Discussions | OlympiadGenie

Number theoryResolved

A fourth-power congruence without dividing by 2 modulo n

Newton identities look useful, but n can be even, so dividing by 2 seems risky. Can we prove the fourth-power statement with integer identities only? Problem: Let

n

be a positive integer and let

a, b, c, d

be integers such that

n|a + b + c + d

and

n|a^2 + b^2 + c^2 + d^2

. Show that

n|a^4 + b^4 + c^4 + d^4 + 4abcd.

Source: Baltic Way 2016 · MNET-01DS

mana_snack · Mar 17, 20262 replies

AlgebraResolved

An inequality for every real x: why the even exponents all work

Odd n fails at x=-1. For even n I can check small cases, but I don't see a uniform proof that avoids differentiating a degree-n polynomial. Problem: Find all positive integers

n

for which

3x^n + n(x + 2) - 3 \ge nx^2

holds for all real numbers

x

. Source: Baltic Way 2016 · MNET-01CU

blue_shell_inc · Mar 4, 20261 reply

CombinatoricsResolved

A divisor-subtraction game where parity is only half the classification

Odd starting numbers look losing, but not every even number wins: 2 and 8 seem to lose as well. What is the pattern, and how can I prove it instead of extending a table? Problem: Players

A

and

B

play a game: given a positive integer

n

, one has to choose a divisor

m

of

n

which is greater than

1

and smaller than

n

and replace

n

with

n - m

. Player

A

makes the first move, players move alternately. The player who can't make a move loses the game. For which starting positive number

n

player

B

has a winning strategy? Source: Baltic Way 2013 · MNET-019Y

pressF_17 · Feb 27, 20266 replies

Geometry

Local concyclicity for every five points — how does it become global?

The hypothesis only gives four concyclic points at a time. I can prove useful things once five points share a circle, but getting that fifth point seems to be the main obstacle. Problem: There are

2021

points on a plane, no three of which are collinear. For every

5

points there exists at least

4

among them which are concyclic. Is it necessarily true that at least

2020

of the points are concyclic? Source: Baltic Way 2021 Shortlist · MNET-01HW

quicksave_q · Feb 12, 20263 replies

AlgebraResolved

The evenness bottleneck in a quadratic functional equation

The candidates f=0 and f(x)=x² work. My problem is proving f is even: comparing y and −y seems to give two alternatives that could change with y. Problem: Find all functions

f: \mathbb{R} \to \mathbb{R}

satisfying, for all

x \neq 0

and all

y

,

f(x + y^2) = f(x) + f(y)^2 + \frac{2f(xy)}{x}.

Source: Baltic Way 2015 Shortlisted Problems · MNET-01CF

bidoof_supreme · Jan 22, 20266 replies

Geometry

Three circles through P: checking the converse of the radical-axis argument

I can see why a second common point should force the three opposite-vertex diagonals to concur. In the converse, how do I know the second intersection is real and distinct from P rather than a tangency? I'd like to settle that before calling the proof complete. Problem: Let

ABCDEF

be a cyclic and convex hexagon. A point

P

is called *admissible* if it does not lie on the circumcircle of

ABCDEF

or on any of the lines

AD

,

BE

and

CF

. An admissible point is called *fantastic* if

\odot(ADP)

,

\odot(BEP)

and

\odot(CFP)

intersect in exactly two points. Prove that if there exists a fantastic point, then all admissible points are fantastic. Source: Baltic Way 2023 Shortlist · MNET-01J8

midnaMain · Jan 8, 20260 replies

Number theoryResolved

An indefinite quadratic Diophantine system that turns out to be finite

The first equation has a negative square, so I expected infinitely many solutions. How does the linear equation cut the possibilities down? Problem: Find all combinations of four integers

(a, b, c, d)

satisfying the equations

\begin{cases} -a^2 + b^2 + c^2 + d^2 = 1 \\ 3a + b + c + d = 1. \end{cases}

Source: Baltic Way 2016 · MNET-01CT

gengarHours · Dec 17, 20253 replies

AlgebraResolved

A fourth-degree-looking inequality after switching to sum and difference

Expanding powers of a and b makes this uglier. Is the substitution

(a+b)/2,(a-b)/2

useful here, or does it only disguise the same algebra? Problem: Consider four positive real numbers

a

,

b

,

c

and

d

, satisfying

a^2 + ab + b^2 = 3c^2 \quad \text{and} \quad a^3 + a^2b + ab^2 + b^3 = 4d^3.

Prove that

a + b + d \le 3c.

Source: Baltic Way 2015 Shortlisted Problems · MNET-01CC

zipline_zebra · Dec 2, 20252 replies

CombinatoricsResolved

Stirling numbers modulo p: why i must stay below p

The recurrence for set partitions doesn't seem to reveal period p−1 in the first argument. Is there a formula where Fermat's theorem can be applied term by term? Problem: For positive integers

n

,

k

let

S(n, k)

be the number of ways to partition a set with

n

elements into

k

non-empty parts. For example

S(3, 2) = 3

, because there are three ways to partition the set

\{1, 2, 3\}

into two parts. For another example,

S(4, 2) = 7

. Let

p

be a prime number, and

m

,

n

positive integers such that

p-1 \mid m-n

. Prove that

S(m, i) \equiv S(n, i) \pmod{p}

for all

i = 1, 2, \dots, p-1

. Source: Baltic Way 2023 Shortlist · MNET-01IR

checkpoint_rat · Nov 23, 20253 replies

Number theoryResolved

A divisor of (4k²−1)² and the descent hidden in the complementary factor

For even k the divisor is obvious, taking

n=k/2

. I need the odd-k obstruction. Direct modular arithmetic doesn't seem to distinguish the cases. Problem: Let

k

be a positive integer. Prove that the number

\left(4 k^{2}-1\right)^{2}

has a positive divisor of the form

8 k n-1

if and only if

k

is even. Source: IMO 2007 Shortlist · MNET-06PT

waddledee42 · Nov 9, 20256 replies

CombinatoricsResolved

Where does the golden-ratio constant in cross-intersecting families come from?

The constant

(3-\sqrt5)/2

seems too specific to come from a crude pigeonhole argument. Is there a construction whose two family sizes naturally balance there? Problem: Let

\alpha < \frac{3 - \sqrt{5}}{2}

be a positive real number. Prove that there exist positive integers

n

and

p > \alpha \cdot 2^{n}

for which one can select

2p

pairwise distinct subsets

S_{1}, \ldots, S_{p}, T_{1}, \ldots, T_{p}

of the set

\{1, 2, \ldots, n\}

such that

S_{i} \cap T_{j} \neq \varnothing

for all

1 \leq i, j \leq p

. Source: IMO 2007 Shortlist · MNET-06PH

last_checkpoint · Oct 20, 20256 replies

CombinatoricsResolved

Greedy laundry bins: why total load gives the wrong capacity bound

The total before a wash stays below 3K, but the proposed answer for each bin is 5K/2. How do we improve a total-load bound into a sharp bound for one bin? Problem: A family wears three colours of clothing: red, blue and green, with a separate laundry bin for each colour. Each week, the family generates a total of

K

kilogrammes of laundry (the proportion of each colour is subject to variation). The laundry is first sorted by colour and disposed of in the bins. Next, the heaviest bin is emptied and its contents washed. What is the storing capacity required of the laundry bins if they must never overflow? Source: Baltic Way 2015 Shortlisted Problems · MNET-01CP

spriggan_8 · Oct 4, 20255 replies

Combinatorics

Adjacent products mod 2013: why should multiples of 3 form a block?

Multiplying all adjacent products only gives a weak congruence because 2013 is composite. The solution apparently uses blocks of multiples of 3, 11 and 61. I'm stuck on the first block claim. Problem: Determine whether it is possible to place the integers

1,2, \ldots, 2012

in a circle in such a way that the 2012 products of adjacent pairs of numbers leave pairwise distinct remainders when divided by 2013. Source: IMO selection problems, 2013 · MNET-0D2V

potatOS.exe · Sep 28, 20252 replies

AlgebraResolved

Divisibility forces a power function — but why must the exponent divide n?

After normalizing

f(0)=0

, I see that

f(p)

is a signed power of p for primes p. I can't see why that information on an infinite set of primes determines f at every integer. Problem: Let

n \geq 1

be an odd integer. Determine all functions

f

from the set of integers to itself such that for all integers

x

and

y

the difference

f(x)-f(y)

divides

x^{n}-y^{n}

. Source: IMO 2011 Shortlist · MNET-06RM

cursed_bonfire · Sep 5, 20257 replies

CombinatoricsResolved

A hidden bijection in the two-office incidence problem

The condition reminds me of a projective plane, but I don't want to assume a classification theorem. Can the degree of Amr be forced directly from the unique-common-neighbor condition? Problem: A Saudi company has two offices. One office is located in Riyadh and the other in Jeddah. To insure the connection between the two offices, the company has designated from each office a number of correspondents so that (a) each pair of correspondents from the same office share exactly one common correspondent from the other office. (b) there are at least 10 correspondents from Riyadh. (c) Zayd, one of the correspondents from Jeddah, is in contact with exactly 8 correspondents from Riyadh. What is the minimum number of correspondents from Jeddah who are in contact with the correspondent Amr from Riyadh? Source: IMO selection problems, 2013 · MNET-0D2J

sleepyVivi · Aug 16, 20254 replies

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