Olympiad problems

2013 Junior Balkan Team Selection Tests - Romania, 2

Tags: combinatorics , weight

Weights of $1$ g, $2$ g,$ ...$ , $200$ g are placed on the two pans of a balance such that on each pan there are $100$ weights and the balance is in equilibrium. Prove that one can swap $50$ weights from one pan with $50$ weights from the other pan such that the balance remains in equilibrium. Kvant Magazine

2014 German National Olympiad, 5

Tags: combinatorics , coin , weight

There are $9$ visually indistinguishable coins, and one of them is fake and thus lighter. We are given $3$ indistinguishable balance scales to find the fake coin; however, one of the scales is defective and shows a random result each time. Show that the fake coin can still be found with $4$ weighings.

Kvant 2020, M2628

Tags: combinatorics , weight

There are $m$ identical two-pan weighting scales. One of them is broken and it shows any outcome, at random. The other scales always show the correct outcome. Moreover, the weight of the broken scale differs from those of the other scales, which are all equal. At a move, we may choose a scale and place some of the other scales on its pans. Determine the greatest value of $m$ for which we may find the broken scale with no more than three moves. [i]Proposed by A. Gribalko and O. Manzhina[/i]

2010 Estonia Team Selection Test, 2

Tags: weight , combinatorics

Let $n$ be a positive integer. Find the largest integer $N$ for which there exists a set of $n$ weights such that it is possible to determine the mass of all bodies with masses of $1, 2, ..., N$ using a balance scale . (i.e. to determine whether a body with unknown mass has a mass $1, 2, ..., N$, and which namely).

1999 Estonia National Olympiad, 4

Tags: combinatorics , algebra , weight

$32$ stones, with pairwise different weights, and lever scales without weights are given. How to determine by $35$ scaling, which stone is the heaviest and which is the second by weight?

2022/2023 Tournament of Towns, P5

Tags: combinatorics , weight

There is a single coin on each square of a $5 \times 5$ board. All the coins look the same. Two of them are fakes and have equal weight. Genuine coins are heavier than fake ones and also weigh the same. The fake coins are on the squares sharing just one vertice. Is it possible to determine for sure a) 13 genuine coins; b) 15 genuine coins; and c) 17 genuine coins in a single weighing on a balance with no unit weights? [i]Rustem Zhenodarov, Alexandr Gribalko, Sergey Tokarev[/i]