Found problems: 9
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.
Find the number of solutions to the given congruence$$x^{2}+y^{2}+z^{2} \equiv 2 a x y z \pmod p$$ where $p$ is an odd prime and $x,y,z \in \mathbb{Z}$.
[i]Proposed by math_and_me[/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, 4.
[b]Q.[/b]Let $ABC$ be a triangle with incenter $I$ and orthocenter $H$. Let $A_1,A_2$ lie on $\overline{BC}$ such that $\overline{IA_1}\perp \overline{IB},\overline{IA_2}\perp\overline{IC}$. $\overline{AA_1},\overline{AA_2}$ cut $\odot(ABC)$ again at $A_3,A_4$. $\overline{A_3A_4}$ cuts $\overline{BC}$ at $A_0$. Similarly, we have $B_0,C_0$. Prove that $A_0,B_0,C_0$ are collinear on a line which is perpendicular to line $\overline{IH}$.
2020 DMO Stage 1, 5.
[b]Q[/b]. $ABC$ is an acute - angled triangle with $\odot(ABC)$ and $\Omega$ as the circumcircle and incircle respectively. Let $D, E, F$ to be the respective intouch points on $\overline{BC}, \overline{CA}$ and $\overline{AB}$. Circle $\gamma_A$ is drawn internally tangent to sides $\overline{AC}, \overline{AB}$ and $\odot(ABC)$ at $X, Y$ and $Z$ respectively. Another circle $(\omega)$ is constructed tangent to $\overline{BC}$ at $\mathcal{T}_1$ and internally tangent to $\odot(ABC)$ at $\mathcal{T}_2$. A tangent is drawn from $A$ such that it touches $\omega$ at $W$ and meets $BC$ at $V$, with $V$ lying inside $\odot(ABC)$. Now if $\overline{EF}$ meets $\odot(BC)$ at $\mathcal{X}_1$ and $\mathcal{X}_2$, opposite to vertex $B$ and $C$ respectively, where $\odot(BC)$ denotes the circle with $BC$ as diameter, prove that the set of lines $\{\overline{B\mathcal{X}_1}, \overline{ZS}, \overline{C\mathcal{X}_2}, \overline{DU}, \overline{YX}, \overline{\mathcal{T}_1W} \}$ are concurrent where $S$ is the mid-point of $\widehat{BC}$ containing $A$ and $U$ is the anti-pode of $D$ with respect to $\Omega$. If the line joining that concurrency point and $A$ meets $\odot(ABC)$ at $N\not = A$ prove that $\overline{AD}, \overline{ZN}$ and $\gamma_A$ pass through a common point.
[i]
Proposed by srijonrick[/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]