2024 Israel Olympic Revenge

Part of Israel Olympic Revenge

Subcontests

(4)

1

Line through O given concurrency

ABC

be an acute triangle. Let

D

be a point inside side

BC

E

be the foot from

D

AC

F

AB

FE\perp AB

. It is given that the lines

AD, BE, CF

M_A, M_B, M_C

are the midpoints of sides

BC, AC, AB

respectively, and

O

is the circumcenter of

ABC

. Moreover, we define

P=EF\cap M_AM_B, S=DE\cap M_AM_C

O, P, S

1

Taking fish in a path

In La-La-Land there are 5784 cities. Alpaca chooses for each pair of cities to either build a road or a river between them, and additionally she places a fish in each city to defend it. Subsequently Bear chooses a city to start his trip. At first, he chooses whether to take his trip in a car or in a boat. A boat can sail through rivers but not drive on roads, and a car can drive on roads but not sail through rivers. When Bear enters a city he takes the fish defending it, and consequently the city collapses and he can't return to it again. What is the maximum number of fish Bear can guarantee himself, regardless of the construction of the paths?
Remarks: Bear takes a fish also from the city he begins his trip from (and the city collapses). All roads and rivers are two-way.

1

Sum over partitions

n\geq 2

be an integer. For each natural

m

and each integer sequence

0<k_1<k_2<\cdots <k_m

k_1+\cdots+k_m=n

, Michael wrote down the number

\frac{1}{k_1\cdot k_2\cdots k_m}

on the board. Prove that the sum of the numbers on the board is less than

1

1

All primes are non-residues

Find all primes

p

, so that for every prime

q<p

x\in \mathbb{Z}

p\nmid x^2-q