If $f(f(n)) = n+2$, then call $n$ superdeficient. How many superdeficient positive integers are there? | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 7$ and $x^2y^2 = 25$. Find the value of $\frac{1}{x} + \frac{1}{y}$. | nvidia |
Let $P$ be a polynomial with real coefficients such that $P(-1) = P(2) = 0$ and $P(-1)P(x) = P(x)P(2)$ for all real numbers $x$. Determine the least possible degree of $P(x)$. | nvidia |
Find the sum of all the even numbers between 200 and 400 that are divisible by 6. | nvidia |
Alice chose five positive integers and found that their product was even. What is the maximum number of odd integers she could have chosen?
New problem:
In a set of five integers, the sum of the reciprocals of any two integers is always an integer. What is the maximum number of non-multiples of 5 that can be in the s... | nvidia |
Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. | nvidia |
Consider the set of numbers $S = \{\frac{1}{2^k}: k~ \text{is an integer}, 0 \le k \le 1000\}$. Given that the sum of all the elements in $S$ can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are coprime positive integers, find the number of digits in the decimal representation of $p$. | nvidia |
The sequence $a_1$, $a_2$, $a_3$, $\ldots$ is defined recursively as follows:
$a_1=1$,
$a_2=4$,
$a_n=a_{n-1}+a_{n-2}$, for all $n>2$.
Find the largest $n$ such that $a_n<100$. | nvidia |
Let $a$, $b$, and $c$ be complex numbers such that
$a+b+c = i^4 + i^8 + i^{12}$,
$ab+ac+bc=i^8+i^{20}+i^{12}$,
and
$abc=i^{12}+i^{24}+i^{36}$.
Determine the value of $|a+b+c|^2$, knowing that
$|a|=|b|=|c|$ and that $a$, $b$, and $c$ form the vertices of an equilateral triangle on the complex plane. | 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 |
How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999?$ For example, both $121$ and $211$ have this property.
$\mathrm{\textbf{(A)} \ }226\qquad \mathrm{\textbf{(B)} \ } 243 \qquad \mathrm{\textbf{(C)} \ } 270 \qquad \mat... | nvidia |
Let $$x,$$ $$y,$$ and $$z$$ be nonnegative real numbers such that $$x+y+z=1.$$ Find the maximum value of
\[\sqrt{\frac{x}{y}} + \sqrt{\frac{y}{z}} + \sqrt{\frac{z}{x}}.\] | nvidia |
Let $P$ be a polynomial of degree 4 with real coefficients such that $P(-1)=P(1)=P(2)=P(4)=0$.
Evaluate $P(-2)P(3)$. | nvidia |
How many odd three-digit numbers are there such that the sum of the hundreds and tens digits is 8? | nvidia |
In the parabola $y=ax^2+bx+c$, it is known that the coordinates of the vertex are $(2,-6)$ and the value of $y$ for $x=-1$ is $-4$.
Determine the equation of the parabola. | nvidia |
Let $\overline{PQ}$ be a diameter of a circle with diameter 2. Let $D$ and $E$ be points on one of the semicircular arcs determined by $\overline{PQ}$ such that $D$ is the midpoint of the semicircle and $PQ = \frac{4}{7}$. Point $F$ lies on the other semicircular arc. Let $x$ be the length of the line segment whose end... | nvidia |
The function $f(x)$ is defined as the product of all the positive odd integers up to $x$. Determine the largest integer $n$ such that $2^n$ divides $f(x)$ for all sufficiently large values of $x$. | nvidia |
A cube is inscribed in a right rectangular prism in such a way that the edges of the cube are parallel to the edges of the prism. If the dimensions of the prism are $x$, $y$, and $z$, and the volume of the cube is 27 cubic units, find the value of $x+y+z$ if the cube touches the sides of the prism at the midpoints of i... | nvidia |
Let $\mathbf{u}$ and $\mathbf{v}$ be vectors such that
$\mathbf{u}\cdot \mathbf{v}=8.$ Find the vector $\mathbf{x}$ such that for any vector $\mathbf{y},$
\[(\mathbf{x}-\mathbf{y})\cdot(\mathbf{u}+\mathbf{v})=(\mathbf{x}+\mathbf{y})\cdot(\mathbf{u}-\mathbf{v}).\] | nvidia |
Find all real values of $x$ that satisfy $\frac{x(x+1)}{(x-4)^2} \ge 12.$ (Give your answer in interval notation.) | nvidia |
The sum of two positive integers is 50 and their difference is 12. What is the value of the positive difference of the squares of the integers? | nvidia |
Let $f(x) = x^2 - \lfloor{x}\rfloor^2$. Find the smallest positive real number $k$ for which there exists a real number $x$ such that $f(x) > k$. | nvidia |
Determine the value of $x$ that satisfies $\sqrt[5]{x\sqrt{x^3}}=3$.
Let's create another problem similar to this one:
Determine the value of $y$ that satisfies $\sqrt[7]{y\sqrt{y^5}}=5$. | nvidia |
Find the area of the triangle with vertices $(3,-5),$ $(-2,0),$ and $(1,-6).$ | nvidia |
Compute $j^{-100}+j^{-99}+j^{-98}+\cdots+j^{-1}+j^0+j^1+\cdots+j^{99}+j^{100}$, where $j$ is a complex number such that $j^2=-1$ and $j^3=j$. | nvidia |
Determine all positive integers less than or equal to $30$ such that when divided by $5$ the remainder is $3$, and when divided by $7$ the remainder is $4$. Find the product of these numbers. | nvidia |
Find the value of $x$ that satisfies the equation $\sin 2x = \cos x$. | nvidia |
Let $a$ and $b$ be two distinct positive integers such that their sum is equal to $n$. Find the smallest value of $n$ for which there exists at least one positive integer $m$ such that the polynomial $x^m - x + a$ is divisible by $x^b - x + b$, and also there exists at least one positive integer $k$ such that $x^m - x ... | nvidia |
Let $f(x)$ be a polynomial such that for all integers $n$, the number $f(n)$ is equal to the LCM of $n$ and $2n+1$ multiplied by the GCD of $n$ and $2n+1$. Find the remainder when $f(100)$ is divided by $5$. | nvidia |
Let $f(x)$ be a function defined for all positive integers $x$ as follows:
\[f(x) = \left\{\begin{aligned} 2x & \quad \text{ if } x \text{ is even} \\ 1 + f(x + 1) & \quad \text{ otherwise}. \end{aligned} \right.\]
Find the smallest positive integer $x$ such that $f(x) > 100$. | nvidia |
Find all possible values of x that satisfy the equation:
∣x + 3∣ + ∣x - 5∣ = 7
Compute the sum of all valid x values. | nvidia |
Find the smallest positive integer $k$ such that the geometric sequence $1$, $k$, $k^2$, $\ldots$ contains two consecutive terms that add up to a perfect square. | nvidia |
Find the number of positive integers $n$ such that $n^2 + 2n + 1$ is a perfect square and $20 < n < 150$. | nvidia |
Find the value of $x$ such that $\sqrt{x+ 7} = 9$. | nvidia |
Find the maximum possible value of the product
\[\frac{3x}{y} \cdot \frac{5y}{7z} \cdot \frac{11z}{13x},\]
where $x,$ $y,$ and $z$ are positive real numbers. | nvidia |
Let \(a\) and \(b\) be positive integers such that \(a>b\) and the only positive divisors of \(a^a b^b\) are \(1,a,b\) and \(ab\). What is the remainder when the sum of all positive integer divisors of \(a+b\) is divided by 7? | nvidia |
Find the number of positive multiples of 11 that are less than 1000 and have a digit sum of 18. | nvidia |
Find the largest three-digit number that is divisible by each of the first three prime numbers. | nvidia |
Find the largest prime factor of the sum of the squares of all positive integers less than or equal to 25. | nvidia |
Find the sum of all integers $a$ such that the polynomial $x^3 + ax^2 + 12$ can be expressed as $(x + r)^3$ for some integer $r$. | nvidia |
How many positive integers less than 1000 can be written in the form $3^n+2^n$, where $n$ is an integer? | nvidia |
In the equation
$$x^3 + bx^2 + cx + 7 = 0$$
the coefficients $b$ and $c$ are both integers such that the roots of the equation are in arithmetic progression and the product of the roots is equal to $7$. Determine $b$. | nvidia |
What is the largest integer that is a divisor of \[
(n+1)(n+3)(n+5)(n+7)(n+9)
\]for all positive even integers $n$? | nvidia |
Determine the number of positive integers n such that the number of positive divisors of n equals the number of positive divisors of n+1. | nvidia |
Find the last two digits of the number $7^{8888}$. | nvidia |
Let $a_n$ be a sequence such that $a_1=0$ and $a_{n+2}=a_{n+1}-\frac{1}{a_n}$ for $n \ge 1$.
Find the number of positive integers that do not appear in the sequence $a_n$ | nvidia |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.