MathDB

4

Part of 2017 Saudi Arabia BMO TST

Problems(4)

every integer can be represented as sum of several distinct Fibonacci numbers

Source: 2017 Saudi Arabia BMO TST I p4

7/24/2020
Fibonacci sequences is defined as f1=1f_1=1,f2=2f_2=2, fn+1=fn+fn−1f_{n+1}=f_{n}+f_{n-1} for n≥2n \ge 2. a) Prove that every positive integer can be represented as sum of several distinct Fibonacci number. b) A positive integer is called Fib-unique if the way to represent it as sum of several distinct Fibonacci number is unique. Example: 1313 is not Fib-unique because 13=13=8+5=8+3+213 = 13 = 8 + 5 = 8 + 3 + 2. Find all Fib-unique.
fibonacci numbernumber theory
GH/GB = tan a/2 wanted, circumcenters related

Source: 2017 Saudi Arabia BMO TST II p4

7/24/2020
Let ABCABC be a triangle with AA is an obtuse angle. Denote BEBE as the internal angle bisector of triangle ABCABC with E∈ACE \in AC and suppose that ∠AEB=45o\angle AEB = 45^o. The altitude ADAD of triangle ABCABC intersects BEBE at FF. Let O1,O2O_1, O_2 be the circumcenter of triangles FED,EDCFED, EDC. Suppose that EO1,EO2EO_1, EO_2 meet BCBC at G,HG, H respectively. Prove that GHGB=tan⁡a2\frac{GH}{GB}= \tan \frac{a}{2}
trigonometrygeometryCircumcentercircumcircle
sum T = \sum^{1009}_{i=1} |a_i - b_i| has the right most digit is $9

Source: 2017 Saudi Arabia Mock BMO I p4

7/25/2020
Consider the set X={1,2,3,...,2018}X =\{1, 2,3, ...,2018\}. How many positive integers kk with 2≤k≤20172 \le k \le 2017 that satisfy the following conditions: i) There exists some partition of the set XX into 10091009 disjoint pairs which are (a1,b1),(a2,b2),...,(a1009,b1009)(a_1, b_1),(a_2, b_2), ...,(a_{1009}, b_{1009}) with ∣ai−bi∣∈{1,k}|a_i - b_i| \in \{1, k\}. ii) For all partitions satisfy the condition (i), the sum T=∑i=11009∣ai−bi∣T = \sum^{1009}_{i=1} |a_i - b_i| has the right most digit is 99
number theorycombinatorics
a table of (p^2+ p+1)x (p^2+p + 1) divided into unit cells.

Source: 2017 Saudi Arabia Mock BMO II p4

7/25/2020
Let pp be a prime number and a table of size (p2+p+1)×(p2+p+1)(p^2+ p+1)\times (p^2+p + 1) which is divided into unit cells. The way to color some cells of this table is called nice if there are no four colored cells that form a rectangle (the sides of rectangle are parallel to the sides of given table). 1. Let kk be the number of colored cells in some nice coloring way. Prove that k≤(p+1)(p2+p+1)k \le (p + 1)(p^2 + p + 1). Denote this number as kmaxk_{max}. 2. Prove that all ordered tuples (a,b,c)(a, b, c) with 0≤a,b,c<p0 \le a, b, c < p and a+b+c>0a + b + c > 0 can be partitioned into p2+p+1p^2 + p + 1 sets S1,S2,...Sp2+p+1S_1, S_2, .. . S_{p^2+p+1} such that two tuples (a1,b1,c1)(a_1, b_1, c_1) and (a2,b2,c2)(a_2, b_2, c_2) belong to the same set if and only if a1≡ka2,b1≡kb2,c1≡kc2a_1 \equiv ka_2, b_1 \equiv kb_2, c_1 \equiv kc_2 (mod pp) for some k∈{1,2,3,...,p−1}k \in \{1,2, 3, ... , p - 1\}. 3. For 1≤i,j≤p2+p+11 \le i, j \le p^2+p+1, if there exist (a1,b1,c1)∈Si(a_1, b_1, c_1) \in S_i and (a2,b2,c2)∈Sj(a_2, b_2, c_2) \in S_j such that a1a2+b1b2+c1c2≡0a_1a_2 + b_1b_2 + c_1c_2 \equiv 0 (mod pp), we color the cell (i,j)(i, j) of the given table. Prove that this coloring way is nice with kmaxk_{max} colored cells
rectanglecombinatoricscombinatorial geometry