2019 IOM

Part of International Olympiad of Metropolises

Subcontests

(6)

1

Remainders of a polynomoal mod p

p

be a prime and let

f(x)

be a polynomial of degree

d

with integer coefficients. Assume that the numbers

f(1),f(2),\dots,f(p)

k

distinct remainders when divided by

p

1<k<p

\frac{p-1}{d}\leq k-1\leq (p-1)\left(1-\frac1d \right) .

Dániel Domán, Gauls Károlyi, and Emil Kiss

1

Unfolding an inscribed pyramid

We are given a convex four-sided pyramid with apex

S

ABCD

such that the pyramid has an inscribed sphere (i.e., it contains a sphere which is tangent to each race). By making cuts along the edges

SA,SB,SC,SD

and rotating the faces

SAB,SBC,SCD,SDA

outwards into the plane

ABCD

, we unfold the pyramid into the polygon

AKBLCMDN

as shown in the figure. Prove that

K,L,M,N

are concyclic.
Tibor Bakos and Géza Kós

1

Tactical examination

There are 100 students taking an exam. The professor calls them one by one and asks each student a single person question: “How many of 100 students will have a “passed” mark by the end of this exam?” The students answer must be an integer. Upon receiving the answer, the professor immediately publicly announces the student’s mark which is either “passed” or “failed.”
After all the students have got their marks, an inspector comes and checks if there is any student who gave the correct answer but got a “failed” mark. If at least one such student exists, then the professor is suspended and all the marks are replaced with “passed.” Otherwise no changes are made.
Can the students come up with a strategy that guarantees a “passed” mark to each of them?
Denis Afrizonov

1

Circumcenter on IO

In a non-equilateral triangle

ABC

I

is the incenter and point

O

is the circumcenter. A line

s

I

is perpendicular to

IO

\ell

symmetric to like

BC

with respect to

s

meets the segments

AB

AC

K

L

, respectively (

K

L

are different from

A

). Prove that the circumcenter of triangle

AKL

lies on the line

IO

.
Dušan Djukić

1

Equal primes

Three prime numbers

p,q,r

and a positive integer

n

are given such that the numbers

\frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq}

are integers. Prove that

p=q=r

.
Nazar Agakhanov

1

Hacking the social network

In a social network with a fixed finite setback of users, each user had a fixed set of followers among the other users. Each user has an initial positive integer rating (not necessarily the same for all users). Every midnight, the rating of every user increases by the sum of the ratings that his followers had just before midnight.
Let

m

be a positive integer. A hacker, who is not a user of the social network, wants all the users to have ratings divisible by

m

. Every day, he can either choose a user and increase his rating by 1, or do nothing. Prove that the hacker can achieve his goal after some number of days.
Vladislav Novikov