Determine all pairs \((a,b)\) of positive integers for which there exist positive integers \(g\) and \(N\) such that

\[ \gcd(a^n+b,b^n+a)=g \]

holds for all integers \(n\ge N\).

Let \(N\) be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers \(1,2,\dots,N\) on a board. They then take turns to make a move, starting with Geoff.

Each move consists of choosing a pair of integers \((k,n)\), where \(k\ge0\) and \(n\) is one of the integers on the board, and then erasing every integer \(s\) on the board such that \(2^k\mid n-s\).

The game continues until the board is empty. The player who erases the last integer on the board loses.

Determine all values of \(N\) for which Geoff can ensure that he wins, no matter how Ceri plays.

Let \(a,b,c,d\) be real numbers such that \(a^2+b^2+c^2+d^2=1\). Determine the minimum value of \((a-b)(b-c)(c-d)(d-a)\), and determine all \((a,b,c,d)\) for which the minimum is attained.

Let \(\mathcal F\) be the set of all functions \(f:\mathbb R\to\mathbb R\) such that

\[ f(x+f(y))=f(x)+f(y) \]

for every \(x,y\in\mathbb R\). Find all rational numbers \(q\) such that for every \(f\in\mathcal F\), there exists some \(z\in\mathbb R\) satisfying \(f(z)=qz\).

Let \(\{a_k\}_{k\ge1}\) be an infinite sequence of positive integers such that, for every \(k\ge1\), \(a_k\) has exactly \(k\) prime divisors, counted with multiplicity.

Must there exist infinitely many \(i\in\mathbb N\) such that \(a_{i+1}\ge2a_i\)?

Let \(a,b,c\) be positive real numbers and let \(x\le y\le z\) be real numbers such that \(a+b+c=1\) and \(ax+by+cz=0\). Prove that

\[ a|x|+b|y|+c|z|\le\frac{z-x}{2}. \]

Let \(m\ge2\) be a positive integer. Suppose \(-1<a_1<a_2<\cdots<a_m<1\) are rational numbers, and let \(T\) be the set of all numbers of the form

\[ \frac{a_i+a_j}{1-a_i a_j},\qquad 1\le i\le j\le m. \]

Determine the minimum possible value of \(|T|\).

A binary word consisting of \(0\)'s and \(1\)'s is called a palindrome if it reads the same forwards and backwards. A binary word is called \(n\)-palindromic if it can be partitioned into \(n\) pairwise disjoint palindromic blocks.

Show that there exists a binary word that is not \(n\)-palindromic for every \(n=1,2,\ldots,26\).

Prove that for all positive real numbers \(a,b,c\) satisfying \(abc=1\),

\[ \frac{a+3}{3b}+\frac{b+3}{3c}+\frac{c+3}{3a}\ge a+b+c+1. \]

Let \(a\) and \(b\) be two distinct positive integers. Prove that

\[ \prod_{i=0}^{2026}\gcd(a+i,b+i)\mid 2026!\,(a-b). \]

Determine all positive real numbers \(a,b,c,d\) satisfying

\[ a+\left\lfloor\frac{b^2}{a}\right\rfloor=c+d,\qquad b+\left\lfloor\frac{c^2}{b}\right\rfloor=d+a, \]

\[ c+\left\lfloor\frac{d^2}{c}\right\rfloor=a+b,\qquad d+\left\lfloor\frac{a^2}{d}\right\rfloor=b+c. \]

In the Mushroom Kingdom, \(20250\) cities are connected by \(202500\) directed roads, each joining two distinct cities. Each road is to be assigned a label from

\[ \{-1000,-999,\ldots,-1,1,\ldots,999,1000\}. \]

Prove that the labels can be chosen so that the sum of the labels along every directed cycle is \(0\).

Prove that there exists a constant \(c>0\) such that, for every positive integer \(n\),

\[ \operatorname{lcm}(n+1,n+4,\ldots,n+2025) \ge c(n+1)(n+4)\cdots(n+2025). \]

Determine all positive integers \(n\) such that, for every positive integer \(m\), there exists a positive integer \(k\ge2025\) satisfying

\[ n+m\mid n^k+m. \]

Let \((a_n)_{n\ge1}\) and \((b_n)_{n\ge1}\) be sequences of positive real numbers such that \(a_1,b_1<5\) and, for every positive integer \(n\),

\[ a_{n+1}=\frac{b_n+\sqrt{a_nb_n}}{2} \qquad\text{and}\qquad b_{n+1}=\sqrt{\frac{a_n(a_n+b_n)}{2}}. \]

Prove that

\[ |a_{20}-b_{20}|<\frac1{2025}. \]

Let \(f:\mathbb R\to\mathbb R\) satisfy:

• \(f(x+f(y))=f(x)+f(y)\) for all \(x,y\in\mathbb R\);
• there exists a real number \(x\) such that \(f(x)=\frac1{2025}\); and
• there is no real number \(x\) such that \(0<f(x)<\frac1{2025}\).

Determine all possible images of \(f\).

Let \(S\) be the set of all triples of positive real numbers \((a,b,c)\) such that \(a+b+c=ab+bc+ca\).

1. Prove that \(\min\{a+b,b+c,c+a\}>1\).
2. Does there exist a triple \((a,b,c)\in S\) such that \(\min\{a+b,b+c,c+a\}<1+\frac1{20^{25}}\)?

Find the number of ordered pairs of positive integers \((a,b)\) with \(1\le a,b\le19^2\) such that \(a^4+b^3\) is divisible by \(19^2\).

Find the number of positive integer pairs \((a,b)\) with \(1\le a,b\le2027\) such that

\[ 2027\mid a^6+b^5+b^2. \]

Find all functions \(f:\mathbb N\to\mathbb N\) such that, for every prime number \(p\) and positive integer \(x\),

\[ \{x,f(x),\dots,f^{p-1}(x)\} \]

is a complete residue system modulo \(p\).

Determine all functions \(f:\mathbb R\to\mathbb R\) such that, for every real \(x,y\),

\[ f(f(x)+y)=\lfloor x+f(f(y))\rfloor. \]

Let \(a,b,c\) be three distinct positive integers. Define \(S(a,b,c)\) as the set of all rational roots of \(px^2+qx+r=0\), over every permutation \((p,q,r)\) of \((a,b,c)\).

For example, \(S(1,2,3)=\{-1,-2,-1/2\}\): the equation \(x^2+3x+2=0\) has roots \(-1,-2\), the equation \(2x^2+3x+1=0\) has roots \(-1,-1/2\), and the other permutations yield no rational roots.

Determine the maximum possible number of elements of \(S(a,b,c)\).