MathDB

2016 Dutch IMO TST

Part of Dutch IMO TST

Subcontests

(4)
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a +\sqrt{ab}+ \sqrt[3]{abc} <= 4/3 (a + b + c)

Prove that for all positive reals a,b,ca, b,c we have: a+ab+abc3≤43(a+b+c)a +\sqrt{ab}+ \sqrt[3]{abc}\le \frac43 (a + b + c)

n boys, n girls, n dancing couples, list with whom a girl wants to dance

Let nn be a positive integer. In a village, nn boys and nn girls are living. For the yearly ball, nn dancing couples need to be formed, each of which consists of one boy and one girl. Every girl submits a list, which consists of the name of the boy with whom she wants to dance the most, together with zero or more names of other boys with whom she wants to dance. It turns out that nn dancing couples can be formed in such a way that every girl is paired with a boy who is on her list. Show that it is possible to form nn dancing couples in such a way that every girl is paired with a boy who is on her list, and at least one girl is paired with the boy with whom she wants to dance the most.
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2

S = {a_i + a_j | 1 <= i, j <= 1000 with i + j \in A} is subset of A

Determine the number of sets A={a1,a2,...,a1000}A = \{a_1,a_2,...,a_{1000}\} of positive integers satisfying a1<a2<...<a1000≤2014a_1 < a_2 <...< a_{1000} \le 2014, for which we have that the set S={ai+aj∣1≤i,j≤1000S = \{a_i + a_j | 1 \le i, j \le 1000 with i+j∈A}i + j \in A\} is a subset of AA.

N lies on a fixed line 2016 Dutch IMO TST3 P4

Let Γ1\Gamma_1 be a circle with centre AA and Γ2\Gamma_2 be a circle with centre BB, with AA lying on Γ2\Gamma_2. On Γ2\Gamma_2 there is a (variable) point PP not lying on ABAB. A line through PP is a tangent of Γ1\Gamma_1 at SS, and it intersects Γ2\Gamma_2 again in QQ, with PP and QQ lying on the same side of ABAB. A different line through QQ is tangent to Γ1\Gamma_1 at TT. Moreover, let MM be the foot of the perpendicular to ABAB through PP. Let NN be the intersection of AQAQ and MTMT. Show that NN lies on a line independent of the position of PP on Γ2\Gamma_2.
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Nice problem

In a 2n×2n2^n \times 2^n square with nn positive integer is covered with at least two non-overlapping rectangle pieces with integer dimensions and a power of two as surface. Prove that two rectangles of the covering have the same dimensions (Two rectangles have the same dimensions as they have the same width and the same height, wherein they, not allowed to be rotated.)

(a^n + b^n + 1) is divisible by d for all positive integers n

Determine all pairs (a,b)(a, b) of integers having the following property: there is an integer d≥2d \ge 2 such that an+bn+1a^n + b^n + 1 is divisible by dd for all positive integers nn.

sorting sums (a_i +a_j) in ascending order, arithmetic progression when?

For distinct real numbers a1,a2,...,ana_1,a_2,...,a_n, we calculate the n(n−1)2\frac{n(n-1)}{2} sums ai+aja_i +a_j with 1≤i<j≤n1 \le i < j \le n, and sort them in ascending order. Find all integers n≥3n \ge 3 for which there exist a1,a2,...,ana_1,a_2,...,a_n, for which this sequence of n(n−1)2\frac{n(n-1)}{2} sums form an arithmetic progression (i.e. the di erence between consecutive terms is constant).
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3

gcm and lcm

Find all positive integers kk for which the equation: lcm(m,n)−gcd(m,n)=k(m−n) \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n) has no solution in integers positive (m,n)(m,n) with m≠nm\neq n.

isosceles AB=AC, BF=BE, angle bisector, BD=EF iff AF=EC 2016 Dutch IMO TST2 P3

Let △ABC\vartriangle ABC be an isosceles triangle with ∣AB∣=∣AC∣|AB| = |AC|. Let D,ED, E and FF be points on line segments BC,CABC, CA and ABAB, respectively, such that ∣BF∣=∣BE∣|BF| = |BE| and such that EDED is the internal angle bisector of ∠BEC\angle BEC. Prove that ∣BD∣=∣EF∣|BD|= |EF| if and only if ∣AF∣=∣EC∣|AF| = |EC|.

numbers satisfying s(n) &lt; s(2n), are as many as ...s(n) &gt; s(2n) [sum of digits\

Let kk be a positive integer, and let s(n)s(n) denote the sum of the digits of nn. Show that among the positive integers with kk digits, there are as many numbers nn satisfying s(n)<s(2n)s(n) < s(2n) as there are numbers nn satisfying s(n)>s(2n)s(n) > s(2n).