P1

Part of 2023 Israel TST

Problems(5)

Five numbers on a pentagon

Source: 2023 Israel TST Test 1 P1

A real number is written next to each vertex of a regular pentagon. All five numbers are different. A triple of vertices is called successful if they form an isosceles triangle for which the number written on the top vertex is either larger than both numbers written on the base vertices, or smaller than both. Find the maximum possible number of successful triples.

TSTcombinatoricspentagon

Ratio of squares to cubes

Source: 2023 Israel TST Test 2 P1

For positive integers

n

f_2(n)

denote the number of divisors of

n

which are perfect squares, and

f_3(n)

denotes the number of positive divisors which are perfect cubes. Prove that for each positive integer

k

there exists a positive integer

n

\frac{f_2(n)}{f_3(n)}=k

ratioTSTnumber theory

Tiling with 6 types of tiles

Source: 2023 Israel TST Test 3 P1

Toph wants to tile a rectangular

m\times n

square grid with the

6

types of tiles in the picture (moving the tiles is allowed, but rotating and reflecting is not). For which pairs

(m,n)

is this possible?

TilingcombinatoricsTST

Perpendicular diagonals in 20-gon

Source: 2023 Israel TST Test 5 P1

A regular polygon with

20

vertices is given. Alice colors each vertex in one of two colors. Bob then draws a diagonal connecting two opposite vertices. Now Bob draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints. He gets a fish from Alice for each such segment he draws. How many fish can Bob guarantee getting, no matter Alice's goodwill?

combinatoricsTSTregular polygon

Source: 2023 Israel TST Test 7 P1

Find all functions

f:\mathbb{R}\to \mathbb{R}

such that for all

x, y\in \mathbb{R}

the following holds:

f(x)+f(y)=f(xy)+f(f(x)+f(y))

TSTalgebrafunctional equationfunction