IMOmath

Functions

1. (31 p.)
Find the number of surjective functions \( f:\{1,2,3,4,5\}\to\{1,2,3\} \).

   A    96

   B    117

   C    120

   D    150

   E    243

   N   

2. (21 p.)
Let \( A\subseteq \mathbb Z \) be the set of all integers that are divisible by \( 5 \). For each \( k\in\{1,2,\dots, 1000\} \) we define the function \( f_k:A\to \mathbb Z \) in the following way: \[ f_k(x)=x^2+k\cdot x.\]

How many functions from the sequence \( f_1 \), \( f_2 \), ..., \( f_{1000} \) are one-to-one?

   A    \( 0 \)

   B    \( 200 \)

   C    \( 400 \)

   D    \( 600 \)

   E    \( 800 \)

   N   

3. (15 p.)
Among the following functions from \( \mathbb N \) to \( \mathbb N \) only one is surjective. Which one? (\( \lfloor x\rfloor \) denotes the largest integer not greater than \( x \), while \( \lceil x\rceil \) is the smallest integer not smaller than \( x \))

   A    \( f(n)=\lfloor\frac n2\rfloor +1 \)

   B    \( g(n)=\lceil\frac1{1+n^2}\rceil \)

   C    \( h(n)=\lfloor\frac{n}{n^2+3}\rfloor+n^3 \)

   D    \( k(n)=n^6-n+1 \)

   E    \( l(n)=11n^4+\lfloor\frac1{n^2+n+1}\rfloor \)

   N   

4. (10 p.)
There are four diagrams in the picture below. They are labeled \( f \), \( g \), \( h \), and \( k \). Which of the diagrams correspond to functions?
picture2

   A    There are no functions among \( f \), \( g \), \( h \), and \( k \).

   B    \( f \) is the only graph that corresponds to a function.

   C    \( g \) and \( h \) are the only graphs that correspond to functions.

   D    \( g \) and \( k \) are the only graphs that correspond to functions.

   E    \( k \) is the only graph that corresponds to a function.

   N   

5. (21 p.)
Determine the number of injective functions \( f:\{1,2,3\}\to\{1,2,3,4,5\} \).

   A    \( 15 \)

   B    \( 24 \)

   C    \( 60 \)

   D    \( 120 \)

   E    \( 125 \)

   N   





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