Practice (TheColoringMethod)
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Let $n$ be a positive integer. When the leftmost digit of (the standard base 10 representation of) $n$ is shifted to the rightmost position (the units position), the result is $n/3$. Find the smallest possible value of the sum of the digits of $n$.
Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively. She creates a sequence by rolling the die and recording the number on its bottom face. However, she discards (without recording) any role such that appending its number to the sequence would result in two consecutive terms that sum to 5. Sabrina stops the moment that all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence.
In the diagram below, the circle with center $A$ is congruent to and tangent to the circle with center $B$. A third circle is tangent to the circle with center $A$ at point $C$ and passes through point $B$. Points $C, A$, and $B$ are collinear. The line segment $\overline{CDEFG}$ intersects the circles at the indicated points. Suppose that $DE=6$ and $FG=9$. Find $AG$.
Let $f(n)$ denote the sum of the digits of $n$. Find $f(f(f(4444^{4444})))$.
Prove that if $p$ and $(p^2 + 8)$ are prime, then $(p^3 + 8p + 2)$ is prime.
Show that if $k \ge 4$, then $lcm(1; 3;\cdots; 2k- 3; 2k- 1) > (2k + 1)^2$ where $lcm$ stands for least common multiple.
Find all positive integers $n$ such that for all odd integers $a$. If $a^2\le n$, then $a|n$.
Find all $n \in \mathbb{Z}^+$ such that $2^n + n | 8^n + n$.
Find all nonnegative integers $n$ such that there are integers $a$ and $b$ with the property:
$$n^2 = a + b \qquad\text{and}\qquad n^3 = a^2 + b^2$$
Find all pairs of positive integers $(n;m)$ satisfying $3n^2 + 3n + 7 = m^3$.
$a, b, c, d$ are integers such that:
$$a < b\le c < d,\qquad ad = bc \qquad\text{and}\qquad \sqrt{d} - \sqrt{a} \le 1$$
Show that $a$ is a perfect square.
- Solve the following diophantine equation in natural numbers: $$y^2 = 1 + x + x^2 + x^3 + x^4$$
Find all pairs of positive integers $(a; b)$ such that $\frac{a}{b} + \frac{21b}{25a}$ is a positive integer.
Let $x; y; z$ be positive integers such that $(x; y; z) = 1$ and $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$. Prove that $x + y; x-z$ and $y-z$ are perfect squares.
Prove that $2^n + 1$ has no prime factors of the form $8k + 7$.
Find all triples $(a; b; c)$ of natural numbers such that $lcm(a; b; c) = a + b + c$.
Find all natural numbers $n$ such that $n$ equals the cube of the sum of its digits.
Find all odd integers $n$ for which $n|3^n + 1$.
If an integer $n$ is such that $7n$ is of the form $a^2 + 3b^2$, prove that $n$ is also of that form.
Find all non-negative solutions to: $43^n-2^x3^y7^z = 1$.
Prove that for every prime $p$, there exists an integer $x$, such that $x^8 \equiv 16 \pmod{p}$
Let $p$ be a prime and $a, b, c \in \mathbb{Z}^+$, such that $p = a+b+c-1$ and $p|a^3+b^3+c^3-1$. Prove that $min (a, b, c) = 1$
Find all primes $p, q$ such that $pq | 2^p + 2^q$.
Given that $9^{4000}$ has $3817$ digits and has a leftmost digit $9$ (base $10$). How many of the number $9^0, 9^1, 9^2, \cdots, 9^{4000}$ have leftmost digit $9$.
Let $A = 6^n$ for real $n$. Find all natural numbers $n$ such that $n^{A+2} + n^{A+1} + 1$ is a prime number.