Found problems: 6
2020 DMO Stage 1, 4.
[b]Q.[/b] We paint the numbers $1,2,3,4,5$ with red or blue. Prove that the equation $x+y=z$ have a monocolor solution (that is, all the 3 unknown there are the same color . It not needed that $x, y, z$ must be different!)
[i]Proposed by TuZo[/i]
2020 DMO Stage 1, 5.
[b]Q.[/b] Let $ABC$ be a triangle, where $L_A, L_B, L_C$ denote the internal angle bisectors of $\angle BAC, \angle ABC, \angle ACB$ respectively and $\ell_A, \ell_B, \ell_C$, the altitudes from the corresponding vertices. Suppose $ L_A\cap \overline{BC} = \{A_1\}$, $\ell_A \cap \overline{BC} = \{A_2\}$ and the circumcircle of $\triangle AA_1A_2$ meets $AB$ and $AC$ at $S$ and $T$ respectively. If $\overline{ST} \cap \overline{BC} = \{A'\}$, prove that $A',B',C'$ are collinear, where $B'$ and $C'$ are defined in a similar manner.
[i]Proposed by Functional_equation[/i]
2020 DMO Stage 1, 3.
[b]Q.[/b] Prove that:
$$\sum_{\text{cyc}}\tan (\tan A) - 2 \sum_{\text{cyc}} \tan \left(\cot \frac{A}{2}\right) \geqslant -3 \tan (\sqrt 3)$$where $A, B$ and $C$ are the angles of an acute-angled $\triangle ABC$.
[i]Proposed by SA2018[/i]
2020 DMO Stage 1, 2.
[b]Q.[/b] Consider in the plane $n>3$ different points. These have the properties, that all $3$ points can be included in a triangle with maximum area $1$. Prove that all the $n>3$ points can be included in a triangle with maximum area $4$.
[i]Proposed by TuZo[/i]
2020 DMO Stage 1, 1.
[b]Q.[/b] Show that for any given positive integers $k, l$, there exists infinitely many positive integers $m$, such that
$i) m \geqslant k$
$ii) \text{gcd}\left(\binom{m}{k}, l\right)=1$
[i]Suggested by pigeon_in_a_hole[/i]
2020 DMO Stage 1, 3.
[b]Q.[/b] Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(x) \geqslant x+1, \forall\ x \in \mathbb{R}\quad \text{and}\quad f(x+y) \geqslant f(x) f(y), \forall\ x, y \in \mathbb{R}$$
[i]Proposed by TuZo[/i]