OFM 2022

find all the integer $n\geq1$ such that $\lfloor\sqrt{n}\rfloor \mid n$

We consider an $n \times n$ table, with $n\ge1$ . Aya wishes to color $k$ cells of this table so that that there is a unique way to place $n$ tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of $k$ for which Aya's wish is achievable?

Let $\triangle ABC$ a triangle, and $D$ the intersection of the angle bisector of $\angle BAC$ and the perpendicular bisector of $AC$ . the line parallel to $AC$ passing by the point $B$ , intersect the line $AD$ at $X$ . the line parallel to $CX$ passing by the point $B$ , intersect $AC$ at $Y$ . $E = (AYB) \cap BX$ . prove that $C$ , $D$ and $E$ collinear.

find the smallest integer $n\geq1$ such that the equation : $$a^2+b^2+c^2-nd^2=0$$ has $(0,0,0,0)$ as unique solution .

find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that $f(m+n)+f(m)f(n)=n^2(f(m)+1)+m^2(f(n)+1)+mn(2-mn)$ holds for all $m,n \in \mathbb{Z}$

To connect to the OFM site, Alice must choose a password. The latter must be consisting of $n$ characters among the following $27$ characters: $$A, B, C, . . ., Y , Z, \#$$ We say that a password $m$ is redundant if we can color in red and blue a block of consecutive letters of $m$ in such a way that the word formed from the red letters is identical to the word formed from blue letters. For example, the password $H\#ZBZJBJZ$ is redundant, because it contains the ZBZJBJ block, where the word $ZBJ$ appears in both blue and red. At otherwise, the $ABCACB$ password is not redundant. Show that, for any integer $n \ge 1$ , there exist at least $18^n$ passwords of length $n$ , that is to say formed of $n$ characters each, which are not redundant.

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Denote $\Delta$ the tangent at $A$ to the circle $\Gamma$ . $\Gamma_1$ is a circle tangent to the lines $\Delta$ , $(AB)$ and $(BC)$ , and $E$ its touchpoint with the line $(AB)$ . Let $\Gamma_2$ be a circle tangent to the lines $\Delta$ , $(AC)$ and $(BC)$ , and $F$ its touchpoint with the line $(AC)$ . We suppose that $E$ and $F$ belong respectively to the segments $[AB]$ and $[AC]$ , and that the two circles $\Gamma_1$ and $\Gamma_2$ lie outside triangle $ABC$ . Show that the lines $(BC)$ and $(EF)$ are parallel.

find all positive integer $a\geq 2$ and $b\geq2$ such that $a$ is even and all the digits of $a^b+1$ are equals.