Discussions | OlympiadGenie

AlgebraResolved

Four trigonometric values in arithmetic progression: avoiding six orderings

I started checking all orders of sin, cos, tan, cot and it became a mess. Is there a symmetry that cuts the number of cases down? Problem: Does there exist an angle

\alpha \in (0, \pi / 2)

such that

\sin \alpha

,

\cos \alpha

,

\tan \alpha

and

\cot \alpha

, taken in some order, are consecutive terms of an arithmetic progression? Source: Baltic Way 2008 · MNET-0153

slime_ranchero · Sep 30, 20263 replies

AlgebraResolved

A two-variable polynomial eigenvalue equation: why only degree two survives

Trying individual quadratic monomials finds examples, but how do we rule out higher-degree terms without assuming the answer is quadratic? Problem: Find all polynomial functions

P(x, y)

with real coefficients which for all real numbers

x

and

y

satisfy

P(x + y, x - y) = 2P(x, y).

Source: Baltic Way 2020 · MNET-01GH

stardew_afterdark · Aug 21, 20263 replies

Algebra

Two polynomial level sets at 54 and 2013: the factorization bound

The difference 1959 factors as

3\cdot653

. I can bound each level set separately, but I'm not convinced that this proves the claimed sharp bound on their product. Problem: A polynomial

P

with integer coefficients satisfies

P(x_1) = P(x_2) = \dots = P(x_k) = 54

and

P(y_1) = P(y_2) = \dots = P(y_n) = 2013

for distinct integers

x_1, \dots, x_k; y_1, \dots, y_n

. Determine the maximal value of

kn

. Source: Baltic Way 2013 · MNET-019N

LobaLootedIt · Aug 9, 20263 replies

AlgebraResolved

Composing a scaling functional equation in two different orders

I tried linear and quadratic functions and got contradictory guesses for a. Is there a substitution that eliminates f before we start guessing examples? Problem: Find all real numbers

a

for which there exists a non-constant function

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

satisfying the equations

1) \quad f(ax) = a^2 f(x)

2) \quad f(f(x)) = a f(x).

Source: Baltic Way 2016 · MNET-01CW

hollowbean · Jun 25, 20263 replies

Algebra

A Fibonacci sum and the number of ones in its binary expansion

The alternating powers of two suggest a closed form before looking at binary digits. Has anyone found a recurrence for the partial sums that controls carries? I don't want to infer a digit pattern from just a few examples. Problem: Let

\mathbb{Z}^+

be the set of positive integers and

(F_n)_{n \in \mathbb{Z}^+}

be the Fibonacci sequence defined by

F_1 = F_2 = 1

and

F_{n+1} = F_n + F_{n-1}

for

n \ge 2

. Consider the number

N = 2^{2022}F_{2023} - 2^{2021}F_{2022} + 2^{2020}F_{2021} + \dots + 2F_2 + F_1.

Prove that the binary expansion of

N

contains more 1's than 0's. Source: Baltic Way 2023 Shortlist · MNET-01JK

not_a_korok · Apr 3, 20260 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

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

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

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

AlgebraResolved

A nested functional equation: getting from involution to identity

Putting x=0 suggests

f(f(y))=y

, but only if I can cancel f(y). I'm worried that this discards a whole branch of solutions. Problem: Let

\mathbb{R}

denote the set of real numbers. Find all functions

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

such that

x f(f(y)) + y f(y - x) = f(f(x + y) - x) f(y)

for all

x, y \in \mathbb{R}

. Source: Baltic Way 2011 Problem Shortlist · MNET-0185

frostbyte_93 · Aug 3, 20257 replies

AlgebraResolved

The exceptional value 2008 in a functional inequality

I found

f\equiv1

and

f(n)=n

. I suspect

f(n)\le n

, but the shortlist claims there is one more possible value at 2007. Where does that extra one come from? Problem: Consider those functions

f: \mathbb{N} \rightarrow \mathbb{N}

which satisfy the condition

f(m+n) \geq f(m)+f(f(n))-1

for all

m, n \in \mathbb{N}

. Find all possible values of

f(2007)

. (

\mathbb{N}

denotes the set of all positive integers.) Source: IMO 2007 Shortlist · MNET-06P7

bonk.dll · Jul 2, 20258 replies

AlgebraResolved

An unexpectedly loose-looking bound that makes the IMO inequality collapse

Bounding each summand by a constant loses the relationship

x^n+y^n=1

. Has anyone found a termwise estimate that actually uses that constraint? Problem: Let

n

be a positive integer, and let

x

and

y

be positive real numbers such that

x^{n}+y^{n}=1

. Prove that

\left(\sum_{k=1}^{n} \frac{1+x^{2 k}}{1+x^{4 k}}\right)\left(\sum_{k=1}^{n} \frac{1+y^{2 k}}{1+y^{4 k}}\right)<\frac{1}{(1-x)(1-y)} .

Source: IMO 2007 Shortlist · MNET-06P8

YoshiStoleMyTax · May 23, 20254 replies

AlgebraResolved

Why doubling the product fixes the splendid-number obstruction

Two is splendid and four isn't, so both addition and ordinary multiplication fail immediately. I don't see why inserting a factor of 2 into the product should repair things. Problem: We call a positive integer

N

splendid, if

N = (a - b)^2 + (b - c)^2 + (c - a)^2

for some integers

a

,

b

and

c

. If

M

and

N

are splendid positive integers, is the sum

M + N

or the product

MN

also necessarily splendid? How about the product

2MN

? Source: Baltic Way 2020 · MNET-01G9

laggyLucio · Apr 26, 20255 replies

AlgebraResolved

Reversing a recurrence: a matching interpretation instead of matrix multiplication

The transfer matrices don't commute, so reversing their order feels dangerous. Is there a combinatorial expression for the final term that is visibly unchanged by reversing the coefficients? Problem: Let

n

be a positive integer and let

a_1, \cdots, a_{n-1}

be arbitrary real numbers. Define the sequences

u_0, \cdots, u_n

and

v_0, \cdots, v_n

inductively by

u_0 = u_1 = v_0 = v_1 = 1

, and

u_{k+1} = u_k + a_k u_{k-1}, \quad v_{k+1} = v_k + a_{n-k} v_{k-1} \text{ for } k = 1, \cdots, n-1.

Prove that

u_n = v_n

. Source: 2014 International Mathematical Olympiad Second Stage Training Camp · MNET-0G86

shulk_said_no · Apr 7, 20254 replies

Math discussions