There are $5$ accents in French, each applicable to only specific letters as follows: [list] [*] The cédille: ç [*] The accent aigu: é [*] The accent circonflexe: â, ê, î, ô, û [*] The accent grave: à, è, ù [*] The accent tréma: ë, ö, ü [/list] Cédric needs to write down a phrase in French. He knows that there are $3$ words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If $n$ is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of $n$.

A number is said to be [i]regular[/i] if when a digit $k$ appears in that number, the digit appears exactly $k$ times. For example, the number $3133$ is a regular number because the digit $1$ appears exactly once and the digit $3$ appears exactly three times. How many regular six-digit numbers are there?

A collection of $n$ positive numbers, where repeats are allowed, adds to $500$. They can be split into $20$ groups each adding to $25$, and can also be split into $25$ groups each adding to $20$. (A group is allowed to contain any amount of integers, even just one integer.) What is the least possible value of $n$? [i]Aaron Wang[/i]

Do there exist two integer-valued functions $f$ and $g$ such that for every integer $x$ we have (a) $f(f(x)) = x, g(g(x)) = x, f(g(x)) > x, g(f(x)) > x$ ? (b) $f(f(x)) < x, g(g(x)) < x, f(g(x)) > x, g(f(x)) > x$ ?

Let $A$ be a point and let k be a circle through $A$. Let $B$ and $C$ be two more points on $k$. Let $X$ be the intersection of the bisector of $\angle ABC$ with $k$. Let $Y$ be the reflection of $A$ wrt point $X$, and $D$ the intersection of the straight line $YC$ with $k$. Prove that point $D$ is independent of the choice of $B$ and $C$ on the circle $k$.

The sum of the following seven numbers is exactly 19: \[a_1=2.56,\qquad a_2=2.61,\qquad a_3=2.65,\qquad a_4=2.71,\] \[a_5=2.79,\qquad a_6=2.82,\qquad a_7=2.86.\] It is desired to replace each $a_i$ by an integer approximation $A_i$, $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and so that $M$, the maximum of the "errors" $|A_i - a_i|$, is as small as possible. For this minimum $M$, what is $100M$?

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be the midpoint of $\overline{AH}$ and let $T$ be on line $BC$ with $\angle TAO=90^{\circ}$. Let $X$ be the foot of the altitude from $O$ onto line $PT$. Prove that the midpoint of $\overline{PX}$ lies on the nine-point circle* of $\triangle ABC$. *The nine-point circle of $\triangle ABC$ is the unique circle passing through the following nine points: the midpoint of the sides, the feet of the altitudes, and the midpoints of $\overline{AH}$, $\overline{BH}$, and $\overline{CH}$. [i]Proposed by Zack Chroman[/i]

[b]p1.[/b] If $a \diamond b = ab - a + b$, find $(3 \diamond 4) \diamond 5$ [b]p2.[/b] If $5$ chickens lay $5$ eggs in $5$ days, how many chickens are needed to lay $10$ eggs in $10$ days? [b]p3.[/b] As Alissa left her house to go to work one hour away, she noticed that her odometer read $16261$ miles. This number is a "special" number for Alissa because it is a palindrome and it contains exactly $1$ prime digit. When she got home that evening, it had changed to the next greatest "special" number. What was Alissa's average speed, in miles per hour, during her two hour trip? [b]p4.[/b] How many $1$ in by $3$ in by $8$ in blocks can be placed in a $4$ in by $4$ in by $9$ in box? [b]p5.[/b] Apple loves eating bananas, but she prefers unripe ones. There are $12$ bananas in each bunch sold. Given any bunch, if there is a $\frac13$ probability that there are $4$ ripe bananas, a $\frac16$ probability that there are $6$ ripe bananas, and a $\frac12$ probability that there are $10$ ripe bananas, what is the expected number of unripe bananas in $12$ bunches of bananas? [b]p6.[/b] The sum of the digits of a $3$-digit number $n$ is equal to the same number without the hundreds digit. What is the tens digit of $n$? [b]p7.[/b] How many ordered pairs of positive integers $(a, b)$ satisfy $a \le 20$, $b \le 20$, $ab > 15$? [b]p8.[/b] Let $z(n)$ represent the number of trailing zeroes of $n!$. What is $z(z(6!))?$ (Note: $n! = n\cdot (n-1) \cdot\cdot\cdot 2 \cdot 1$) [b]p9.[/b] On the Cartesian plane, points $A = (-1, 3)$, $B = (1, 8)$, and $C = (0, 10)$ are marked. $\vartriangle ABC$ is reflected over the line $y = 2x + 3$ to obtain $\vartriangle A'B'C'$. The sum of the $x$-coordinates of the vertices of $\vartriangle A'B'C'$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Compute $a + b$. [b]p10.[/b] How many ways can Bill pick three distinct points from the figure so that the points form a non-degenerate triangle? [img]https://cdn.artofproblemsolving.com/attachments/6/a/8b06f70d474a071b75556823f70a2535317944.png[/img] [b]p11.[/b] Say piece $A$ is attacking piece $B$ if the piece $B$ is on a square that piece $A$ can move to. How many ways are there to place a king and a rook on an $8\times 8$ chessboard such that the rook isn't attacking the king, and the king isn't attacking the rook? Consider rotations of the board to be indistinguishable. (Note: rooks move horizontally or vertically by any number of squares, while kings move $1$ square adjacent horizontally, vertically, or diagonally). [b]p12.[/b] Let the remainder when $P(x) = x^{2020} - x^{2017} - 1$ is divided by $S(x) = x^3 - 7$ be the polynomial $R(x) = ax^2 + bx + c$ for integers $a$, $b$, $c$. Find the remainder when $R(1)$ is divided by $1000$. [b]p13.[/b] Let $S(x) = \left \lfloor \frac{2020}{x} \right\rfloor + \left \lfloor \frac{2020}{x + 1} \right\rfloor$. Find the number of distinct values $S(x)$ achieves for integers $x$ in the interval $[1, 2020]$. [b]p14.[/b] Triangle $\vartriangle ABC$ is inscribed in a circle with center $O$ and has sides $AB = 24$, $BC = 25$, $CA = 26$. Let $M$ be the midpoint of $\overline{AB}$. Points $K$ and $L$ are chosen on sides $\overline{BC}$ and $\overline{CA}$, respectively such that $BK < KC$ and $CL < LA$. Given that $OM = OL = OK$, the area of triangle $\vartriangle MLK$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are positive integers, $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p15.[/b] Euler's totient function, $\phi (n)$, is defined as the number of positive integers less than $n$ that are relatively prime to $n$. Let $S(n)$ be the set of composite divisors of $n$. Evaluate $$\sum^{50}_{k=1}\left( k - \sum_{d\in S(k)} \phi (d) \right)$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the sequence $(x_n)_{n\in\mathbb{N^*}}$ such that $$x_0=0,\quad x_1=2024,\quad x_n=x_{n-1}+x_{n-2}, \forall n\geq2.$$ Prove that there is an infinity of terms in this sequence that end with $2024.$

The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$

In the sequence $00$, $01$, $02$, $03$, $\cdots$, $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$, $39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could remain on their places?

Suppose that k and x are positive integers such that $$\frac{k}{2}=\left( \sqrt{1 +\frac{\sqrt3}{2}}\right)^x+\left( \sqrt{1 -\frac{\sqrt3}{2}}\right)^x.$$ Find the sum of all possible values of $k$

Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \dfrac{193}{137}.\] If $a+b=m/n$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$.

Four odd positive intgers $a, b, c, d (a\leq b \leq c\leq d)$ are given. Choose any three numbers among them and divide their sum by the un-chosen number, and you will always get the remainder as $1$. Find all $(a, b, c, d)$ that satisfies this.

The product $a_1 \cdot a_2 \cdot ... \cdot a_{100}$ is written on the board , where $a_1$, $a_2$, $ ... $, $a_{100}$, are natural numbers. Let's consider $99$ expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that the values of exactly $32$ of these expressions are even. What is the largest number of even numbers among $a_1$, $a_2$, $ ... $, $a_{100}$ could it be?

Given a,b c are lenth of a triangle (If ABC is a triangle then AC=b, BC=a, AC=b) and $a+b+c=2$. Prove that $1+abc<ab+bc+ca\leq \frac{28}{27}+abc$

Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly $ 5\over 4$ times greater than Vasya's result. What minimum number of portraits can be on the wall? [i]Author: V. Frank[/i]

Determine all integers $n > 1$ such that \[\frac{2^{n}+1}{n^{2}}\] is an integer.

Find all pairs $(p,q)$ of prime numbers which $p>q$ and $$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$ is an integer.

a) The digits of a natural number were rearranged. Prove that the sum of given and obtained numbers can't equal $999...9$ ($1967$ of nines). b) The digits of a natural number were rearranged. Prove that if the sum of the given and obtained numbers equals $1010$, than the given number was divisible by $10$.

Let $a$, $b$ and $c$ be positive real numbers and $a^3+b^3+c^3=3$. Prove $$\frac{1}{3-2a}+\frac{1}{3-2b}+\frac{1}{3-2c}\geq 3$$

Let the functions $f(\alpha,x)$ and $g(\alpha)$ be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then $g(\alpha)$ is a polynomial is $\alpha$. Find the leading coefficient of $g(\alpha)$.

For what value of $n$ is $\frac{1}{2\cdot5}+\frac{1}{5\cdot8}+\frac{1}{8\cdot 11}+\frac{1}{n(n+3)}=\frac{25}{154}$?

How many integers between $2$ and $100$ have only odd numbers in their prime factorizations?

Find all integers $k\ge 3$ with the following property: There exist integers $m,n$ such that $1<m<k$, $1<n<k$, $\gcd (m,k)=\gcd (n,k) =1$, $m+n>k$ and $k\mid (m-1)(n-1)$.