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Consider the following sequence: $a_1, a_2, a_3, \dots, a_n$, where each term $a_i$ equals either 1 or -1. The sequence satisfies the condition that the sum of any three consecutive terms is always 0 or 1. What is the minimum number of terms $n$ such that the sum of all the terms in the sequence is 0, and $n$ is divisi... | nvidia |
Let $x$ be a real number such that $4x^{\frac{1}{3}} + 9x^{\frac{2}{5}} = 13$. Find the value of $x^{\frac{3}{5}} + x^{\frac{2}{3}}$. | nvidia |
Find the largest integer $n$ such that the number of ones in the binary representation of $n$ is less than or equal to 20 and the number of zeros in the binary representation of $n$ is greater than or equal to 10. | nvidia |
Two fair, standard six-sided dice are rolled. Determine the smallest positive integer $n$ such that the probability that the sum of the numbers on the top faces is greater than or equal to $n$ is less than 0.25. | nvidia |
Simplify the expression $\sqrt{36}+\frac{\sqrt{288}}{\sqrt{2}}+5\sqrt{3}$ and represent it as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not a perfect square. | nvidia |
Let $R(x)=b_0+b_1x+\dots+b_mx^m$ be a polynomial with rational coefficients, where each coefficient can be written in the form $\frac{c}{2^d}$ with $c$ and $d$ integers and $0\le b_i<2$ for all $0\le i\le m$.
Given that $R(\frac{1+\sqrt{2}}{2})=5+4\sqrt{2}$, find $R(\frac{3}{4})$. | nvidia |
Find the value of $x$ such that the polynomial \[ x^3 - 3x^2 + 5x + 7 \] is equal to its own derivative. | nvidia |
Let $f(x)$ and $g(x)$ be two functions such that $f(2) = 8$, $g(3) = 27$, and $f(g(x)) = g(f(x))$ for all positive integers $x$. Find the value of $g(2) + f(3)$. | nvidia |
Let $g(x)$ be a function defined for all real numbers, and consider the equation
$$g(x) - 3g(-x) = x^2 + 1.$$
Find all real values of $x$ for which $g(x) = g(2x)$. | nvidia |
Find all real values of $x$ that satisfy the equation
\[\sin^2 \left(\frac{x^2 + 2}{x + 1}\right) = \frac{1}{2}.\] | nvidia |
In a set of 12 cards, there are 4 cards with the number 1, 4 cards with the number 2, and 4 cards with the number 3. If 6 cards are drawn randomly, what is the probability that there are exactly two numbers that appear an odd number of times? | nvidia |
Let $x,y,z,$ and $w$ be positive real numbers such that $x^3 + y^3 = z^3 + w^3$, $xy = zw$, and $x+y+z+w=18$. Find the maximum value of $(x+y)^2(z+w)^2$. | nvidia |
A circle with center $(2,9)$ passes through the point $(4,3)$. Find the sum of the coordinates of the point where the circle intersects the line $y=x$. | nvidia |
Given the vertices of a triangle as (0, 0), (4, 6), and (3, k), find the value of k such that the area of the triangle is 10 square units. | nvidia |
The sum of the absolute values of the roots of the equation x^2 + ax + b = 0 is equal to the square of the sum of their squares. Determine the value of a. | nvidia |
Let $a$, $b$, $c$ be positive integers such that $a^2b^3c^4 = 6^9$.
Find $a+b+c$. | nvidia |
Define a regular $n$-sided polygonal chain to be a sequence of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that
the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear,
each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint... | nvidia |
Problem:
Consider a matrix of the form
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}\]
where $a$, $b$, $c$, and $d$ are real numbers. Find the values of $a$, $b$, $c$, and $d$ such that the matrix is invertible, but its square is not.
Hint: Use the property that a matrix is invertible if and only if its determinant i... | nvidia |
A portion of the graph of $y = G(x)$ is shown in red below. The distance between grid lines is $1$ unit.
Compute $G(G(G(G(G(1)))))$.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tick... | nvidia |
The function
\[f(x,y,z)=x^2+y^2+z^2-kxyz\]
has a minimum value of 1 over all real numbers $x, y, z$ satisfying the condition
$x+y+z=3$.
Find the largest possible value of $k$. | nvidia |
Find the smallest positive integer $x$ which is greater than $1$ and relatively prime to $120$ (recall that relatively prime means that the GCD of $x$ and $120$ is $1$) | nvidia |
Find the smallest positive integer that is divisible by exactly five distinct prime numbers. | nvidia |
The polynomial
$$g(x)=x^{2007}+17x^{2006}+1$$
is divisible by $f(x) = x^3 + ax^2 + bx + c.$ Find the number of distinct values of positive integer $c$ such that
$$\max(a,b,c)\le 100.$$ | nvidia |
There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number? | nvidia |
The sum of the three smallest numbers in a set of consecutive positive integers is 72. If the largest number in the set is twice the smallest number, how many integers are in the set? | nvidia |
Determine $x$ such that $\sqrt[4]{(\sqrt{x})^3}=18$. | nvidia |
What is the smallest positive integer $n$ such that $\sin(\frac{\pi}{n})$ is less than 0.1? | nvidia |
Let $P(x)$ be a quartic polynomial with rational coefficients such that the product of the non-real zeros of $P$ is 56. Determine the minimum possible sum of the absolute values of the real zeros of $P$ if $P(x)$ satisfies $|P(1)| = 12$ and $|P(-1)| = 12$. | nvidia |
The decimal $0.\overline{27}$ can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find the value of $a^2 + b^2$. | nvidia |
Find the number of $y$-intercepts of the graph of the equation $x = \frac{1}{y^2+1}$. | nvidia |
Let $a=3^x$ and $b=2^x$, where $x$ is a positive integer. How many positive integers $n$ less than $ab$ have the property that $n$ is divisible by $a$, but not by $b$, and $ab-n$ is divisible by $b$, but not by $a$? | nvidia |
The lengths of the sides of a triangle with positive area are $\log_5 8$, $\log_5 27$, and $\log_5 m$, where $m$ is a positive integer. Find the number of possible values for $m$. | nvidia |
Find the number of unordered pairs of prime numbers that have a difference of 18. | nvidia |
The sum of two numbers is 14, and the difference of the cubes of each number is 204. What is the value of the sum of the squares of the two numbers? | nvidia |
Given that the determinant of a 2x2 matrix is 14, find the determinant of the matrix that results when the rows and columns of the original matrix are swapped, and then the matrix is multiplied by a scalar value of 3. | nvidia |
Find the sum of all positive divisors of $50$ that are also divisors of $15$. | nvidia |
Find the sum of all positive integral values of $m$ such that $m^2-22m+117=q$ for some prime number $q$. Note that there is at least one such $m$. | nvidia |
The function $g(x)$ satisfies
\[g(x \cdot y) = g(x) + g(y)\]for all positive real numbers $x$ and $y.$ If $g(2) = 1$ and $g(3) = 2,$ find $g(12).$ | nvidia |
For any finite sequence $A = (a_1, a_2, \ldots, a_n)$ of real numbers, define $S(A)$ to be the sum of all the elements of $A$, and $D(A)$ to be the sum of the squares of the elements of $A$. Find the smallest real number $k$ such that the inequality \[D(A) + D(B) \geq k(S(A) - S(B))^2\]holds for all pairs $(A, B)$ of s... | nvidia |
The least common multiple of two positive integers is $x^2+3x+2$ and their greatest common divisor is $x+1$, where $x$ is a positive integer. If the sum of the two integers is 48, what is the largest possible value of $x$? | nvidia |
Given that $a$ and $b$ are positive integers less than or equal to 12, find the greatest possible difference between the number of positive factors of $a^b$ and the number of positive factors of $a + b$. | nvidia |
Problem:
A certain function $f(x)$ has the property that $f(x) \cdot f(x+1) = f(x+2)$ for all real numbers $x$. If $f(0) = 2$ and $f(1) = 3$, find the sum of all positive integer values of $x$ such that $f(x) = 5$. | nvidia |
Find the number of distinct ordered-pair solutions (x, y) to the equation x + 2y = 20, where x and y are positive integers and x < y. | nvidia |
Find the smallest positive $a$ such that $a$ is a multiple of $4$ and $a$ is a multiple of $14.$ | nvidia |
Find the smallest value of $a$ such that the quadratic expression $x^2 + ax + b$ can be factored as the product of two linear factors with rational coefficients, where $b$ is a prime number greater than 1000. | nvidia |
Find the number of positive integers $m \le 500$ such that $12m$ is a perfect cube. | nvidia |
Let $w$ be a complex number such that $w^3=64+64i$. Find the modulus of $w$. | nvidia |
Let $P$ be a point on the unit circle, and let $Q$ be the point on the circle that is symmetric to $P$ with respect to the line $x=y$. If the midpoint $M$ of $PQ$ lies on the line $y=x+\frac{1}{2}$, find the distance between $P$ and the origin. | nvidia |
Let $a$ and $b$ be positive integers such that $a^2+b^2=37$.
Determine the smallest possible value of $ab$. | 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 |
Find the values of $p$, $q$, and $r$ such that the quadratic $px^2+qx+r$ can be written in the form $2(x-1)(x+2)$, and then determine $p+q+r$. | nvidia |
Let $x$ and $y$ be real numbers such that $2(x^2 + y^2) = x + y.$ Find the maximum value of $x - y.$ | nvidia |
Find the sum of the lengths of the two longest altitudes in a triangle with sides $9$, $20$, and $22$. | nvidia |
Find the smallest positive integer $N$ such that $N^3$ ends with the same last three digits as $N$, and $N^4$ ends with the same last four digits as $N^2$. | nvidia |
Consider the quadratic equation $ax^2+bx+c=0$, where $a$, $b$, and $c$ are positive integers. Given that the sum of the coefficients is equal to $25$ ($a+b+c=25$) and the product of the roots is equal to $9$ ($c/a=9$), find the ordered triple $(a,b,c)$. | nvidia |
How many positive two-digit integers have an odd number of positive factors? | nvidia |
Find the greatest integer n such that 3^n is less than 1000. | nvidia |
The graph of the equation $x^2+y^2=-2x-10y-16$ is a circle. Find the distance from the center of the circle to the point $(1, 1)$. | nvidia |
Find the sum of all the distinct three-digit numbers that can be formed by rearranging the digits of the number 345, if no digit is repeated in any number.
(End of response) | nvidia |
Let $f(x)$ be a function such that $f(x+y) = f(x) + f(y)$ for all real numbers $x$ and $y,$ and $f(x) \neq 0$ for at least one real number $x.$ Find $f(0).$ | nvidia |
Find the largest possible value of $x+y+z$ for the three-digit number $\overline{xyz}$, given that it can be expressed as the product of a single-digit integer $a$ and a two-digit integer $\overline{bc}$. | nvidia |
Find the smallest positive integer $M$ with the following property: of the four numbers $M$, $M+1$, $M+2$, and $M+3$, one of them is divisible by $2^2$, one of them is divisible by $3^2$, one is divisible by $5^2$, and one is divisible by $7^2$, and the product of these four numbers is a perfect square. | nvidia |
Let $M$ be a point on the parabola $y=x^2$, such that the point $(1,-2)$ is equidistant from $M$ and the focus of the parabola. Find the coordinates of $M$. | nvidia |
A sequence of integers is formed by writing the sum of any two consecutive numbers starting from $0$ and $1$, then writing the sum of consecutive numbers in the resulting sequence, and so on. What is the probability that the number $11$ will be included in the sequence after $5$ iterations? | nvidia |
Find the sum of all positive divisors of 60 that are also divisors of 24, and that can be expressed as the product of two distinct prime numbers. | nvidia |
Find the number of positive integers between 100 and 1000 that are divisible by neither 5 nor 7. | nvidia |
Find all real numbers $a$ for which there exists a nonzero, 2-dimensional vector $\mathbf{w}$ such that
\[\begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix} \mathbf{w} = a \mathbf{w}^T.\]Enter all the solutions, separated by commas. | nvidia |
Determine the smallest integer $n$ such that $n\cdot10^{20}+2^{20}$ is divisible by 11. | nvidia |
Find the number of possible values of $m$ such that both roots of the quadratic equation $x^2 - 50x + m = 0$ are prime numbers. | nvidia |
Determine the smallest positive integer that has exactly 10 digits in its base-4 representation. | nvidia |
Find the smallest positive real number $x$ such that $\cos x = \sin 2x$. | nvidia |
Given that $x$ is real and $x^3+\frac{1}{x^3}=52$, find $x+\frac{1}{x}$. | nvidia |
Find all values of $a$ so that
\[x^2 + ax + a > 0\]for all $x > -2.$ | nvidia |
Find the positive real number $x$ such that $\sqrt[x]{4}\cdot\sqrt[x]{9}\cdot\sqrt[x]{16} = 24$ | nvidia |
A rectangle is formed by connecting the points where two diagonals of a square intersect the sides of the square. If the sides of the rectangle are in the ratio of 4:3, what is the ratio of the area of the rectangle to the area of the square? | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $x+y+z=1$. Find the minimum value of $\frac{1}{x} + \frac{4}{y} + \frac{9}{z}$. | nvidia |
Find the real number $x$ such that $\lfloor x \rfloor \cdot (x-\lfloor x \rfloor)=19$. Express $x$ as a decimal. | nvidia |
Find the number of integers $n$ such that $\sqrt{10} < n < \sqrt{100}$ and $\frac{n}{\lfloor n \rfloor}$ is a perfect square. | nvidia |
Find the largest positive integer $n$ such that there exist distinct positive integers $a$ and $b$, where
$a \equiv 0 \pmod{n}$ and $b \equiv 3 \pmod{n}$, and
$a^2 + b^2$ is not divisible by $n$. | nvidia |
Determine the sum of all positive integers $n$ such that the equation $x^2 + 3x + n = 0$ has no real solutions in $x$ and the equation $x^2 + 4x + n = 0$ has exactly one real solution in $x$. | nvidia |
A polynomial $f(x)$ has integer coefficients and can be factored as $f(x)=(x-a_1)(x-a_2)...(x-a_n)$ where all $a_i$ are integers. If $f(x)$ has a factor of the form $(x-b)^2$ for some integer $b$, what is the smallest integer $k$ that can be multiplied with $f(x)$ such that the resulting polynomial can be factored into... | nvidia |
Find the smallest value of $a$ such that the function $f(x) = |x-a| + |x+2|$ takes on the value $10$ for some real number $x$. | nvidia |
Find the sum of all positive integral values of $a$ such that $a^3-12a^2+47a-60= b^2$ for some perfect square $b^2$. Note that there is at least one such $a$. | nvidia |
What is the smallest positive integer $k$ such that $k!$ is divisible by $2^8$? | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(xf(y) + x) = xy + f(x)\]for all $x,$ $y.$
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 |
The difference between two perfect squares
\[(n+11)^2-(n-6)^2\]
increases with the increasing of the positive integer $n$. What is the smallest positive integer $n$ for which this difference is greater than 300? | nvidia |
Find the smallest positive integer that has exactly four different divisors, including itself, and whose prime factorization includes at least one repeated prime factor. | nvidia |
In the expansion of $(1+x)^n$ using the binomial theorem, where $n$ is a positive integer and $x$ is a real number, what is the sum of the coefficients of the even-powered terms? | nvidia |
Consider a number N such that the number of its divisors is a divisor of N. What could be the possible values of N? | nvidia |
Find the number of positive integers $n,$ $1 \le n \le 2000,$ for which the polynomial $x^2 + 2x - n$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
The sum of the digits of the number $16^n$ written in base 10 is $n^2$. What is the largest power of 16 such that the sum of the base-10 digits of the result is a perfect square? | nvidia |
Let $f(n)$ be the sum of the digits of $n$. Determine the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $f(n)$ is a multiple of 4. | nvidia |
Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. Express your answer in interval notation. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(0) = 0$ and
\[g(x^2 + y^2) = g(x)^2 + g(y)^2 + x^2y^2\]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 |
Find the number of divisors of 180 that are also multiples of 3. | nvidia |
What is the sum of the digits of the product of the first three positive prime numbers that are also perfect squares? | nvidia |
Let $p, q, r,$ and $s$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}p^2+q^2&=&r^2+s^2&=&2012,\\ pr&=&qs&=&1001.\end{array}\]
If $T=p+q+r+s$, compute the value of $\lceil T\rceil$. | nvidia |
Find all positive values of $d$ so that the inequality $x^2-8x+d<0$ has real solutions for $x$. Express your answer in interval notation. | nvidia |
What is the units digit of the number that is the sum of all natural numbers from 1 to 1000000? | nvidia |
Find the value of $a$ for which the equation $\frac{x^2}{a} + \frac{x}{a^2} + 1 = 0$ has a unique solution for $x$, given that $a$ is a positive real number and $x$ is a non-zero real number. | nvidia |
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