IN-TST 2001

Let $x$ , $y$ , $z>0$ . Prove that if $xyz\geq xy+yz+zx$ , then $xyz \geq 3(x+ y+z)$ .

In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that: $$AL+BM+CN \leq 3(AD+BE+CF)$$ When does equality occur?

For any positive integer $n$ , show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$ .

Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ , $2$ , $3$ , $4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$ . Prove that $p$ divides all the coefficients of $Q(x)$ .

Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$ -element set, such that $A\cap B \neq \emptyset$ and $\left |A \right | \neq \left |B \right |$ .

Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : $$f ( f (x)-x) = 2x$$ for all $x > 0$ .

Points $B = B_1 , B_2, \cdots , B_n , B_{n+1} = C$ are chosen on side $BC$ of a triangle $ABC$ in that order. Let $r_j$ be the inradius of triangle $AB_jB_{j+1}$ for $j = 1, \cdots, n$ , and $r$ be the inradius of $\triangle ABC$ . Show that there is a constant $\lambda$ independent of $n$ such that : $$(\lambda -r_1)(\lambda -r_2)\cdots (\lambda -r_n) =\lambda^{n-1}(\lambda -r)$$

Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$ . Prove that the polynomial $$(x-\alpha)(x-\beta )(x-\gamma )$$ has integer coefficients.

Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$ , define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$ . Prove that : $$\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}$$

Each vertex of an $m\times n$ grid is colored blue, green or red in such a way that all the boundary vertices are red. We say that a unit square of the grid is properly colored if: $(i)$ all the three colors occur at the vertices of the square, and $(ii)$ one side of the square has the endpoints of the same color. Show that the number of properly colored squares is even.

A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$ . Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$ . Prove that $r | k$ and $k | s$ .

Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$ . Prove that $$\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1$$