Practice (90/1000)
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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 $pq | 2^p + 2^q$.
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.
Find all non-negative integers $n$ such that $2^{200}+2^{192}\cdot 15+2^n$ is a perfect square
Prove that $\frac{5^{125}-1}{5^{25}-1}$ is composite.
Find all integers $a$, $b$, $c$ with $1 < a < b < c$ such that the number $(a-1)(b-1)(c-1)$ is a divisor of $(abc-1)$.