Olympiad problems

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Found problems: 7

2023 IRN-SGP-TWN Friendly Math Competition, 3

Tags: Taiwan , combinatorics , Iran , singapore

Let $N$ and $d$ be two positive integers with $N\geq d+2$. There are $N$ countries connected via two-way direct flights, where each country is connected to exactly $d$ other countries. It is known that for any two different countries, it is possible to go from one to another via several flights. A country is \emph{important} if after removing it and all the $d$ countries it is connected to, there exist two other countries that are no longer connected via several flights. Show that if every country is important, then one can choose two countries so that more than $2d/3$ countries are connected to both of them via direct flights. [i]Proposed by usjl[/i]

2019 Singapore Senior Math Olympiad, 4

Tags: singapore , number theory , partition

Positive integers $m,n,k$ satisfy $1+2+3++...+n=mk$ and $m \ge n$. Show that we can partite $\{1,2,3,...,n \}$ into $k$ subsets (Every element belongs to exact one of these $k$ subsets), such that the sum of elements in each subset is equal to $m$.

2019 Singapore Senior Math Olympiad, 5

Tags: combinatorics , square grid , singapore

Determine all integer $n \ge 2$ such that it is possible to construct an $n * n$ array where each entry is either $-1, 0, 1$ so that the sums of elements in every row and every column are distinct

2019 Singapore Senior Math Olympiad, 2

Tags: graph theory , singapore , combinatorics

Graph $G$ has $n$ vertices and $mn$ edges, where $n>2m$, show that there exists a path with $m+1$ vertices. (A path is an open walk without repeating vertices )

2021 SG Originals, Q5

Tags: Diophantine equation , number theory , algebra , singapore

Find all $a,b \in \mathbb{N}$ such that $$2049^ba^{2048}-2048^ab^{2049}=1.$$ [i]Proposed by fattypiggy123 and 61plus[/i]

2016 Singapore MO Open, 1

Tags: Inversion , similarity , geometry , Triangle , singapore

Let $D$ be a point in the interior of $\triangle{ABC}$ such that $AB=ab$, $AC=ac$, $BC=bc$, $AD=ad$, $BD=bd$, $CD=cd$. Show that $\angle{ABD}+\angle{ACD}=60^{\circ}$. Source: 2016 Singapore Mathematical Olympiad (Open) Round 2, Problem 1

2018 Singapore Senior Math Olympiad, 4

Tags: algebra , inequalities , High school olympiad , singapore

Let $a,b,c,d$ be positive integers such that $a+c=20$ and $\frac{a}{b}+\frac{c}{d}<1$. Find the maximum possible value of $\frac{a}{b}+\frac{c}{d}$.