ITOT 42 Spring 2021 - Junior O (THCS, Basic) - additive practice

Is it possible that the product of nine consecutive positive integers equals the sum of nine consecutive positive integers (not necessarily the same nine integers)? (4 points)



Editorial illustration; no extra mathematical assumptions.

Practice scoring: add all five whole-question scores (maximum 24). Original ITOT scoring adds only the three highest whole-question scores (maximum 16); sum each question’s subparts before ranking. The importer does not enforce best-three scoring. Duration: 300 minutes, based on historical organizer general rules, not an individually printed time on this paper.

Let $AX$ and $BZ$ be altitudes of triangle $ABC$, and let $AY$ and $BT$ be its angle bisectors. Given $\angle XAY=\angle ZBT$, must triangle $ABC$ be isosceles? (4 points)



Editorial illustration; no extra mathematical assumptions.

Maria has four weights that look the same but have masses 1001 g,1002 g,1004 g and 1005 g. Using a balance scale that distinguishes a heavier pan from equal weights, can she certainly determine the mass of each weight in four weighings? Later weighings may depend on earlier results. (4 points)



Editorial illustration; no extra mathematical assumptions.

(a) Can a square be split into four isosceles triangles, no two of which are congruent? (3 points)

(b) Can an equilateral triangle be split into four isosceles triangles, no two of which are congruent? (3 points)

A rectangular grid consists of square cells. Some dominoes occupy two side-adjacent cells each; no two dominoes touch, even at a vertex. The bottom-left and top-right cells are unoccupied. A token starts at the bottom-left and may move one cell right or one cell up through side-adjacent unoccupied cells. Is it always possible to reach the top-right if the board has:

(a)101 columns and 100 rows? (2 points)

(b)$100\times100$ cells? (4 points)