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Given a positive integer $n$, find the sum of the squares of all positive integer divisors of $n$ if the sum of the divisors of $n$ is 78 and the sum of the divisors of $n$ excluding 1 and $n$ is 54. | nvidia |
Determine the number of distinct pairs of integers $(x, y)$ that satisfy the equation $5^x - 7^y = 27$. | nvidia |
For how many positive integral values of $b$ is it true that $y = 3$ is the only positive integer solution of the system of inequalities
\begin{cases}
3y>2y-4\\
2y-b>-5
\end{cases} | nvidia |
The set $\mathcal{S}$ consists of all non-negative real numbers. The function $f$ assigns to each finite subset $X$ of $\mathcal{S}$ a real number $f(X)$ so that the following two properties hold:
(i) $f(X)=f(Y)$ whenever $X$ and $Y$ are translates of each other.
(ii) For all nonempty subsets $X$ and $Y$ of $\mathcal{S... | nvidia |
Find the largest positive integer $m$ such that the polynomial $x^3+4x^2+mx+4$ is divisible by $x+2$. | nvidia |
Find the smallest integer $n$ such that the prime factorization of $n!$ contains exactly 10 different prime factors. | nvidia |
Let $a, b, c$ be complex numbers such that $a+b+c=2$, $ab+bc+ca=-1$, and $abc=5$. Find $a^2+b^2+c^2$. | nvidia |
Find the smallest positive integer $n$ such that the difference between the largest and smallest values of $\lfloor \sin(x) + \sin(2x) \rfloor$ on the interval $[0, n\pi]$ is greater than 10. | nvidia |
Find all values of $r$ such that $\lfloor r^2 \rfloor + r = 30.8$. | nvidia |
How many different 5-digit positive integers exist such that the first digit is not 0 and the last digit is even? | nvidia |
Find all real values of $x$ that satisfy $x + \frac{45}{x-4} = -10.$
Enter all the solutions, separated by commas. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x^2 + yf(z)) = xf(x) + zf(y)\]for all real numbers $x,$ $y,$ and $z.$
Let $n$ be the number of possible values of $f(5),$ and let $s$ be the sum of all possible values of $f(5).$ Find $n \times s.$ | nvidia |
For all positive integers $n$, what is the largest integer that always divides into the sum of $n$, $n^2$, and $n^3$? | nvidia |
Let $f(x)$ be a function such that $f(x) + f(y) = f(x + y) + 2xy$ for all real numbers $x$ and $y$. Find $f(5)$, given that $f(1) = 1$. | nvidia |
A rectangular prism has the property that the sum of the lengths of its three dimensions is equal to the sum of the areas of its three pairs of adjacent faces. What is the ratio of the length of the longest dimension to the length of the shortest dimension? | nvidia |
Find the smallest positive integer $n$ such that the product of all its positive divisors is equal to $2^{10}$. | nvidia |
Find the number of positive integers less than 100 that have a remainder of 3 when divided by 11. | nvidia |
Let the squares be colored in white and black in such a way that each column and row contains an even number of black squares (not necessarily different). Prove that the number of black squares is divisible by four. | nvidia |
Which of the following could NOT be the lengths of the external diagonals of a right regular prism [a "box"]? (An $\textit{external diagonal}$ is a diagonal of one of the rectangular faces of the box.)
$\text{(A) }\{4,5,6\} \quad \text{(B) } \{4,5,7\} \quad \text{(C) } \{4,6,7\} \quad \text{(D) } \{5,6,7\} \quad \text... | nvidia |
Find the number of integers less than 1000 that are divisible by 7 and have a sum of digits equal to 11. | nvidia |
Given that the point $(4,3)$ 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 sum of coordinates of that point? | nvidia |
Consider a set of seven distinct positive integers, where four of the integers are known: $10$, $21$, $15$, and $13$. The remaining three integers must be chosen from a separate pool of five positive integers, where the sum of these five integers is known to be $93$. What is the maximum possible median of the resulting... | nvidia |
Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 11? | nvidia |
In a triangle $ABC$, lines $AX$ and $BY$ intersect at $M$. How many such intersections of lines $AX$, $BY$, $CZ$, $DM$, $EN$, and $FP$ can exist at a single point $M$ in the triangle $ABC$, where $X$, $Y$, $Z$, $D$, $E$, $F$, and $P$ are all points on the sides of the triangle? (Assume that no two of the seven points... | nvidia |
Let $p(x)$ be a polynomial with integer coefficients such that $p(1)=7$, $p(2)=13$, and $p(3)=19$.
Find $p(5)+p(7)+p(11)$. | nvidia |
The vertices of a square are located at the points (1,0), (2,0), (2,1) and (1,1). A triangle is formed by connecting the bottom-left corner of the square to the top-right corner and a point on the diagonal of the square. Find the length of the longest side of the triangle if its area is equal to one-quarter of the area... | nvidia |
Determine all positive integers $n$ such that the quadratic equation $x^2 + (n - 1)x + n^2 - n + 1 = 0$ has roots that are conjugate complex numbers with rational imaginary parts. | nvidia |
Find the product of the distinct prime factors of the number that is the sum of all positive divisors of 72 excluding 1 and 72 itself. | nvidia |
Let $C$ equal the number of three-digit numbers that are divisible by 3. Let $D$ equal the number of three-digit numbers that are divisible by 7. Find $C + D$. | nvidia |
A sphere intersects a right circular cone of radius $2$ forming a circle. If the diameter of the circle is $25\%$ shorter than the radius of the cone, the length of the diameter is
$\textbf{(A)}\ 1\qquad \textbf{(B)}\ \frac{3}{2}\qquad \textbf{(C)}\ 1.5\qquad \textbf{(D)}\ \frac{15}{4}\qquad \textbf{(E)}\ 2$ | nvidia |
In a right triangle $DEF$ with right angle at $E$, a circle with diameter $EF$ intersects side $DF$ at point $G$. Given that the area of $\triangle DEF$ is $108$ and $DF = 18$, find the length of $EG$. | nvidia |
What is the least positive integer $n$ such that $4125$ is a factor of $n!$? | nvidia |
Determine $k$ such that the product
$$\log_a(a+k) \cdot \log_{a+k} (a+2k) \dotsm \log_{a+2k} (a+3k)$$
contains exactly $3$ terms and its value is $\log_a(2a).$ | nvidia |
In a certain organization, each employee has a unique ID number that is a positive integer between 1 and 1000. The IDs are assigned such that no ID is the sum of two other IDs. What is the maximum number of employees that can be assigned an ID under these conditions? | nvidia |
Find the smallest three-digit number that is divisible by 7 and has the sum of its digits equal to 9. | nvidia |
Given a positive integer n, find the smallest positive integer k such that 2^k - 1 is divisible by n. | nvidia |
Find the smallest positive integer $n$ such that the product of all positive divisors of $n$ is equal to $2^n$. | nvidia |
Find the value of $x$ such that $\frac{\log_{2} x + \log_{2} 5}{\log_{2} 3} = 3$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $3x + 2y = 24.$ Find the maximum value of $2xy.$ | nvidia |
Find the number of positive integer divisors of $5^3 \cdot 11^2$ that are also multiples of 25. | nvidia |
Find the largest value of $x$ that satisfies the equation $|x-5|=12$. | nvidia |
Find the value of $x$ such that $\log_{x^2+1} (3x-2) = 1/3$. | nvidia |
Consider the congruence $ax+b\equiv 0\pmod{m}$ where a, b, m are positive integers such that a and m are coprime. Find the least natural number n for which the given congruence has exactly n incongruent solutions modulo mn. | nvidia |
Find the number of positive integers n such that $n^2 + n$ is less than 100, but greater than 20. | nvidia |
Consider the function $f(x)=x^3+2x^2-5x+1$ and $g(x)=x^2-3x-2$. Find the value of $x$ such that $f(g(x))=g(f(x))$. | nvidia |
Find the value of $x$ such that $0.125\cdot\sqrt{x}=0.0625\cdot\sqrt[3]{x^2}$. | nvidia |
Find the smallest positive integer $x$ such that any $y$ that is relatively prime to $x$ is also relatively prime to $x^2+x+1$.
Express your answer in the simplest form. | nvidia |
Let $ABC$ be a triangle with vertices at points $A(2,3)$, $B(4,5)$, and $C(1,1)$. The centroid of the triangle is at point $G(x,y)$. If point $G$ lies on the line $y=2x-3$, find the coordinates of the midpoint of $\overline{AC}$. | nvidia |
For all integers $n$, what is the smallest integer that always divides into the sum of $n$ and its square, when this sum is greater than $2n$? | nvidia |
Let $A$ and $B$ be two points in the plane such that the midpoint of segment $AB$ lies on the unit circle. Given that the line through $A$ and $B$ is tangent to the circle, what is the maximum value of $\frac{1}{|AB|}$? | nvidia |
Find the smallest positive integer that is divisible by four distinct prime numbers. | nvidia |
Find all real values of $x$ and $y$ such that
\begin{align*}
\frac{1}{x}+\frac{1}{y}&=xy,\\
x+y&=\frac{1}{xy}.
\end{align*}
Determine the value of $x^2y^2$. | nvidia |
Let $S$ be the set of all positive integers that can be written in the form $a_4 \cdot 5^4 + a_3 \cdot 5^3 + a_2 \cdot 5^2 + a_1 \cdot 5 + a_0$, where the $a_i$'s are integers, and $0 \le a_i \le 4$. Determine the number of elements in the set $S$ such that the sum of their digits is divisible by $5$. | nvidia |
The function $g(n)$ is defined on the positive integers such that $g(g(n)) = n^2$ and $g(2n) = 2g(n) + 1$ for all positive integers $n.$ Find $g(256).$ | nvidia |
If $x$ is a positive number such that \[\sqrt{8x}\cdot\sqrt{10x}\cdot\sqrt{3x}\cdot\sqrt{15x}=15,\]find all possible values for $x$. | nvidia |
When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of five consecutive positive integers all of which are nonprime? | nvidia |
An integer $n$ is called a "divisor of all integers" if for every integer $k$, $k$ divided by $n$ results in an integer. Find the sum of all positive integers that are divisors of all integers. | nvidia |
Find the maximum possible value of $|\text{Re}(z_1z_2)+\text{Im}(z_3z_4)|$ where $z_1,z_2,z_3,z_4$ are the four roots of the polynomial $g(x)=x^4-4x^3+6x^2-4x+4$. | nvidia |
Find the largest positive integer $x$ which is less than $1000$ and has a GCD of $12$ with $180$.
(Note: This problem requires a different approach than the original problem, as it involves finding a number with a specific GCD rather than a number that is relatively prime.) | nvidia |
A square is inscribed in a circle with radius 5 cm. One of the vertices of the square lies on the positive x-axis. Find the side length of the square. | nvidia |
Let $P(x)$ be a monic quartic polynomial such that $P(\sqrt[3]{10}+1)=0.$ Let $Q(x)$ be the monic polynomial such that $Q(x)=(x-1)^4-P(x).$ Determine the sum of the coefficients of $Q(x)$. | nvidia |
Let $a, b,$ and $c$ be positive real numbers such that $a+b+c=1.$ Find the maximum value of
\[\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}.\] | nvidia |
How many four-digit numbers are there such that the sum of their digits is 10? | nvidia |
Find the maximum value of $2^x - 4^x$, over all real numbers $x$. | nvidia |
Find the minimum value of
\[x^2 + xy + y^2\]over all real numbers $x$ and $y.$ | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=18$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=6$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$?
Another problem:
Given that $$(x+y+z)(xy+xz+yz)=24$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=12$$for real numbers $x$, $y$, and $z$, find the value of $xyz$ such that $x$, $y$, and ... | nvidia |
Let $P$ be a point on the line
$$-5x+3y=25$$
such that $P$ is equidistant from the points
$(0,0)$ and $(4,12)$.
Find the coordinates of $P$. | nvidia |
Find the least positive integer $y$ that satisfies $y+1912 \equiv 1876 \pmod{15}$. | nvidia |
Let $p, q, r,$ and $s$ be positive integers such that $\gcd(p, q)=40$, $\gcd(q, r)=48$, $\gcd(r, s)=72$, and $110<\gcd(s, p)<130$. Which of the following must be a divisor of $p$?
$\textbf{(A)} \text{ 3} \qquad \textbf{(B)} \text{ 5} \qquad \textbf{(C)} \text{ 7} \qquad \textbf{(D)} \text{ 11} \qquad \textbf{(E)} \text... | nvidia |
Let $S_n$ denote the sum of the digits of the integer $n$. For example, if $n=145$, then $S_n=1+4+5=10$. There exists a smallest positive integer $N$ such that for every integer $m\ge N$, we have $S_m > S_{S_m}$. Find $N$. | nvidia |
How many positive three-digit integers have an even number of positive factors? | nvidia |
Find the sum of all integer divisors of $180$ that are not multiples of $3$. (Recall that the divisors of an integer may be positive or negative.) | nvidia |
Determine the product of all real numbers $x$ satisfying
\[(x^2-4x+2)^{x^2-5x+2} + (x^2-4x+2)^{x^2-5x+1} = 2.\] | nvidia |
Determine the complex number $z$ satisfying the equation $2z-3\bar{z}=-2-30i$. Note that $\bar{z}$ denotes the conjugate of $z$. | nvidia |
Find the number of real solutions of the equation
\[\frac{x^2}{100} = \cos x.\] | nvidia |
Find the value of $x$ such that $\left(\sqrt{3}\right)^x + \left(\frac{1}{\sqrt{3}}\right)^x = 2$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x) f(y) - f(xy) = x + y\]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 |
For each integer $n\geq 4$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. The product $a_4a_5 \dotsm a_{99}$ can be expressed as $\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$? | nvidia |
On the unit circle in the coordinate plane, point $A$ is at $(1,0)$, and point $B$ is at $(-1,0)$. There exists a constant $d$ such that for all points $P$ on the unit circle that are not $(1,0)$ or $(-1,0)$, the equation
$$\frac{|PA|}{\sin(\angle PBA)} + \frac{|PB|}{\sin(\angle PAB)} = d$$
holds. Find $d$. | nvidia |
In a triangular array of numbers, each row starts and ends with 1, and each internal element is the sum of the two elements diagonally above it. Find the sum of all the numbers in the row that starts with 1, then 5, if we know that the row contains an even number of elements. | nvidia |
Consider the operation $*$ such that for all $x$ and $y$, $x*y=x^2-2y$. If $(a*b)*c=4$ and $a*b=3$, determine the value of $c$. | nvidia |
Let $a$ and $b$ be the roots of the equation
$$x^2-4x+1=0.$$
Prove that $$\frac{a^2+b^2}{ab}+2=5.$$ | nvidia |
The integers $p$ and $q$ are such that $p$ is the smallest positive integer that can be expressed as the sum of two distinct perfect squares, and $q$ is the largest integer less than $200$ that can be expressed as the difference of two distinct perfect squares. What is $p+q$? | nvidia |
How many different positive, three-digit integers can be formed using the digits 1, 2, and 3, with the condition that the sum of the digits is 7? | nvidia |
Determine the number of real solutions to the equation
$$\cos\left(\ln x+\frac{\pi}{2}\right)=-\cos(\ln x).$$
where $x>0$. | nvidia |
Find the value of $b$ that satisfies the equation $253_{b}+427_{b}=1012_{b}$. | nvidia |
Compute the value of $x$ such that $\sqrt[4]{x^4+12960000}=2x+3$. | nvidia |
In the equation $x^3 + bx^2 + cx + d = 0$, find the sum of all possible values of $b$ for which the equation has exactly two distinct real solutions for $x$. | nvidia |
The diagonals of a rhombus $ABCD$ intersect at the point $O$. If the diagonals are perpendicular and one of them is twice as long as the other, find the side length of the rhombus, given that the area of the rhombus is $180$ square units. | nvidia |
Determine the number of positive divisors of $n=6^6\cdot8^8$ that are divisible by $6$ but not by $8$. | nvidia |
Find the sum of all values of $x$ such that $3^{x^2+2x-5} = 9^{x - 2}$. | nvidia |
Let $x$ be a real number such that $0 \leq x < 1$. Find the minimum value of $|\lfloor 1/x \rfloor - \lceil x \rceil|$. | nvidia |
Find the units digit of the sum of all the even positive integers between 20 and 120. | nvidia |
The numbers 18, 24, and 30 are all divisible by 6 and their prime factorizations have only 2's and/or 3's. Find the sum of all positive divisors of these numbers that are also divisible by 6. | nvidia |
Let $a$ and $b$ be complex numbers such that
$a^3 b^5 = a^7 b^2 = 1$ and $a \ne b$.
Compute the smallest positive integer $k$
such that $a^k = b^k = 1$. | nvidia |
What is the sum of all the positive integers less than or equal to $20$ that are relatively prime to $20$? | nvidia |
The equation $2x^2+ax+b=0$ has two distinct real roots, and the sum of the absolute values of the roots is $3$. If the sum of the squares of the roots is $5$, find the value of $b$. | nvidia |
Find the number of integers $n$ that satisfy
\[10 < n^2 < 99.\] | nvidia |
Find the sum of all positive integers that can be expressed as the product of two distinct prime numbers whose sum is 10. | nvidia |
Find the smallest positive integer $n$ such that there exist real numbers $a$ and $b$ satisfying the equations $\tan a=n$ and $\sec a + \tan b=3n$. | nvidia |
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