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Find the product of all the even divisors of a number that can be expressed as the sum of two consecutive perfect squares. | nvidia |
Given an 8-digit integer $ABCDEFGH$ where each letter represents a distinct digit between 1 and 9, find the number of ways to arrange the digits so that the numbers formed by $AB$, $CD$, $EF$, and $GH$ are all distinct and represent valid dates in the month-day format, where none of the formed dates are in the future a... | nvidia |
How many of the positive integers less than 100 have a prime factor that is one less than a perfect square? | nvidia |
Determine all real numbers x that satisfy the inequality
$$|x^2-2x+2|<2x.$$ | nvidia |
Count the number of nonnegative integer solutions to $x_1+x_2+\cdots+x_n=k$, where $n$ and $k$ are positive integers. | nvidia |
Determine the value of $a$ such that $a^{a^{a^{a}}}=a^{4a}$ | nvidia |
What is the largest integer that can be written as a sum of distinct powers of 2, such that each power of 2 is used exactly once and the sum is equal to a given decimal number with a repeating block of digits? | nvidia |
Find the smallest positive integer $n$ such that $2n^2+3n+1 \equiv 0 \pmod{11}$. | nvidia |
If the positive real numbers $a$, $b$, and $c$ satisfy $abc=1000$, compute \[ \dfrac {ab+bc+ca}{a^2+b^2+c^2}. \] | nvidia |
Compute the number of ordered pairs of integers $(a,b)$ with $1\le a<b\le 50$ such that $\sqrt{a}+\sqrt{b}$ is an integer. | nvidia |
Find the smallest positive integer $k$ such that for any two relatively prime positive integers $a$ and $b$ with $a < b$, the number $a^{bk} + b^{ak}$ is divisible by $a + b$. | nvidia |
A sequence of real numbers $a_1,a_2,a_3,...$ satisfies the condition that every term $a_n$ is equal to the sum of all previous terms that are smaller than $a_n$, except the first term $a_1$. How many different sequences of length 5 can be generated that satisfy the condition and contain only positive integers from 1 to... | nvidia |
Determine the value of $x$ in the expression $\frac{4^{6x}}{12^{2x}}$ such that the result is an integer. | nvidia |
Find the value of $x$ such that the sum of the squares of the roots of the equation $x^2 - 4x + 5 = 0$ is equal to 7. | nvidia |
Find the positive integer $n$ such that
$$\arctan\frac {1}{3} + \arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.$$ | nvidia |
Find the sum of all $y$ that satisfy the equation $\frac{4y}{y^2-4} = \frac{3y}{y-2} - \frac{1}{y+2}.$ | nvidia |
Find the greatest possible value of $x+y$ given that $x$ and $y$ are positive integers such that $\frac{1}{x} - \frac{1}{y} = \frac{1}{9}$. | nvidia |
Find the number of positive integers $n$ less than or equal to $30$ such that there exists a positive integer $m$ less than or equal to $30$ satisfying $nm \equiv 1 \pmod{31}$. | nvidia |
Let $a_1,$ $a_2,$ $\dots,$ $a_n$ be positive real numbers such that $\sum_{i=1}^na_i=1$. Find the minimum value of
\[\left(\sum_{i=1}^na_i\right)\left(\sum_{i=1}^n\frac{1}{a_i}\right)-n^2,\]over all positive integers $n.$ | nvidia |
Let $\mathbf{P}$ be a $3 \times 3$ matrix, with real entries, such that $\mathbf{P}^4 = \mathbf{I}$ and the trace of $\mathbf{P}$ is 0. Find the number of different possible determinants that $\mathbf{P}^2$ can have. | nvidia |
Find the smallest positive integer $n$ such that there exist distinct positive integers $a$, $b$, and $c$ satisfying $a + b + c = n$ and $ab + bc + ca = a^2 + b^2 + c^2$. | nvidia |
Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. Find the number of such distinct triangles whose area is a positive integer.
| nvidia |
Given the function $f(x)=3x^3+2$, find the value of $x$ so that $f^{-1}(x)=4$. | nvidia |
Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$. | nvidia |
Find the smallest positive integer $x$ such that $5x \equiv 1 \pmod{27}$. | 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 $xyz$? | nvidia |
Simplify $((5p+1)-2p\cdot4)(3)+(4-1\div3)(6p-9)$ to a much simpler expression of the form $ap-b$ , where $a$ and $b$ are positive integers. | nvidia |
Consider a sequence $b_1, b_2, \dots$ where $b_n$ is defined as the sum of all the divisors of $n$. Starting from $n=3$, if $b_n$ is odd, then the next term $b_{n+1}$ is defined as the sum of all the odd divisors of $n+1$, otherwise $b_{n+1}$ is defined as the sum of all the even divisors of $n+1$. Given that $b_1=1$ a... | nvidia |
How many ten-digit sequences are there such that every two adjacent digits are different and no digit is repeated? | nvidia |
Consider the set of all real numbers x that satisfy the following equation:
\[ \sqrt{(x-1)^2 + (x-2)^2} = \sqrt{(x-3)^2 + (x-4)^2} \]
Determine the number of elements in this set. | nvidia |
Let $(a, b, c)$ be an ordered triple of positive integers such that the number of divisors of $a$ is $1000$, the number of divisors of $b$ is $2000$, and the number of divisors of $c$ is $2000$. Find the number of such ordered triples. | 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 $a,b,c,d,e,f,g$ be positive integers such that $$\begin{split}
&\gcd(a,b,c,d,e,f,g)=1\\
&a+b+c+d+e+f+g=16
\end{split}$$
If $p(x)$ is a polynomial such that $$\begin{split}
p(x)&=(x-a)(x-b)(x-c)(x-d)(x-e)(x-f)(x-g)\\
&=x^7+Bx^6+Cx^5+Dx^4+Ex^3+Fx^2+Gx+H,
\end{split}$$ determine... | nvidia |
The sum of the squares of two consecutive integers is 61. What are these integers? | nvidia |
Given that $\ds \lfloor \ds\f{3x+5}{x+2}\rfloor=4$,
determine the smallest integer $x$ that satisfies the inequality. | nvidia |
Determine the largest positive integer $n$ such that the numbers $4n+3$ and $7n+1$ have exactly three common divisors. | nvidia |
Let $a$, $b$, $c$, $d$, and $e$ be positive real numbers such that $a+b+c+d+e=25$. Find the minimum value of the expression $a^2+b^2+c^2+d^2+e^2$ if we also know that $a^3+b^3+c^3+d^3+e^3=400$. | nvidia |
Find the equation of the curve that is the locus of all points equidistant from the origin and the line $y = 2x$. | nvidia |
Let $m$ be a positive integer. We define the function $\phi(m)$ to be the minimum positive integer $n$, if it exists, such that for all positive integers $a$ there exists an integer $b$ such that $ab \equiv 1 \pmod{n}$ implies $ab \equiv 1 \pmod{m}$. Otherwise we let $\phi(m) = -1$.
Prove that for every positive integ... | nvidia |
Determine the smallest whole number greater than 50 that has exactly 5 positive factors. | nvidia |
Find the largest positive integer $n$ such that the product $n(n+1)(n+2)(n+3)$ is divisible by 16. | nvidia |
Find all 6-digit numbers of the form $a\!b\!c\!d\!e\!f$ where each letter is a digit, that can be formed by using the digits 1, 2, 3, 4, 5, and 6 exactly once, and that are divisible by 11.
What is the largest possible value of $a+b+c+d+e+f$? | nvidia |
Given that $x$ is a real number, calculate the value of $\frac{x^5+5x^4+10x^3+10x^2+5x+1}{x^5+5x^4+10x^3+10x^2+5x}$ if $x\neq -1$. | nvidia |
Let $n$ be a positive integer, and let $b_0, b_1, \dots, b_n$ be a sequence of real numbers such that $b_0 = 25$, $b_1 = 48$, $b_n=0$, and $$ b_{k+1} = b_{k-1} - \frac{2}{b_k} $$for $k = 1, 2, \dots, n-1$. Find $n$. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that the quantity $(a^2+b^2+c^2)^2-(a^4+b^4+c^4)$ is fixed and equals $k$, where $k>0$. Find the maximum possible value of $a^2b^2+b^2c^2+c^2a^2$. Express your answer in simplest form in terms of $k$. | nvidia |
Find the product of all real values of $r$ for which $\frac{1}{2x}=\frac{r-x}{7}$ has exactly one real solution. | nvidia |
Point $P$ lies on the line $x= -3$ and is 10 units from the point $(5,2)$. Find the product of all possible $y$-coordinates that satisfy the given conditions. | nvidia |
What is the sum of all integer values $n$ for which $\binom{26}{13}+\binom{26}{n}=\binom{27}{14}$? | nvidia |
What fraction of the two-digit numbers that are divisible by 5 are also divisible by 3? | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 4f(x) y\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$
New problem:
Let $g : \mathbb{R} \to \mathbb{R}$ be a fu... | nvidia |
Find the number of four-digit numbers greater than 2000 that have at least one pair of consecutive digits that are the same. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 2.$ Find the minimum value of
\[\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}.\] | nvidia |
How many five-digit positive integers have a digit that appears at least twice? | nvidia |
Let $g$ be a function defined on the positive integers, such that
\[g(x+y) = g(x) \cdot g(y)\]for all positive integers $x$ and $y.$ Given $g(3) = 5$ and $g(6) = 20,$ find $g(12).$ | nvidia |
The cubic polynomial $ax^3+bx^2+cx+d$ has three real roots $x_1$, $x_2$, and $x_3$. If the product of the roots is equal to $-1$, the sum of the squares of the roots is equal to $4$, and the sum of the roots is equal to $-1$, determine the value of $b/a$. | nvidia |
Let H be the set of polynomials of the form
$$ Q(w)=w^n+d_{n-1}w^{n-1}+\cdots+d_2w^2+d_1w+25, $$
where $ d_1,d_2,\dots, d_{n-1} $ are integers and $Q(w)$ has distinct roots of the form $\frac{a}{b}+i\frac{c}{d}$ with $a, b, c, d$ integers and $b, d \neq 0$. How many polynomials are in $H$? | nvidia |
In a regular polygon with $n$ sides, each vertex is connected to its two adjacent vertices by two sides of the polygon. If the polygon is inscribed in a circle of radius $r$, and each side of the polygon has length $s$, find the least value of $n$ for which the length of the segment joining two opposite vertices is gre... | nvidia |
For a certain positive integer $n,$ there exist real numbers $x_1,$ $x_2,$ $\dots,$ $x_n$ such that
\begin{align*}
x_1 + x_2 + x_3 + \dots + x_n &= 1000, \\
x_1^4 + x_2^4 + x_3^4 + \dots + x_n^4 &= 512000.
\end{align*}Find the smallest positive integer $n$ for which this is possible. | nvidia |
Find the largest positive real number $d$, such that for all nonnegative real numbers $u$ and $v$,
\[\sqrt{uv} - d |u - v| \le \frac{u + v}{3}.\] | nvidia |
Find the number of ways to arrange the digits 1, 2, 3, 4, and 5 to form a five-digit number such that the sum of the digits in the even positions (second and fourth digits) is greater than the sum of the digits in the odd positions (first, third, and fifth digits). | nvidia |
The graph of the equation $4x^2-24x+5y^2+30y=0$ is a conic section. Determine the conic section that this equation represents, and then find the coordinates of its center. | nvidia |
Find the smallest positive integer $b$ such that $2^x + b^y$ is not prime for any positive integers $x$ and $y$. | nvidia |
Let A, B, C, and D be four complex numbers such that $|A|=|B|=|C|=|D|=1$. If the area of the quadrilateral ABCD is $\frac{35}{37}$, find the smallest possible value of $|A+B+C+D|^2$. | nvidia |
In the plane, there are two congruent, regular pentagons, $ABCDE$ and $FGHIJ$, such that $\overline{AB}\parallel \overline{FG}, \overline{BC}\parallel \overline{GH}, \overline{CD}\parallel \overline{HI}, \overline{DE}\parallel \overline{IJ},$ and $\overline{EA}\parallel \overline{JF}$. The x-coordinates of the vertices... | nvidia |
Let $x\Diamond n = \sqrt{x+\sqrt{x+\sqrt{x+...}}}$, where the expression contains $n$ nested square roots. Find the smallest positive integer $n$ such that $15\Diamond n = 16\Diamond (n-1)$. | nvidia |
A bag contains ten balls, some of which are red and the rest of which are yellow. When two balls are drawn at random at the same time, the probability that both balls are red is $\frac{1}{15}$. How many balls in the bag are red? | nvidia |
Find all real numbers $x$ which satisfy \[\frac{x-1}{x-3} \ge 2.\](Give your answer in interval notation.) | nvidia |
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 5. If $\frac{S_{2n}}{S_n}$ is a constant that does not depend on $n$, for all positive integers $n$, then find the first term. | nvidia |
Determine the number of integer values of $n$ between 1 and 1000 inclusive for which the decimal representation of $\sqrt{n}$ is a repeating decimal. | nvidia |
Given the quadratic equation $ax^2+bx+c=0$, determine $a$ and $b$ such that the sum and product of the roots are equal in magnitude but opposite in sign. | nvidia |
Find the minimum value of the expression $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a},$ where $a,$ $b,$ and $c$ are positive real numbers, and $abc = 8.$ | nvidia |
Given a triangle $ABC$, point $P$ lies on side $\overline{BC}$ such that $BP:PC = 1:3$. The area of triangle $ABP$ is 120 square units. The triangle $ABC$ is then reflected over line $AP$, resulting in triangle $A'C'B'$. Find the area of quadrilateral $ABB'A'$ formed by connecting corresponding vertices of the two tria... | nvidia |
Let $x$, $y$, and $z$ be real numbers such that
$\sin x + \sin y + \sin z = \cos x + \cos y + \cos z = 0.$
Find the sum of all possible values of $\cos (x+y+z)$. | nvidia |
Let $x$, $y$, and $z$ be three distinct positive integers such that $x+y+z=12$, and
$$xyz=\left(\prod_{\text{cyc}}(x+y)\right)-\left(\prod_{\text{cyc}}(x-y)\right).$$
Determine $x^2+y^2+z^2$. | nvidia |
What is the smallest positive integer $k$ such that the equation $12x \equiv 15x \pmod{k}$ has exactly three solutions in the range $0 \leq x < k$? | nvidia |
Let $f(x)$ be a quadratic polynomial such that $f(1)=0, f(f(2))=1$, and there exists $c\in\mathbb R$ satisfying $f(c)=c^2.$ Find all possible values of $f(0).$ | nvidia |
Let $S$ be the set of all sequences $(a_1,a_2,...,a_{2013})$ with $a_i\in\{1,2,3\}$ for each $i$. For each such sequence, let $n_i$ be the number of $i$'s in the sequence. Determine the smallest positive integer $k$ such that if $x,y,z$ are nonnegative real numbers satisfying
\[n_1x+n_2y+n_3z=k\]
for every $(a_1,a_2,..... | nvidia |
Evaluate the product \[ (n-1) \cdot n \cdot (n+1) \cdot (n+2) \cdot (n+3), \] where $n=2$. | nvidia |
Find the number of positive integers that are common factors of $32$ and $54$, excluding $1$ and the greatest common factor of the two numbers. | nvidia |
Of the numbers 1, 2, 3, ..., 15, which number has the greatest number of divisors (the dots mean that we are including all the integers between 1 and 15)? | nvidia |
Find all values of $x$ such that $\arctan x > \arccos x.$ | nvidia |
There are 3 complex numbers $a+bi$, $c+di$, and $e+fi$. If $b=1$, $e=-a-c$, and the sum of the numbers is $-i$, find $d+f$. | nvidia |
Find the smallest positive integer that has exactly three distinct prime factors, and when divided by the sum of these prime factors, the quotient is a prime number. | nvidia |
Find the largest integer $n$ such that $n^5$ divides the product of the first $n^2$ positive integers. | nvidia |
In a right triangle, the length of the hypotenuse is 20 feet and the area of the triangle is 30 square feet. Find the length of the shorter leg of the triangle. | nvidia |
Find the number of positive integers less than or equal to 100 that are not divisible by any of the prime numbers between 30 and 40. | nvidia |
A sequence of integers $b_1, b_2, b_3, \ldots$ is chosen so that $b_n = b_{n - 1} + b_{n - 2}$ for each $n \ge 3.$ The sum of the first $m$ terms of this sequence is $m$ for every positive integer $m.$ Determine the sum of the squares of the first $m$ terms of this sequence for every positive integer $m.$ | nvidia |
The points $(2,3)$, $(14,6)$ and $(8,k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the perimeter of the triangle is a minimum? | nvidia |
There are four complex numbers $z$ such that
\[z \overline{z}^3 + \overline{z} z^3 = 350,\]and both the real and imaginary parts of $z$ are integers. These four complex numbers are plotted in the complex plane. Find the area of the quadrilateral formed by the four complex numbers as vertices. | nvidia |
A circle intersects the line segment from $(1,2)$ to $(3,6)$ at its midpoint. Determine the radius of the circle, if the center of the circle is $2$ units away from the line that passes through the endpoints of the line segment. | nvidia |
Let $N$ be a three-digit number whose digits are distinct. When the digits of $N$ are reversed, a number $M$ is obtained such that $M-N=99k$, where $k$ is an integer. What is the smallest possible value of $k$? | nvidia |
Determine the remainder when the sum of the squares of the first 50 positive integers is divided by 4. | nvidia |
Problem:
A circle is inscribed in a regular hexagon with side length $\sqrt{3}$. The circle is tangent to all sides of the hexagon and the vertices of the hexagon are $(1,0)$, $(-1,0)$, $(0,1)$, $(0,-1)$, $(\frac{1}{2},\frac{\sqrt{3}}{2})$, and $(-\frac{1}{2},\frac{\sqrt{3}}{2})$. Find the area of the region inside the... | nvidia |
Let $x$ be a positive real number. Define $a_n = \sqrt{x+\sqrt{x+\ldots+\sqrt{x}}}$ where there are $n$ square root signs. Determine the value of $x$ for which the sum $\sum_{n=1}^\infty \frac{1}{a_n}$ converges to a finite value. | nvidia |
Determine the maximum number of acute interior angles that can be found in a convex polygon with 11 sides. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x + 2y = 1.$ Find set of all possible values of $\frac{1}{x} + \frac{1}{y}.$ | nvidia |
For the polynomial function \(g(x) = x^3 + 2x^2 + x - 1\) defined for all real numbers \(x\), find the number of distinct real solutions for the equation \(g(g(x)) = 2\). | nvidia |
Determine the number of five-dollar bills and ten-dollar bills in a wallet, given that there are 30 bills in total and the sum of the amounts on all the bills is $190. | nvidia |
Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$ Find the maximum value of $xy.$ | nvidia |
Find the largest positive integer n such that 1^2 + 2^2 +... + n^2 is the product of three distinct integers, one of which is a perfect square. | nvidia |
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