MathDB

4

Part of 2016 Dutch IMO TST

Problems(2)

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

Source: Dutch IMO TST day2 p4

8/30/2019
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.
Setscombinatorics
N lies on a fixed line 2016 Dutch IMO TST3 P4

Source:

8/4/2019
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.
geometrycirclesfixed