Find the strictly increasing functions $f : \{1,2,\ldots,10\} \to \{ 1,2,\ldots,100 \}$ such that $x+y$ divides $x f(x) + y f(y)$ for all $x,y \in \{ 1,2,\ldots,10 \}$.

[i]Cristinel Mortici[/i]
1. **Initial Setup and Assumptions:**
   We are given a function \( f : \{1,2,\ldots,10\} \to \{1,2,\ldots,100\} \) that is strictly increasing. We need to find such functions \( f \) such that \( x + y \) divides \( x f(x) + y f(y) \) for all \( x, y \in \{1,2,\ldots,10\} \).

2. **Special Case Analysis:**
   Consider the case when \( y = x + 1 \). We need \( 2x + 1 \) to divide \( x f(x) + (x + 1) f(x + 1) \).

3. **Divisibility Condition:**
   Since \( 2x + 1 \) also divides \( x f(x) + (x + 1) f(x) \), it follows that:
   \[
   2x + 1 \mid (x + 1) f(x + 1) - (x + 1) f(x)
   \]
   Simplifying, we get:
   \[
   2x + 1 \mid (x + 1) (f(x + 1) - f(x))
   \]

4. **GCD Analysis:**
   Since \( \gcd(2x + 1, x + 1) = 1 \) (as \( 2x + 1 \) is odd and \( x + 1 \) is even), it follows that:
   \[
   2x + 1 \mid f(x + 1) - f(x)
   \]

5. **Inequality Derivation:**
   Given that \( f \) is strictly increasing, \( f(x + 1) > f(x) \). Therefore, we have:
   \[
   f(x + 1) - f(x) \geq 2x + 1
   \]

6. **Summing Inequalities:**
   Summing these inequalities for \( x = 1, 2, \ldots, 9 \), we get:
   \[
   \sum_{x=1}^{9} (f(x + 1) - f(x)) \geq \sum_{x=1}^{9} (2x + 1)
   \]
   The left-hand side simplifies to \( f(10) - f(1) \), and the right-hand side is the sum of the first 9 odd numbers:
   \[
   f(10) - f(1) \geq 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81
   \]

7. **Boundary Conditions:**
   Since \( f \) maps to \(\{1, 2, \ldots, 100\}\), the maximum possible value for \( f(10) \) is 100 and the minimum possible value for \( f(1) \) is 1. Therefore:
   \[
   f(10) - f(1) \leq 100 - 1 = 99
   \]
   Combining this with the previous inequality, we get:
   \[
   81 \leq f(10) - f(1) \leq 99
   \]

8. **Equality Condition:**
   For the inequalities to hold strictly, we must have \( f(10) = 100 \) and \( f(1) = 1 \). This implies that all intermediate inequalities must be equalities:
   \[
   f(x + 1) - f(x) = 2x + 1
   \]

9. **Function Determination:**
   We can now determine \( f \) by solving the recurrence relation:
   \[
   f(x + 1) = f(x) + 2x + 1
   \]
   Starting with \( f(1) = 1 \):
   \[
   f(2) = f(1) + 3 = 1 + 3 = 4
   \]
   \[
   f(3) = f(2) + 5 = 4 + 5 = 9
   \]
   \[
   f(4) = f(3) + 7 = 9 + 7 = 16
   \]
   \[
   f(5) = f(4) + 9 = 16 + 9 = 25
   \]
   \[
   f(6) = f(5) + 11 = 25 + 11 = 36
   \]
   \[
   f(7) = f(6) + 13 = 36 + 13 = 49
   \]
   \[
   f(8) = f(7) + 15 = 49 + 15 = 64
   \]
   \[
   f(9) = f(8) + 17 = 64 + 17 = 81
   \]
   \[
   f(10) = f(9) + 19 = 81 + 19 = 100
   \]

10. **General Form:**
    Observing the pattern, we see that \( f(x) = x^2 \) satisfies all conditions.

The final answer is \( \boxed{ f(x) = x^2 } \)