MathDB

2013 Dutch IMO TST

Part of Dutch IMO TST

Subcontests

(5)
5
2

a + b + c >= \sqrt{\frac13 (a + 2)(b + 2)(c + 2)} when a,b,c>0 and abc=1

Let a,ba, b, and cc be positive real numbers satisfying abc=1abc = 1. Show that a+b+c≥13(a+2)(b+2)(c+2)a + b + c \ge \sqrt{\frac13 (a + 2)(b + 2)(c + 2)}

Cyclic hexagon with two equal sides

Let ABCDEFABCDEF be a cyclic hexagon satisfying AB⊥BDAB\perp BD and BC=EFBC=EF.Let PP be the intersection of lines BCBC and ADAD and let QQ be the intersection of lines EFEF and ADAD.Assume that PP and QQ are on the same side of DD and AA is on the opposite side.Let SS be the midpoint of ADAD.Let KK and LL be the incentres of △BPS\triangle BPS and △EQS\triangle EQS respectively.Prove that ∠KDL=900\angle KDL=90^0.
2
2
3
2

3 circles, tangents, intersections, midpoint wanted

Fix a triangle ABCABC. Let Γ1\Gamma_1 the circle through BB, tangent to edge in AA. Let Γ2\Gamma_2 the circle through C tangent to edge ABAB in AA. The second intersection of Γ1\Gamma_1 and Γ2\Gamma_2 is denoted by DD. The line ADAD has second intersection EE with the circumcircle of △ABC\vartriangle ABC. Show that DD is the midpoint of the segment AEAE.

Sequence of integers and prime numbers

Fix a sequence a1,a2,a3…a_1,a_2,a_3\ldots of integers satisfying the following condition:for all prime numbers pp and all positive integers kk,we have apk+1=pak−3ap+13a_{pk+1}=pa_k-3a_p+13.Determine all possible values of a2013a_{2013}.
4
2

m x 1 and 1 x m rectangles in an nxn board, equal no of type I,II, min N

Let n≥3n \ge 3 be an integer, and consider a n×nn \times n-board, divided into n2n^2 unit squares. For all m≥1m \ge 1, arbitrarily many 1×m1\times m-rectangles (type I) and arbitrarily many m×1m\times 1-rectangles (type II) are available. We cover the board with NN such rectangles, without overlaps, and such that every rectangle lies entirely inside the board. We require that the number of type I rectangles used is equal to the number of type II rectangles used.(Note that a 1×11 \times 1-rectangle has both types.) What is the minimal value of NN for which this is possible?

Congruence relation with binomial coefficients

Determine all positive integers n≥2n\ge 2 satisfying i+j≡(ni)+(nj)(mod2)i+j\equiv\binom ni +\binom nj \pmod{2} for all ii and jj with 0≤i≤j≤n0\le i\le j\le n.
1
2