TH-TMO 2020

Show that $\varphi(2n)\mid n!$ for all positive integer $n$ .

There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf $E_n$ takes $n$ minutes to travel $1 \text{ km}$ for each $n = 1,2,...,63$ , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?

Suppose that $f : \mathbb{R}^+\to\mathbb R$ satisfies the equation $$f(a+b+c+d) = f(a)+f(b)+f(c)+f(d)$$ for all $a,b,c,d$ that are the four sides of some tangential quadrilateral. Show that $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb{R}^+$ .

Let $\triangle ABC$ be a triangle with altitudes $AD,BE,CF$ . Let the lines $AD$ and $EF$ meet at $P$ , let the tangent to the circumcircle of $\triangle ADC$ at $D$ meet the line $AB$ at $X$ , and let the tangent to the circumcircle of $\triangle ADB$ at $D$ meet the line $AC$ at $Y$ . Prove that the line $XY$ passes through the midpoint of $DP$ .

You have an $n\times n$ grid and want to remove all edges of the grid by the sequence of the following moves. In each move, you can select a cell and remove exactly three edges surrounding that cell; in particular, that cell must have at least three remaining edges for the operation to be valid. For which positive integers $n$ is this possible?

Let the incircle of an acute triangle $\triangle ABC$ touches $BC,CA$ , and $AB$ at points $D,E$ , and $F$ , respectively. Place point $K$ on the side $AB$ so that $DF$ bisects $\angle ADK$ , and place point $L$ on the side $AB$ so that $EF$ bisects $\angle BEL$ . [list=a] []Prove that $\triangle ALE\sim\triangle AEB$ . []Prove that $FK=FL$ . [/list]

Determine all functions $f:\mathbb{R}\to\mathbb{Z}$ satisfying the inequality $(f(x))^2+(f(y))^2 \leq 2f(xy)$ for all reals $x,y$ .

For all positive real numbers $a,b,c$ with $a+b+c=3$ , prove the inequality $$\frac{a^6}{c^2+2b^3} + \frac{b^6}{a^2+2c^3} + \frac{c^6}{b^2+2a^3} \geq 1.$$

Let $n,k$ be positive integers such that $n>k$ . There is a square-shaped plot of land, which is divided into $n\times n$ grid so that each cell has the same size. The land needs to be plowed by $k$ tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side until it reaches the upper-right corner cell. In addition, each tractor can only move in two directions: up and right. Determine the minimum possible number of unplowed cells.

Determine all polynomials $P(x)$ with integer coefficients which satisfies $P(n)\mid n!+2$ for all postive integer $n$ .