MathDB

1

{1, 2, . . . , 20} has at least 2018 sumfree subsets

\Omega

formed by the first twenty natural numbers,

\Omega = \{1, 2, . . . , 20\}

. A nonempty subset

A

of

\Omega

is said to be sumfree [/i ] if for all pair of elements

x, y \in A

, the sum

(x + y)

is not in

A

, (

x

can be equal to

y

). Prove that

\Omega

has at least

2018

sumfree subsets.

4

1

[n/2] [n/3] [n/4] =n^2 , floor function

Find all postitive integers n such that

\left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor=n^2

where

\lfloor x \rfloor

represents the largest integer less than the real number

x

.

6

1

chilean collinearity , tangents to (H,HE), (B,BE) , orthocenter related

Consider an acute triangle

ABC

and its altitudes from

A

,

B

that intersect the respective sides at

D ,E

. Let us call the point of intersection of the altitudes

H

. Construct the circle with center

H

and radius

HE

. From

C

draw a tangent line to the circle at point

P

. With center

B

and radius

BE

draw another circle and from

C

another tangent line is drawn to this circle in the point

Q

. Prove that the points

D, P

, and

Q

are collinear.

3

1

a bee and a fly painting points on a network of connected hexagons

With

2018

points, a network composed of hexagons is formed as a sample the figure:
[asy] unitsize(1 cm);
int i;
path hex = dir(30)--(0,1)--dir(150)--dir(210)--(0,-1)--dir(330)--cycle;
draw(hex); draw(shift((sqrt(3),0))*(hex)); draw(shift((2*sqrt(3),0))*(hex)); draw(shift((4*sqrt(3),0))*(hex)); draw(shift((5*sqrt(3),0))*(hex));
dot((3*sqrt(3) - 0.3,0)); dot((3*sqrt(3),0)); dot((3*sqrt(3) + 0.3,0));
dot(dir(150)); dot(dir(210));
for (i = 0; i <= 5; ++i) { if (i != 3) { dot((0,1) + i*(sqrt(3),0)); dot(dir(30) + i*(sqrt(3),0)); dot(dir(330) + i*(sqrt(3),0)); dot((0,-1) + i*(sqrt(3),0)); } }
dot(dir(150) + 4*(sqrt(3),0)); dot(dir(210) + 4*(sqrt(3),0)); [/asy]
A bee and a fly play the following game: initially the bee chooses one of the

2018

dots and paints it red, then the fly chooses one of the

2017

unpainted dots and paint it blue. Then the bee chooses an unpainted point and paints it red and then the fly chooses an unpainted one and paints it blue and so they alternate. If at the end of the game there is an equilateral triangle with red vertices, the bee wins, otherwise Otherwise the fly wins. Determine which of the two insects has a winning strategy.

2

1

area of a ... square inside a square wanted (Chile 2018 L2 P1)

Consider

ABCD

a square of side

1

. Points

P,Q,R,S

are chosen on sides

AB

,

BC

,

CD

and

DA

respectively such that

|AP| = |BQ| =|CR| =|DS| = a

, with

a < 1

. The segments

AQ

,

BR

,

CS

and

DP

are drawn. Calculate the area of the quadrilateral that is formed in the center of the figure.
[asy] unitsize(1 cm);
pair A, B, C, D, P, Q, R, S;
A = (0,3); B = (0,0); C = (3,0); D = (3,3); P = (0,2); Q = (1,0); R = (3,1); S = (2,3);
draw(A--B--C--D--cycle); draw(A--Q); draw(B--R); draw(C--S); draw(D--P);
label("

A

", A, NW); label("

B

", B, SW); label("

C

", C, SE); label("

D

", D, NE); label("

P

", P, W); label("

Q

", Q, dir(270)); label("

R

", R, E); label("

S

", S, N); label("

a

", (A + P)/2, W); label("

a

", (B + Q)/2, dir(270)); label("

a

", (C + R)/2, E); label("

a

", (D + S)/2, N); [/asy]

1

sum of any 3 from 5 is prime number? (Chile 2018 L2 P1)

Is it possible to choose five different positive integers so that the sum of any three of them is a prime number?

2018 Chile National Olympiad

Subcontests

{1, 2, . . . , 20} has at least 2018 sumfree subsets

[n/2] [n/3] [n/4] =n^2 , floor function

chilean collinearity , tangents to (H,HE), (B,BE) , orthocenter related

a bee and a fly painting points on a network of connected hexagons

area of a ... square inside a square wanted (Chile 2018 L2 P1)

sum of any 3 from 5 is prime number? (Chile 2018 L2 P1)