MathDB

2019 Dutch IMO TST

Part of Dutch IMO TST

Subcontests

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max (a^2/b,b^2/a) max (c^2/d,d^2/c) =(min (a + b, c + d))^4

Determine all 44-tuples (a,b,c,d)(a,b, c, d) of positive real numbers satisfying a+b+c+d=1a + b +c + d = 1 and max⁡(a2b,b2a)⋅max⁡(c2d,d2c)=(min⁡(a+b,c+d))4\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4

exactly one of the equalities f(g(x)) = x and g(f(x)) = x holds

Write SnS_n for the set {1,2,...,n}\{1, 2,..., n\}. Determine all positive integers nn for which there exist functions f:Sn→Snf : S_n \to S_n and g:Sn→Sng : S_n \to S_n such that for every xx exactly one of the equalities f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x holds.

n^2 + n + 1 cannot be product of 2 integers with difference <2\sqrt{n}

Let nn be a positive integer. Prove that n2+n+1n^2 + n + 1 cannot be written as the product of two positive integers of which the difference is smaller than 2n2\sqrt{n}.
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max of gcd(a, b) + gcd(b, c) + gcd(c, a) when a + b + c = 5n.

Let nn be a positive integer. Determine the maximum value of gcd(a,b)+gcd(b,c)+gcd(c,a)gcd(a, b) + gcd(b, c) + gcd(c, a) for positive integers a,b,ca, b, c such that a+b+c=5na + b + c = 5n.

2 lines and a circle concurrent, 2 circumcircles and parallelogram related

Let ABCABC be an acute angles triangle with OO the center of the circumscribed circle. Point QQ lies on the circumscribed circle of △BOC\vartriangle BOC so that OQOQ is a diameter. Point MM lies on CQCQ and point NN lies internally on line segment BCBC so that ANCMANCM is a parallelogram. Prove that the circumscribed circle of △BOC\vartriangle BOC and the lines AQAQ and NMNM pass through the same point.

functional in Z, f(f(x)) = x, if x + y is odd then f(x) + f(y) \ge x + y

Find all functions f:Z→Zf : Z \to Z satisfying the following two conditions: (i) for all integers xx we have f(f(x))=xf(f(x)) = x, (ii) for all integers xx and y such that x+yx + y is odd, we have f(x)+f(y)≥x+yf(x) + f(y) \ge x + y.
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|a|,|b| >= 2017 , P(a^2+b^2) >= P(2ab) , P trinomial

Let P(x)P(x) be a quadratic polynomial with two distinct real roots. For all real numbers aa and bb satisfying ∣a∣,∣b∣≥2017|a|,|b| \ge 2017, we have P(a2+b2)≥P(2ab)P(a^2+b^2) \ge P(2ab). Show that at least one of the roots of PP is negative.

pupils and best friends go to either Rome or Paris

In each of the different grades of a high school there are an odd number of pupils. Each pupil has a best friend (who possibly is in a different grade). Everyone is the best friend of their best friend. In the upcoming school trip, every pupil goes to either Rome or Paris. Show that the pupils can be distributed over the two destinations in such a way that (i) every student goes to the same destination as their best friend; (ii) for each grade the absolute difference between the number of pupils that are going to Rome and that of those who are going to Paris is equal to 11.

Tangent circles to random line yielding cyclic quadrilateral

Let ABCDABCD be a cyclic quadrilateral (In the same order) inscribed into the circle ⊙(O)\odot (O). Let AC‾\overline{AC} ∩\cap BD‾\overline{BD} == EE. A randome line ℓ\ell through EE intersects AB‾\overline{AB} at PP and BCBC at QQ. A circle ω\omega touches ℓ\ell at EE and passes through DD. Given, ω\omega ∩\cap ⊙(O)\odot (O) == RR. Prove, Points B,Q,R,PB,Q,R,P are concyclic.
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Parallel through Random points yield cyclic quadrilateral

Let ΔABC\Delta ABC be a scalene triangle. Points D,ED,E lie on side AC‾\overline{AC} in the order, A,E,D,CA,E,D,C. Let the parallel through EE to BCBC intersect ⊙(ABD)\odot (ABD) at FF, such that, EE and FF lie on the same side of ABAB. Let the parallel through EE to ABAB intersect ⊙(BDC)\odot (BDC) at GG, such that, EE and GG lie on the same side of BCBC. Prove, Points D,F,E,GD,F,E,G are concyclic

divisibility functional with primes , f(p) > 0, p| (f(x) + f(p))^{f(p)}- x

Find all functions f:Z→Zf : Z \to Z satisfying ∙\bullet f(p)>0 f(p) > 0 for all prime numbers pp, ∙\bullet p∣(f(x)+f(p))f(p)−xp| (f(x) + f(p))^{f(p)}- x for all x∈Zx \in Z and all prime numbers pp.

300 participants, n games of chess, maximum n wanted

There are 300300 participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of nn for which it is possible to satisfy the following conditions at the same time: each contestant plays at most nn games of chess, and for each mm with 1≤m≤n1 \le m \le n, there is a contestant playing exactly mm games of chess.