Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed. Determine the number of ways in which this can be done. [i]Proposed by Morteza Saghafian, Iran[/i]

There are $256$ lumps of metal that have different weights in pairs. With the help of a beam balance , one may now compare every two lumps. Find the smallest number $m$ such that you can be sure to find the heaviest as well as the lightest lump with the weighing process.

Fix a positive integer $n$ and a finite graph with at least one edge; the endpoints of each edge are distinct, and any two vertices are joined by at most one edge. Vertices and edges are assigned (not necessarily distinct) numbers in the range from $0$ to $n-1$, one number each. A vertex assignment and an edge assignment are [i]compatible[/i] if the following condition is satisfied at each vertex $v$: The number assigned to $v$ is congruent modulo $n$ to the sum of the numbers assigned to the edges incident to $v$. Fix a vertex assignment and let $N$ be the total number of compatible edge assignments; compatibility refers, of course, to the fixed vertex assignment. Prove that, if $N \neq 0$, then the prime divisors of $N$ are all at most $n$.

Consider $n$ weights, $n \ge 2$, of masses $m_1, m_2, ..., m_n$, where $m_k$ are positive integers such that $1 \le m_ k \le k$ for all $k \in \{1,2,...,n\} $: Prove that we can place the weights on the two pans of a balance such that the pans stay in equilibrium if and only if the number $m_1 + m_2 + ...+ m_n$ is even. Estonian Olympiad

Seven identical-looking coins are given, of which four are real and three are counterfeit. The three counterfeit coins have equal weight, and the four real coins have equal weight. It is known that a counterfeit coin is lighter than a real one. In one weighing, one can select two sets of coins and check which set has a smaller total weight, or if they are of equal weight. How many weightings are needed to identify one counterfeit coin?

We are given a balance with two bowls and a number of weights. (a) Give an example of four integer weights using which one can measure any weight of $1,2,...,40$ grams. (b) Are there four weights using which one can measure any weight of $1,2,...,50$ grams?

$6$ indistinguishable coins are given, $4$ are authentic, all of the same weight, and $2$ are false, one is more light than the real ones and the other one, heavier than the real ones. The two false ones together weigh same as two authentic coins. Find two authentic coins using a balance scale twice only by two plates, no weights. Clarification: A two-pan scale only reports if the left pan weighs more, equal or less that right.

We have 101 coins and a two-pan scale. In one weighing, we can compare the weights of two coins. What is the smallest number of weighings required in order to decide whether there exist 51 coins which all have the same weight?

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).

Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed. Determine the number of ways in which this can be done. [i]Proposed by Morteza Saghafian, Iran[/i]

$32$ 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?

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]

Six people, with distinct weights, want to form a triangular position where there are three people in the bottom row, two in the middle row, and one in the top row, and each person in the top two rows must weigh less than both of their supports. How many distinct formations are there?

We have $n{}$ coins, one of which is fake, which differs in weight from the real ones and a two-pan scale which works correctly if the weights on the pans are different, but can show any outcome if the weights on the pans are equal. For what $n{}$ can we determine which coin is fake and whether it is lighter or heavier than the real coins, in at most $k{}$ weightings? [i]Proposed by A. Zaslavsky[/i]

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.

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).

Let $S$ be the sum of integer weights that come with a two pan balance Scale, say $\omega_1 \le \omega_2 \le \omega_3 \le ... \le\omega_n$. Show that all integer-weighted objects in the range $1$ to $S$ can be weighed exactly if and only if $\omega_1=1$ and $$\omega_{j+1} \le 2 \left( \sum_{l=1}^{j} \omega_l\right) +1$$

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

Kostya had two sets of $17$ coins: in one set all the coins were real, and in the other set there were exactly $5$ fakes (all the coins look the same; all real coins weigh the same, all fake coins also weigh the same, but it is unknown lighter or heavier than real ones). Kostya gave away one of the sets friend, and subsequently forgot which of the two sets had stayed. With the help of two weighings, can Kostya on a cup scale without weights, find out which of the two did he give away the sets?

Given $6$ balls: $2$ white, $2$ green, $2$ red, it is known that there is a white, a green and a red that weigh $99$ g each and that the other balls weigh $101$ g each. Determine the weight of each ball using two times a two-plate scale . Clarification: A two-pan scale only reports if the left pan weighs more than, equal to or less than the right.

A postman wants to divide $n$ packages with weights $1, 2, 3, 4, n$ into three groups of exactly the same weight. Can he do this if (a) $n = 2011$ ? (b) $n = 2012$ ?

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]

Consider a $N\times N$ square board where $N\geq 3$ is an odd integer. The caterpillar Carl sits at the center of the square; all other cells contain distinct positive integers. An integer $n$ weights $1\slash n$ kilograms. Carl wants to leave the board but can eat at most $2$ kilograms. Determine whether Carl can always find a way out when a) $N=2003.$ b) $N$ is an arbitrary odd integer.

Let $N$ be a positive integer. It is given set of weights which satisfies following conditions: $i)$ Every weight from set has some weight from $1,2,...,N$; $ii)$ For every $i\in {1,2,...,N}$ in given set there exists weight $i$; $iii)$ Sum of all weights from given set is even positive integer. Prove that set can be partitioned into two disjoint sets which have equal weight

It is known that in a set of five coins three are genuine (and have the same weight) while two coins are fakes, each of which has a different weight from a genuine coin. What is the smallest number of weighings on a scale with two cups that is needed to locate one genuine coin?