Deduce that, for all $n \in \mathbb { N } ^ { * }$,
$$n ^ { ( \pi ( n ) - \pi ( \sqrt { n } ) ) / 2 } < 4 ^ { n }$$

Let $n \in \mathbb { N } , n \geqslant 2$. Justify that
$$\pi ( \sqrt { n } ) \leqslant \sqrt { n } < \frac { n } { \ln ( n ) }$$
then deduce that
$$\pi ( n ) \leqslant 4 \frac { \ln ( n ) } { n }$$
One may note that $2 > \ln ( 4 )$.

Let $x \geqslant 3$. Using the monotonicity of the function $t \mapsto \frac { t } { \ln ( t ) }$ on the interval $[ \mathrm { e } , + \infty [$, show that
$$\pi ( x ) \leqslant 4 \frac { x } { \ln ( x ) }$$

Let $n \in \mathbb { N } ^ { * }$. Show that
$$\binom { 2 n } { n } \leqslant ( 2 n ) ^ { \pi ( 2 n ) }$$

Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\}.$$ We assume $k = 2$. Prove that $$P(n) = \frac{n}{ab} - \left\{\frac{b^* n}{a}\right\} - \left\{\frac{a^* n}{b}\right\} + 1$$ for all integer $n \geq 0$, where $a^*$ and $b^*$ are integers satisfying $a^* a = 1$ modulo $b$ and $b^* b = 1$ modulo $a$ respectively.
Hint. One may use the formula $(*)$ for $b = 1$.

Let $r \in \mathbb { N } ^ { \star }$. Let $a _ { 1 } , \ldots , a _ { r }$ be non-zero natural integers. Justify that there exists a unique natural integer $d \left( a _ { 1 } , \ldots , a _ { r } \right)$ such that
$$a _ { 1 } \mathbb { Z } \cap a _ { 2 } \mathbb { Z } \cap \cdots \cap a _ { r } \mathbb { Z } = d \left( a _ { 1 } , \ldots , a _ { r } \right) \mathbb { Z }$$

Let $r \in \mathbb { N } ^ { * }$. Let $a _ { 1 } , \ldots , a _ { r }$ be non-zero natural integers. Show that $d \left( a _ { 1 } , \ldots , a _ { r } \right)$ is the smallest non-zero natural integer that is divisible by $a _ { 1 } , \ldots , a _ { r }$.

Calculate $d _ { 2 } , d _ { 3 }$ and $d _ { 4 }$, then show that $d _ { n } \leqslant n !$ for all natural integer $n \in \mathbb { N } ^ { * }$.

For all $n \in \mathbb { N } ^ { * }$, we denote by $d _ { n }$ the LCM of the natural integers between 1 and $n$, in other words: $d _ { n } = \operatorname { LCM } ( 1,2 , \ldots , n )$. For all prime number $p$, we denote by $k _ { p }$ the largest natural integer such that $p ^ { k _ { p } } \leqslant n$.
Show that $d _ { n } = \prod _ { \substack { p \leqslant n \\ p \text { prime } } } p ^ { k _ { p } }$.

For all $n \in \mathbb { N } ^ { * }$, we denote by $d _ { n }$ the LCM of the natural integers between 1 and $n$, in other words: $d _ { n } = \operatorname { LCM } ( 1,2 , \ldots , n )$. For all prime number $p$, we denote by $k _ { p }$ the largest natural integer such that $p ^ { k _ { p } } \leqslant n$.
For all prime number $p$, show that $k _ { p } = \left\lfloor \frac { \ln ( n ) } { \ln ( p ) } \right\rfloor$. Deduce that $d _ { n } \leqslant n ^ { \pi ( n ) }$.

For all $n \in \mathbb { N } ^ { * }$, we denote by $d _ { n }$ the LCM of the natural integers between 1 and $n$. Deduce that there exists a non-zero natural integer $N$ such that, for all $n \geqslant N , d _ { n } \leqslant 3 ^ { n }$.
One may use the prime number theorem: $\pi ( x ) \underset { x \rightarrow + \infty } { \sim } \frac { x } { \ln ( x ) }$.

Let $\alpha \in \mathbb { R } _ { + }$. We assume that there exist two sequences of non-zero natural integers $\left( p _ { n } \right) _ { n \in \mathbb { N } }$ and $\left( q _ { n } \right) _ { n \in \mathbb { N } }$ such that
$$\lim _ { n \rightarrow + \infty } \frac { p _ { n } } { q _ { n } } = \alpha \quad \text { and } \quad \left| \alpha - \frac { p _ { n } } { q _ { n } } \right| \underset { n \rightarrow + \infty } { = } o \left( \frac { 1 } { q _ { n } } \right)$$
We further assume that for all $n \in \mathbb { N } , \frac { p _ { n } } { q _ { n } } \neq \alpha$.
Show that $\alpha$ is an irrational number.

Let $\beta = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { 10 ^ { n ! } }$.
Justify that $\beta$ is well-defined, then show that $\beta$ is an irrational number.

Let $n \in \mathbb { N } ^ { * }$. Justify that $\zeta ( 2 ) = \sum _ { k = 1 } ^ { + \infty } \frac { 1 } { k ^ { 2 } }$ is well-defined, then show that we can write
$$\sum _ { k = 1 } ^ { n } \frac { 1 } { k ^ { 2 } } = \frac { p _ { n } } { q _ { n } }$$
with $p _ { n } \in \mathbb { N } ^ { * }$ and $q _ { n } = d _ { n } ^ { 2 }$.

Can we apply the result of Q20 to these sequences $\left( p _ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ and $\left( q _ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ to conclude about the irrationality of $\zeta ( 2 )$ ?

Let $r$ and $s$ be two strictly positive natural integers such that $r > s$, and
$$J _ { r , s } = \frac { 1 } { r - s } \sum _ { k = s + 1 } ^ { r } \frac { 1 } { k }$$
Deduce that we can write
$$J _ { r , s } = \frac { p _ { r , s } } { q _ { r , s } }$$
with $p _ { r , s }$ and $q _ { r , s }$ natural integers and $q _ { r , s }$ dividing $d _ { r } ^ { 2 }$.

Let $X = \{0,1,2,3,\ldots,99\}$. For $a, b$ in $X$, we define $a * b$ to be the remainder obtained by dividing the product $ab$ by 100. For example, $9 * 18 = 62$ and $7 * 5 = 35$. Let $x$ be an element in $X$. An element $y$ in $X$ is called the inverse of $x$ if $x * y = 1$. Find all elements of $X$ that have inverses and write down their inverses.

Let $g(n) = 5^k$ where $k$ is the number of distinct primes dividing $n$, and let $h(n) = 0$ if $n$ is divisible by $k^2$ for some integer $k > 1$, and $h(n) = 1$ otherwise.
a) Show that $g(mn) = g(m)g(n)$ does not hold in general, and determine when it holds.
b) Show that $h(mn) = h(m)h(n)$ for all positive integers $m, n$.

a) Show that among any 2-coloring (Red/Blue) of the vertices of a regular hexagon and its center, there must exist a monochromatic equilateral triangle.
b) Extend the argument to show the result holds for a larger configuration of points.

a) $n$ lines are drawn through a point $A$ inside a circle, creating chords. Show that the number of regions created inside the circle is $(n+1)^2$, assuming no three lines meet at any point other than $A$.
b) Using the result from part a), find the total number of regions when additional lines are drawn, and show the total is $(n+1)^2 + (2n+1)n$.

Find a 4-digit number $N$ such that $N = 4M$, where $M$ is the number obtained by reversing the digits of $N$.

Let $n_1, n_2, \ldots, n_k$ be positive integers with $\gcd(n_1, n_2, \ldots, n_k) = 1$. Show that every sufficiently large positive integer $n$ can be represented as $n = \sum_i c_i n_i$ with $c_i \geq 0$ integers.

Find the $n$-th non-square positive integer, and show that it equals $n + \lfloor \sqrt{n} + \frac{1}{2} \rfloor$.

The sum of all even positive divisors of $1000$ is
(a) $2170$
(b) $2184$
(c) $2325$
(d) $2340$

Let $n$ be an integer. The number of primes which divide both $n^{2}-1$ and $(n+1)^{2}-1$ is
(a) At most one.
(b) Exactly one.
(c) Exactly two.
(d) None of the above.