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In the equation $2x^3-3x^2-3x+1=0$, determine the value of $\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}$, where $\alpha$, $\beta$, and $\gamma$ are the roots of the equation. | nvidia |
How many different 3-digit numbers can be formed using the digits 1, 2, 3, 4, and 5, with the condition that the first and last digits must be the same, and the middle digit can be any one of the remaining four digits? | nvidia |
Determine the smallest positive integer $n$ for which there are three distinct triples $(a, b, c)$ of positive integers such that for each triple, $a^2 + b^2 + c^2 = n$ and $a \le b \le c$. | nvidia |
The rectangular sheet of paper $ABCD$ has the following property: if $P$ is an arbitrary point on $AB$ and $Q$ is an arbitrary point on $CD$, then the quadrilateral $BPCQ$ is folded along the diagonal $PQ$ so that the point $B$ maps to the point $B'$ on the line $AD$ and the point $C$ maps to the point $C'$ on the line... | nvidia |
Determine the smallest positive integer that, when multiplied by 1,493,824, results in a product that is a perfect square. | nvidia |
Consider the polynomial function:
$Q(x) = x^4 + px^3 + qx^2 + rx + s$
The graph of this polynomial has four distinct $x$-intercepts, one of which is at $(1,0)$. Which of the following coefficients cannot be zero?
$\textbf{(A)}\ p \qquad \textbf{(B)}\ q \qquad \textbf{(C)}\ r \qquad \textbf{(D)}\ s$ | nvidia |
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \neq 0$$ | nvidia |
Let $m$ and $n$ be two relatively prime positive integers. The number of positive divisors of $mn$ is $32$. Given that $m$ has only two distinct prime divisors, determine the maximum number of distinct prime divisors that $n$ could have. | nvidia |
Let $g(x)$ be a function that satisfies the equation $g(x+y) = g(x) + g(y)$ for all real numbers $x$ and $y$. Find all possible values of $g(0)$. | nvidia |
Let $a$ and $b$ be nonnegative real numbers such that
\[\cos (ax + b) = \cos 17x\]for all integers $x.$ Find the smallest possible value of $b$. | nvidia |
Two numbers are independently selected from the set of positive integers less than or equal to 5. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | nvidia |
Determine the value of $b$ such that the equation $x^3-bx^2+bx-1=0$ has a root $\frac{1+\sqrt{3}i}{2}$. | nvidia |
Find the sum of all $x$ such that the expression $\frac{x^2+5x+6}{x^2-7x-18}$ is undefined. | nvidia |
Find the number of divisors of 180 that are also multiples of 3. | nvidia |
Let $a$, $b$, $c$, and $d$ be positive integers such that $a$ is divisible by $15$, $b$ is divisible by $18$, $c$ is divisible by $24$, and $d$ is divisible by $30$. Find the smallest possible value of $\lcm(a,b,c,d)$ that is also divisible by $5$. | nvidia |
Find the smallest positive integer n such that n is a multiple of 4 and the sum of its digits is also a multiple of 4. | nvidia |
Find all positive real numbers $x$ and $y$ that satisfy
\[\frac{x^2 + y^2}{x + y} \le \sqrt{x^2 + y^2 + xy}.\] | nvidia |
Problem:
Determine the number of ways to arrange the letters of the word PROOF such that no two vowels are adjacent. | nvidia |
Find the largest positive integer n such that there exists a polynomial of degree at most n whose coefficients are all integers and whose roots are precisely all of the primitive n-th roots of unity.
Note: By "primitive n-th roots of unity" we mean the complex numbers e^(2πik/n) for k = 1, 2, ..., n-1 such that k and ... | nvidia |
Let $f(x)$ be a function such that $f(f(x)) = 2$ for all real numbers $x$. Find the value of $f(2)$ if $f(2) \neq 2$. | nvidia |
Consider the set of all positive integers $n$ such that $n$ can be represented as a sum of distinct powers of $3$, with each power of $3$ appearing at most once in the representation. For example, the number $45$ can be represented as $3^3 + 3^2 + 3^0$.
Find the largest number in this set that is less than $1000$. | nvidia |
The function $f(x)$ satisfies
\[f(2^x) + xf(2^{-x}) = 1\]for all real numbers $x.$ Find $f(2).$ | nvidia |
Find the area of the rectangle with integer sides such that the
length is three times as large as the width, and the perimeter
of the rectangle is equal to the perimeter of a square with
side length 24 cm. | nvidia |
Determine the number of zeros at the end of the expansion of $333,\!333,\!333,\!332^3$. | nvidia |
Suppose $x$ and $y$ are positive integers such that $\gcd(x,y)$ is divisible by exactly $4$ distinct primes and $\mathop{\text{lcm}}[x,y]$ is divisible by exactly $15$ distinct primes.
If $x$ has fewer distinct prime factors than $y$, then $x$ has at most how many distinct prime factors? | nvidia |
Let $\mathbf{A}$ be a $2 \times 2$ matrix, with real entries, such that $\mathbf{A}^3 = \mathbf{0}.$ Find the number of different possible matrices that $\mathbf{A}^2$ can be. If you think the answer is infinite, then enter "infinite". | nvidia |
Find the value of $x$ such that $2^{x+3} \cdot 2^{2x-1} = 64$. | nvidia |
How many different 6-digit numbers can be formed by rearranging the digits of the number $125,\!125$? (Note that numbers cannot start with 0.) | nvidia |
Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $3(6x+1)\equiv 4\pmod p$. | nvidia |
Let's call the row of Pascal's triangle with 41 numbers the "first row". The sum of the numbers in the first row is equal to 2^40. Now consider a "second row" of Pascal's triangle whose sum is twice the sum of the "first row". What is the first number from the left that appears in both rows? | nvidia |
Find the product of the roots of $\cos^2x-5\cos x-1=0$ that are between $x=0$ and $x=4\pi$ radians. | nvidia |
Find the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=72$. | nvidia |
Find the smallest positive integer $n$ such that $n$ is a multiple of 7 and the sum of its digits is also a multiple of 7. | nvidia |
Consider the function $$f(x) = \frac{x}{x+1} + \frac{x}{x+2} - kx$$ for a given complex number $k$. Suppose that the graph of $y=f(x)$ intersects the $x$-axis exactly twice. Find all possible values of $k$ such that the two intersection points are symmetric with respect to the $y$-axis. | nvidia |
Given positive integers $m$ and $n$ such that $\frac{1}{m} + \frac{1}{3n} = \frac{1}{11}$, what is the least possible value of $mn$? | nvidia |
A five-digit number $M$ has the property that the four-digit number obtained by removing the leftmost digit is the number obtained by reversing the digits of $M$.
How many such numbers $M$ exist? | nvidia |
Determine the number of rows in Pascal's Triangle that contain the number 67. | nvidia |
A regular hexagon is inscribed in a circle of radius 10. The sum of the lengths of all sides and diagonals of the hexagon can be written in the form
\[m + n \sqrt{3},\]where $m$ and $n$ are positive integers. Find $m+n$. | nvidia |
Find the number of three-digit numbers that are multiples of both 3 and 11. | nvidia |
The sum of the positive whole number divisors of 210 is equal to the sum of the positive whole number divisors of 315 minus the sum of the positive whole number divisors of 105. Find the sum of the positive whole number divisors of 315. | nvidia |
Find the absolute value of the difference of single-digit integers $A$ and $B$ such that $$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & & B& B & A_6\\ & & & \mathbf{4} & \mathbf{1} & B_6\\& & + & A & \mathbf{1} & \mathbf{5_6}\\ \cline{2-6} & & A & \mathbf{1} & \mathbf{5} & \mathbf{2_6} \\ \end{array} $$Express your ans... | nvidia |
Let $f(x)$ be a polynomial with integer coefficients such that $f(1)=1$, $f(2)=3$, $f(3)=7$, and $f(4)=13$. Determine the number of possible values of $f(5)$. | nvidia |
What is the sum of all integer values $n$ for which $\binom{24}{n}+\binom{24}{12}=\binom{25}{13}$? | nvidia |
A box contains three coins, one of which is a fair coin, one is a biased coin that comes up heads with probability 0.7, and the other is a biased coin that comes up tails with probability 0.8. A coin is chosen at random from the box and flipped three times. Given that the first two flips are both heads, what is the pro... | nvidia |
Consider the cubic polynomial
\[ax^3+bx^2+cx+d,\]
where $a$, $b$, $c$ and $d$ are real numbers. We know that the linear binomials
$(x-1)$, $(x-2)$ and $(x-3)$ are factors of this polynomial and
that the constant term is equal to $6$. Find the value of $a+b+c+d$. | nvidia |
Given a triangle with vertices at $(2,3)$, $(5,7)$, and $(1,9)$, find the coordinates of the point where the medians of the triangle intersect. | nvidia |
Determine the number of six-digit positive integers that can be formed using exactly two 1s, three 2s, and one 3, such that the resulting number is divisible by 3. | nvidia |
Find the number of polynomials $f(x)$ of degree at most 3 with real coefficients such that the polynomial obtained by replacing each coefficient with the digit sum of its integer part has degree 3 and is divisible by $(x-1)(x+1)$. | nvidia |
Find the value of $c$ such that the equation $x^2-4x+c=0$ has two real roots and the sum of their squares is 32. | nvidia |
Determine the value of $y$ such that the five-digit number $\underline{24y24}$ is divisible by 36. | nvidia |
Find the value of $a$ that satisfies the equation $293_{a}+468_{a}=73B_{a}$, where $B_{a}=11_{10}$. | nvidia |
Let $f:\mathbb{R}\to\mathbb{R}$ be a function satisfying $$f(x+y)+f(xy)=f(x)f(y)-1$$ for all real numbers $x$ and $y.$ Determine the possible values of $f(1)$. | nvidia |
In the diagram, the area of the shaded region is equal to the sum of the areas of the two unshaded regions. Given that $CD=8$ and $AB=10$, determine the length of $EF$.
[asy]
draw((0,0)--(10,0)--(10,8)--(0,8)--cycle);
draw((0,4)--(2,4)--(2,0));
draw((2,0)--(8,0));
draw((8,0)--(10,0));
draw((8,0)--(8,8));
draw((8,4)--(... | nvidia |
$\triangle ABC$ is inscribed in a circle of radius $r$ so that its base $AB$ is a chord of length $c$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:
$\textbf{(A)}\ s^2\le4r^2+c^2\qquad \textbf{(B)}\ s^2=4r^2+c^2 \qquad \textbf{(C)}\ s^2 \ge 4r^2+c^2 \qqua... | nvidia |
The number of solutions of $3^x+2^y=1$, in which $x$ and $y$ are positive integers, is:
\[0, 1, 2, \text{or} \infty\] | nvidia |
Find the area of the triangle formed by connecting the midpoints of the three sides of a triangle whose vertices are $(0,0)$, $(3,4)$, and $(6,0)$. | 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 |
How many integers are between $(10.2)^3$ and $(10.3)^3$?
Another problem inspired by this one:
Let $a$ and $b$ be positive real numbers such that $a^2+b^2=1$. Find the minimum value of $\left(\frac{a+b}{2}\right)^3+\left(\frac{a-b}{2}\right)^3$. | nvidia |
Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$
\text A. \ \ 21 \qquad \text B. \ \ 60 \qquad \text C. \ \ 119 \qquad \text D. \ \ 180 \qquad \text E. \ \ 231
$$ | nvidia |
Find all values of $x$ such that $\left|\frac{x-1}{x+1}\right| = \frac{1}{2}$. | nvidia |
Find the sum of all the positive integer divisors of 100 that are not divisible by 3.
(Remark: Instead of finding the product, we ask for the sum, and add an additional constraint on the divisors. This requires a different approach and reasoning.) | nvidia |
Let $a,b,c$ be positive real numbers such that $a^2b^3c^4=1$, $b^5c^6a^7=1$ and $c^8a^9b^{10}=1$. Find $a+b+c$. | nvidia |
Determine the number of positive real numbers $x$ such that $x$, $\sqrt{x}$, and $\ln x$ form an arithmetic sequence in some order. | nvidia |
If the quadratic $2x^2-5nx+n$ has exactly one real root, find the positive value of $n$. | nvidia |
Find the maximum value of $c$ such that the equation $\sqrt{(x-1)^2+(y-c)^2}+\sqrt{(x+1)^2+(y-2)^2}=4$ represents an ellipse. | nvidia |
Let $S$ be the sum of the squares of all the positive integer divisors of 99. Determine $\sqrt{S}$. | nvidia |
Find the smallest positive integer with exactly three distinct prime factors. | nvidia |
For a positive integer $m$, let $r_7(m)$ denote the remainder when $m$ is divided by $7$.
Find the $15^{\text{th}}$ entry in an ordered list of all positive integers $m$ that satisfy the condition $$r_7(3m) \leq 2.$$
Note: The first entry in this list is $1$. | nvidia |
The product of two positive integers $x$ and $y$ is $432$. What is the smallest possible value of the sum $x + y$? | nvidia |
A sequence of digits is called "alternating" if each digit is different from the one preceding it and the one following it. How many positive four-digit integers are alternating? | nvidia |
If we let $f(n)$ denote the sum of all the positive divisors of the integer $n$, how many integers $i$ exist such that $1 \le i \le 2010$ and $f(i) = 1 + \sqrt{i} + i$? | nvidia |
Find the largest positive integer that is both a factorial number and is not a multiple of 5. | nvidia |
Determine the value of $a$ such that the system of equations
\begin{align*}
2x+ay&=7\\
ax+3y&=11
\end{align*}
has infinitely many solutions. | nvidia |
Determine the last two digits of the sum $2^{20} + 2^{21} + \cdots + 2^{30}$. | nvidia |
Find the sum of all possible values of $x$ that satisfy the equation $\sqrt{x+1} - 2 = \sqrt{x-3}$. | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that $xyz = 8.$ Find the minimum value of $x + 2y + 4z.$ | nvidia |
We define a function $f(x)$ such that $f(11)=34$, and if there exists an integer $a$ such that $f(a)=b$, then $f(b)$ is defined and
$f(b)=3b+1$ if $b$ is odd
$f(b)=\frac{b}{2}$ if $b$ is even.
What is the smallest possible number of integers in the domain of $f$? | nvidia |
Given that $x$ and $y$ are positive integers satisfying the equation $5x + 2y = 11$, how many of the first five prime numbers must divide $x + 3y$? | nvidia |
Find the sum of all the perfect squares between 1 and 20. | nvidia |
For how many positive integer values of $n$ is there a function $f$ from $\{1,2,\ldots,n\}$ to $\{-1,1\}$ such that for any three values $i$, $j$, and $k$ in the domain of $f$, the product $f(i)f(j)f(k)$ is not equal to 1? | nvidia |
Let $g(n)$ be the sum of the digits of $\frac{1}{3^n}$ that are to the right of the decimal point. What is the smallest positive integer $n$ such that $g(n) > 15$? | nvidia |
How many four-digit numbers have the property that the sum of their digits is equal to 10? | nvidia |
Find the smallest positive integer that has at least one prime factor in common with 16, 24, and 28, and is divisible by a prime number that is not a factor of any of these numbers. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the maximum value of $x^3 y^2 z.$
New problem:
Find the maximum value of $x^2 + y^2 + z^2 + 2xyz$ given that $x,$ $y,$ and $z$ are positive real numbers satisfying $x + y + z = 3.$ | nvidia |
Determine the smallest five-digit palindrome that is a perfect square, knowing that its third digit is a prime number. | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $[a, b] = 1815$ and $[a, c] = 294$. Determine the largest possible value of $[b, c]$. | nvidia |
Problem:
Find the value of $x$ such that $\log_{2x}4=\log_{x}2$. | nvidia |
What is the smallest positive integer that can be written in the form $2002m + 44444n$, where $m$ and $n$ are integers? | nvidia |
In a triangle with vertices at $(a, b)$, $(c, d)$, and $(e, f)$, the coordinates of the centroid are $(2, 3)$. Find the sum of the $y$-coordinates of the points where the medians of the triangle intersect the $y$-axis. | nvidia |
Compute the product of all the fractions of the form $\dfrac{n^3-1}{n^3+1}$ where $n$ is a positive integer, such that the product of the numerators is equal to the product of the denominators. | nvidia |
Problem:
Let $S_n$ be the sum of all the interior angles of a convex polygon with $n$ sides. What is the least positive integer $k$ such that $S_{k-1} + S_{k+1} = 2S_k$? | nvidia |
Find all values of $b$ for which the points $(0,0,0),$ $(2,b,0),$ $(0,2,b),$ and $(b,0,2)$ are coplanar. | nvidia |
Find the number of complex numbers $z$ such that $z$, $z^2$, and $z^3$ form an arithmetic sequence in some order. | nvidia |
Find the sum of the nine smallest distinct positive integer multiples of 5. | nvidia |
Given a right triangle with legs of length 4 and 5, a rhombus is inscribed in the triangle such that one of its vertices is the right angle of the triangle and its sides are parallel to the legs of the triangle. Find the length of one side of the rhombus. | nvidia |
Consider a binary operation $\otimes$ with the properties $a\otimes(a\otimes b)=a\cdot b$ and $a\otimes e=a$ for all real numbers $a$ and $b$, where $e$ is a fixed nonzero real number. Determine the value of $x$ in the equation $(x\otimes 5)\otimes 2 = x + 2$, given that $e=3$. | nvidia |
How many positive integers less than $500$ can be written as the sum of two positive perfect cubes?
Here is another problem:
The number $n$ can be expressed as the sum of two perfect cubes, and also as the sum of two perfect squares. What is the minimum value of $n$ that satisfies both conditions? | nvidia |
Find the smallest positive integer such that its sum of divisors is a perfect square. | nvidia |
In the triangle $XYZ$, the angle bisector of $\angle XYZ$ intersects the circumcircle at point $P$. If the diameter of the circumcircle is equal to the perimeter of the triangle, and the area of the triangle is equal to $k\sqrt{2}$, where $k$ is an integer, find $k$. | nvidia |
In the equation
$$5x^2-kx+1=0$$
determine $k$ such that the difference of the roots be equal to unity.
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 $a$.
To create a new problem inspired by this one, we can change th... | nvidia |
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