Subcontests
(5)Arranging beads around a necklace; no three labls in a row form a triangle
Zaineb makes a large necklace from beads labeled 290,291,…,2023. She uses each bead exactly once, arranging the beads in the necklace any order she likes. Prove that no matter how the beads are arranged, there must be three beads in a row whose labels are the side lengths of a triangle. Two triangles have equal area x, find x
In the following figure—not drawn to scale!—E is the midpoint of BC, triangle FEC has area 7, and quadrilateral DBEG has area 27. Triangles ADG and GEF have the same area, x. Find x.
[asy]
unitsize(2cm);
pair A = (0,38/16);
pair B = (0,0);
pair C = (38/16,0);
pair D = (0,25/16);
pair E = (19/16,0);
pair F = .4*D+.6*C;
draw(D -- C -- B -- A -- E -- F);
label("A", A, W);
label("B", B, W);
label("C", C, S);
label("D", D, W);
label("E", E, S);
label("F", F, N);
label("G", (17*F-8*C)/9, NE);
[/asy] Substrings of integer are multiples of 9
Given a positive integer N (written in base 10), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from N, including N itself. For example, the integer substrings of 3208 are 3, 2, 0, 8, 32, 20, 320, 208, 3208. (The substring 08 is omitted from this list because it is the same integer as the substring 8, which is already listed.)What is the greatest integer N such that no integer substring of N is a multiple of 9? (Note: 0 is a multiple of 9.) Students lost trying to find pizza
Mr. Murgatroyd decides to throw his class a pizza party, but he's going to make them hunt for it first. He chooses eleven locations in the school, which we'll call 1,2,…,11. His plan is to tell students to start at location 1, and at each location n from 1 to 10, they will find a message directing them to go to location n+1; at location 11, there's pizza!Mr. Murgatroyd sends his teaching assistant to post the ten messages in locations 1 to 10. Unfortunately, the assistant jumbles up the message cards at random before posting them. If the students begin at location 1 as planned and follow the directions at each location, show that they will still get to the pizza.