MathDB

2018 Dutch IMO TST

Part of Dutch IMO TST

Subcontests

(4)
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a pawn moves on 2m x 2n grid, a snail wants to avoid the pawn's square

Suppose a grid with 2m2m rows and 2n2n columns is given, where mm and nn are positive integers. You may place one pawn on any square of this grid, except the bottom left one or the top right one. After placing the pawn, a snail wants to undertake a journey on the grid. Starting from the bottom left square, it wants to visit every square exactly once, except the one with the pawn on it, which the snail wants to avoid. Moreover, it wants to fi nish in the top right square. It can only move horizontally or vertically on the grid. On which squares can you put the pawn for the snail to be able to finish its journey?

c(a^3+b^3)= a(b^3+c^3)= b(c^3+a^3), a(a^3+b^3)= b(b^3+c^3)= c(c^3+a^3)

(a) If c(a3+b3)=a(b3+c3)=b(c3+a3)c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3) with a,b,ca, b, c positive real numbers, does a=b=ca = b = c necessarily hold? (b) If a(a3+b3)=b(b3+c3)=c(c3+a3)a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3) with a,b,ca, b, c positive real numbers, does a=b=ca = b = c necessarily hold?

nice set of lines if every lines intersects odd no of other lines of set

A set of lines in the plan is called nice i f every line in the set intersects an odd number of other lines in the set. Determine the smallest integer k≥0k \ge 0 having the following property: for each 20182018 distinct lines ℓ1,ℓ2,...,ℓ2018\ell_1, \ell_2, ..., \ell_{2018} in the plane, there exist lines ℓ2018+1,ℓ2018+2,...,ℓ2018+k\ell_{2018+1},\ell_{2018+2}, . . . , \ell_{2018+k} such that the lines ℓ1,ℓ2,...,ℓ2018+k\ell_1, \ell_2, ..., \ell_{2018+k} are distinct and form a nice set.
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f_1(f_2(y) - x)+ 2x = f_3(x + y) => f(x - f(x))= 0

Let AA be a set of functions f:R→Rf : R\to R. For all f1,f2∈Af_1, f_2 \in A there exists a f3∈Af_3 \in A such that f1(f2(y)−x)+2x=f3(x+y)f_1(f_2(y) - x)+ 2x = f_3(x + y) for all x,y∈Rx, y \in R. Prove that for all f∈Af \in A, we have f(x−f(x))=0f(x - f(x))= 0 for all x∈Rx \in R.

students in a classroom sit in a round table, possible to split into 3 groups

In the classroom of at least four students the following holds: no matter which four of them take seats around a round table, there is always someone who either knows both of his neighbours, or does not know either of his neighbours. Prove that it is possible to divide the students into two groups such that in one of them, all students know one another, and in the other, none of the students know each other.
(Note: if student A knows student B, then student B knows student A as well.)

FN is tangent to the circle through B, I,C, incenter and circumcircle related

In a non-isosceles triangle ABCABC the centre of the incircle is denoted by II. The other intersection point of the angle bisector of ∠BAC\angle BAC and the circumcircle of △ABC\vartriangle ABC is DD. The line through II perpendicular to ADAD intersects BCBC in FF. The midpoint of the circle arc BCBC on which AA lies, is denoted by MM. The other intersection point of the line MIMI and the circle through B,IB, I and CC, is denoted by NN. Prove that FNFN is tangent to the circle through B,IB, I and CC.
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a_k is smallest integer > a_{k-1} for which a_k +a_{k-1} is perfect square

Let n≥0n \ge 0 be an integer. A sequence a0,a1,a2,...a_0,a_1,a_2,... of integers is de fined as follows: we have a0=na_0 = n and for k≥1,akk \ge 1, a_k is the smallest integer greater than ak−1a_{k-1} for which ak+ak−1a_k +a_{k-1} is the square of an integer. Prove that there are exactly ⌊2n⌋\lfloor \sqrt{2n}\rfloor positive integers that cannot be written in the form ak−aℓa_k - a_{\ell} with k>ℓ≥0k > \ell\ge 0.

AE=BE, AF =CF, <BTE= <CTF=90^o, prove TA^2 =TB \cdot TC

Let ABCABC be an acute triangle, and let DD be the foot of the altitude through AA. On ADAD, there are distinct points EE and FF such that ∣AE∣=∣BE∣|AE| = |BE| and ∣AF∣=∣CF∣|AF| =|CF|. A pointT≠D T \ne D satis es ∠BTE=∠CTF=90o\angle BTE = \angle CTF = 90^o. Show that ∣TA∣2=∣TB∣⋅∣TC∣|TA|^2 =|TB| \cdot |TC|.

(a+b)^3-2a^3-2b^3 is a power of two

Determine all pairs (a,b)(a,b) of positive integers such that (a+b)3−2a3−2b3(a+b)^3-2a^3-2b^3 is a power of two.
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for every positive divisor d of n, exists (d-k) divisor of n

Find all positive integers nn, for which there exists a positive integer kk such that for every positive divisor dd of nn, the number d−kd - k is also a (not necessarily positive) divisor of nn.

tangent of circumcircle perpenciular to line, right triangle starting with

Suppose a triangle △ABC\vartriangle ABC with ∠C=90o\angle C = 90^o is given. Let DD be the midpoint of ACAC, and let EE be the foot of the altitude through CC on BDBD. Show that the tangent in CC of the circumcircle of △AEC\vartriangle AEC is perpendicular to ABAB.

f(x^2) - f(y^2) <= (f(x)+y) (x-f(y))

Find all functions f:R→Rf : R \to R such that f(x2)−f(y2)≤(f(x)+y)(x−f(y))f(x^2)-f(y^2) \le (f(x)+y) (x-f(y)) for all x,y∈Rx, y \in R.