IMSO 2011 International Mathematics — Exploration

A $5\times5$ grid is divided into five regions as shown. Fill the empty cells with digits 1,2,3,4,5 so each row, each column and each diagonal contains each digit once. The sums of all digits in each region are equal. Two top-row cells already contain 1 and 2. Complete the grid. What are A,B,C,D,E?

The diagram has nine circles connected by six line segments. One circle contains 6. Choose nine consecutive positive integers including 6 and put each integer in a circle so the sum along every line segment is 23.

Four teams A,B,C,D play a one-week double round-robin tournament: every pair plays twice and each team plays at most once a day. Part of the final scoreboard is shown. The missing part was split into four fragments with scores on only one side. A black circle indicates a win and a white circle a loss. Arrange the fragments in the scoreboard. Which team is champion?

The figure can be formed using some or all of the distinct pieces shown. Pieces can be rotated and reversed, but each may be used only once. Which combination forms the target? Use colors to show the pieces in the figure provided on the answer sheet.

A dartboard is divided into 20 numbered sectors. Each turn uses 3 darts. A bull (smallest central circle) scores 50; the surrounding half-bull ring scores 25. The unshaded part of a sector scores its number, the outer gray ring doubles it, and the middle gray ring triples it. The black outer area scores 0. The illustrated darts score $25+3\times13+12=76$.

(a) Ruthee scores 137 with two triples and one single, neither bull nor half-bull. Find all ways.

(b) Archie scores 137 with two triples and a double. Find all ways.

(c) Archie throws three darts, each scoring a double or triple. What total scores are impossible?

The arms of the balance are divided into equal sections. A weight two sections from a pivot has twice the turning effect of an equal weight one section away. In which pan should you put each of the weights shown to balance the whole system? Write the weight in grams in each corresponding pan. The printed weights are 1,2,2,5,7,9,10,11,11,15,20,22.