MathDB

6

Part of 2024 Malaysian IMO Training Camp

Problems(2)

Three murderous tangent circles

Source: Own. Malaysian IMO TST 2024 P6

4/21/2024
Let ω1\omega_1, ω2\omega_2, ω3\omega_3 are three externally tangent circles, with ω1\omega_1 and ω2\omega_2 tangent at AA. Choose points BB and CC on ω1\omega_1 so that lines ABAB and ACAC are tangent to ω3\omega_3. Suppose the line BCBC intersect ω3\omega_3 at two distinct points, and XX is the intersection further away to BB and CC than the other one.
Prove that one of the tangent lines of ω2\omega_2 passing through XX, is also tangent to an excircle of triangle ABCABC.
Proposed by Ivan Chan Kai Chin
geometry
Pushing triominoes without undo-ing

Source: Own. Malaysian SST 2024 P6

9/5/2024
Let nn be a positive integer, and Megavan has a (3n+1)×(3n+1)(3n+1)\times (3n+1) board. All squares, except one, are tiled by non-overlapping 1×31\times 3 triominoes. In each step, he can choose a triomino that is untouched in the step right before it, and then shift this triomino horizontally or vertically by one square, as long as the triominoes remain non-overlapping after this move.
Show that there exist some kk, such that after kk moves Megavan can no longer make any valid moves irregardless of the initial configuration, and find the smallest possible kk for each nn.
(Note: While he cannot undo a move immediately before the current step, he may still choose to move a triomino that has already been moved at least two steps before.)
Proposed by Ivan Chan Kai Chin
combinatorics