MathDB

2021 Dutch IMO TST

Part of Dutch IMO TST

Subcontests

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2-player game on a m x n board

Stekel and Prick play a game on an m×n m \times n board, where mm and nn are positive are integers. They alternate turns, with Stekel starting. Spine bets on his turn, he always takes a pawn on a square where there is no pawn yet. Prick does his turn the same, but his pawn must always come into a square adjacent to the square that Spike just placed a pawn in on his previous turn. Prick wins like the whole board is full of pawns. Spike wins if Prik can no longer move a pawn on his turn, while there is still at least one empty square on the board. Determine for all pairs (m,n)(m, n) who has a winning strategy.

PQ passes through centroid, 3 centroids, <ACB =<CEP=<CFQ=90^o, CP=AP, CQ=BQ

Let ABCABC be a right triangle with ∠C=90o\angle C = 90^o and let DD be the foot of the altitude from CC. Let EE be the centroid of triangle ACDACD and let FF be the centroid of triangle BCDBCD. The point PP satisfies ∠CEP=90o\angle CEP = 90^o and ∣CP∣=∣AP∣|CP| = |AP|, while point QQ satisfies ∠CFQ=90o\angle CFQ = 90^o and ∣CQ∣=∣BQ∣|CQ| = |BQ|. Prove that PQPQ passes through the centroid of triangle ABCABC.
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a_{n+1} - a_n = n(a_n - 1)

The sequence of positive integers a0,a1,a2,...a_0, a_1, a_2, . . . is defined by a0=3a_0 = 3 and an+1−an=n(an−1)a_{n+1} - a_n = n(a_n - 1) for all n≥0n \ge 0. Determine all integers m≥2m \ge 2 for which gcd(m,an)=1gcd (m, a_n) = 1 for all n≥0n \ge 0.

covering a m x n board with dominos

Let mm and nn be natural numbers with mnmn even. Jetze is going to cover an m×nm \times n board (consisting of mm rows and nn columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer VV (in terms of mm and nn) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than VV, no matter how Jetze covers the board.

EX=EY wanted, 3 circles related

Let Γ\Gamma be the circumscribed circle of a triangle ABCABC and let DD be a point at line segment BCBC. The circle passing through BB and DD tangent to Γ\Gamma and the circle passing through CC and DD tangent to Γ\Gamma intersect at a point E≠DE \ne D. The line DEDE intersects Γ\Gamma at two points XX and YY . Prove that ∣EX∣=∣EY∣|EX| = |EY|.
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at least 2/3 of the squares of the m x n board are covered with dominos

On a rectangular board with m×nm \times n squares (m,n≥3m, n \ge 3) there are dominoes (2×12 \times 1 or 1×21\times 2 tiles), which do not overlap and do not extend beyond the board. Every domino covers exactly two squares of the board. Assume that the dominos cover the has the property that no more dominos can be added to the board and that the four corner spaces of the board are not all empty. Prove that at least 2/32/3 of the squares of the board are covered with dominos.

an^k, bn^{\ell} , cn^m are sidelengths of a triangle

Determine all positive integers nn with the following property: for each triple (a,b,c)(a, b, c) of positive real numbers there is a triple (k,ℓ,m)(k, \ell, m) of non-negative integer numbers so that ankan^k, bnℓbn^{\ell} and cnmcn^m are the lengths of the sides of a (non-degenerate) triangle shapes.

p is a divisor of 5^m7^n-1

Let p>10p > 10 be prime. Prove that there are positive integers mm and nn with m+n<pm + n < p exist for which pp is a divisor of 5m7n−15^m7^n-1.
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