MathDB

2017 Dutch IMO TST

Part of Dutch IMO TST

Subcontests

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problem involving tangencyand midpoints

Let ABCABC be a triangle, let MM be the midpoint of ABAB, and let NN be the midpoint of CMCM. Let XX be a point satisfying both ∠XMC=∠MBC\angle XMC = \angle MBC and ∠XCM=∠MCB\angle XCM = \angle MCB such that XX and BB lie on opposite sides of CMCM. Let ω\omega be the circumcircle of triangle AMXAMX. (a)(a) Show that CMCM is tangent to ω\omega. (b)(b) Show that the lines NXNX and ACAC intersect on ω\omega

functional equation

Find all functions f:R→Rf : \mathbb{R} \rightarrow \mathbb{R} such that (y+1)f(x)+f(xf(y)+f(x+y))=y(y + 1)f(x) + f(xf(y) + f(x + y))= y for all x,y∈Rx, y \in \mathbb{R}.

combinatorial geometry

Let n≥2n \geq 2 be an integer. Find the smallest positive integer mm for which the following holds: given nn points in the plane, no three on a line, there are mm lines such that no line passes through any of the given points, and for all points X≠YX \neq Y there is a line with respect to which XX and YY lie on opposite sides
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geometry problem

The incircle of a non-isosceles triangle ABCABC has centre II and is tangent to BCBC and CACA in DD and EE, respectively. Let HH be the orthocentre of ABIABI, let KK be the intersection of AIAI and BHBH and let LL be the intersection of BIBI and AHAH. Show that the circumcircles of DKHDKH and ELHELH intersect on the incircle of ABCABC.

combinatorial nT in 2n-gon

Let n≥4n \geq 4 be an integer. Consider a regular 2n−2n-gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this 2n−2n-gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum. Determine for which (not necessarily positive) integers mm the integers m+1,m+2,...,m+2nm + 1, m + 2, . . . , m + 2n can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)

algebra problem

let a1,a2,...ana_1,a_2,...a_n a sequence of real numbers such that a1+....+an=0a_1+....+a_n=0. define bi=a1+a2+....aib_i=a_1+a_2+....a_i for all 1≤i≤n1 \leq i \leq n .suppose bi(aj+1−ai+1)≥0b_i(a_{j+1}-a_{i+1}) \geq 0 for all 1≤i≤j≤n−11 \leq i \leq j \leq n-1. Show that max⁡1≤l≤n∣al∣≥max⁡1≤m≤n∣bm∣\max_{1 \leq l \leq n} |a_l| \geq \max_{1 \leq m \leq n} |b_m|
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disks in combinatorics

Let nn be a positive integer. Suppose that we have disks of radii 1,2,...,n.1, 2, . . . , n. Of each size there are two disks: a transparent one and an opaque one. In every disk there is a small hole in the centre, with which we can stack the disks using a vertical stick. We want to make stacks of disks that satisfy the following conditions: i)i) Of each size exactly one disk lies in the stack. ii)ii) If we look at the stack from directly above, we can see the edges of all of the nn disks in the stack. (So if there is an opaque disk in the stack,no smaller disks may lie beneath it.) Determine the number of distinct stacks of disks satisfying these conditions. (Two stacks are distinct if they do not use the same set of disks, or, if they do use the same set of disks and the orders in which the disks occur are different.)

geometry problem

A circle ω\omega with diameter AKAK is given. The point MM lies in the interior of the circle, but not on AKAK. The line AMAM intersects ω\omega in AA and QQ. The tangent to ω\omega at QQ intersects the line through MM perpendicular to AKAK, at PP. The point LL lies on ω\omega, and is such that PLPL is tangent to ω\omega and L≠QL\neq Q. Show that K,LK, L, and MM are collinear.

Trivial NT in TST

Let a,b,ca, b,c be distinct positive integers, and suppose that p=ab+bc+cap = ab+bc+ca is a prime number. (a)(a) Show that a2,b,c2a^2,b^,c^2 give distinct remainders after division by pp. (b) Show that a3,b3,c3a^3,b^3,c^3 give distinct remainders after division by pp.