answer:Part (1): Proving ( S(n) leq n^2 - 14 ) for every integer ( n geq 4 ) 1. **Assume the contrary:** Suppose for some ( n geq 4 ), ( S(n) > n^2 - 14 ). This means there exist ( k = n^2 - 13 ) integers (( a_1, a_2, ldots, a_k )) such that [ n^2 = a_1^2 + a_2^2 + cdots + a_k^2. ] 2. **Rewrite the above equation:** [ a_1^2 + a_2^2 + cdots + a_k^2 = k + 13 ] implies [ sum_{i=1}^{k} (a_i^2 - 1) = 13 ] Since ( a_i ) are integers, we know that ( 0 leq a_i^2 - 1 leq 13 ). 3. **Possible values of ( a_i^2 - 1 ):** [ a_i^2 - 1 in {0, 3, 8} ] This implies ( a_i ) can only be ( 1, 2, 3 ). 4. **Let ( a ) be the number of 0s, ( b ) be the number of 3s, and ( c ) be the number of 8s:** [ 3b + 8c = 13 ] 5. **Checking non-negative integer solutions:** [ text{If } c = 0, quad 3b = 13 quad text{(no integer solution for ( b ))} text{If } c = 1, quad 3b = 5 quad text{(no integer solution for ( b ))} ] Thus, no non-negative integer solutions satisfy the equation. 6. **Conclusion:** The initial assumption that ( S(n) > n^2 - 14 ) is incorrect. Hence, for all ( n geq 4 ), [ S(n) leq n^2 - 14. ] Part (2): Find an integer ( n ) such that ( S(n) = n^2 - 14 ) 1. **Verify for ( n = 13 ):** We need ( S(13) = 13^2 - 14 = 155 ). 2. **We know that:** Each integer ( k leq 13^2 - 14 ) can be expressed as a sum of the squares. Particularly, ensure with specific numbers ( l ): [ l = 3s + 8t ] with non-negative ( s ) and ( t ), holds for numbers larger than 13. 3. **Some verifications:** [ 13^2 = 12^2 + 5^2, quad 13^2 = 12^2 + 4^2 + 3^2, ] [ 13^2 = 8^2 + 8^2 + 5^2 + 4^2, quad 13^2 = 8^2 + 8^2 + 4^2 + 4^2 + 3^2 ] Breaking down into smaller squares repeatedly until representations as sums of squares. 4. **Conclusion:** Therefore, ( n = 13 ) satisfies ( S(n) = n^2 - 14 ), Part (3): Infinite integers ( n ) such that ( S(n) = n^2 - 14 ) 1. **Assume ( S(n) = n^2 - 14 ) holds for some ( n geq 13 ):** 2. **Proving for ( 2n ):** [ S(2n) = (2n)^2 - 14 = 4n^2 - 14 ] 3. **For any ( 1 leq k leq n^2 - 14 ):** By assumption, ( n^2 = sum_{i=1}^k a_i^2 ). Then [ (2n)^2 = sum_{i=1}^k (2a_i)^2 ] meaning for each ( k ), ( (2n)^2 ) can be expressed as a sum of ( k ) squares. Additionally, [ (2n)^2 = 4n^2 Rightarrow k leq 4(n^2 - 14). ] 4. **For ( k geq 1/4 (2n)^2 and k leq 4n^2 - 14 ):** By induction with existing known proofs, ( S(n) = n^2 - 14 ) continuing for each doubled parameter. 5. **Conclusion:** If ( S(n) = n^2 - 14 ) for some ( n geq 13 ), ( n = 13 ) and its further sequences, then there exist infinitely many integers ( n ) such that [ S(n) = n^2 - 14 ] (blacksquare)

answer:1. Firstly, note that: [ angle I F K = 90^circ ] This implies that I K is the diameter of the circumcircle of triangle C F I. Since IK is a diameter, it follows that: [ angle I C K = 90^circ ] 2. Similarly, since: [ angle I G L = 90^circ ] I L is the diameter of the circumcircle of triangle B G I. This implies: [ angle I B L = 90^circ ] Therefore, observe that C K parallel G L and also B L parallel F K. 3. Let the lines C K and B L intersect at point D. From our previous observations, we note that D K A L forms a parallelogram. This is because in a parallelogram, opposite sides are parallel and equal in length. Here, since: [ C K parallel G L quad text{and} quad B L parallel F K ] We conclude that D K A L is indeed a parallelogram. 4. Additionally, note that D is the excenter with respect to vertex A of triangle A B C. This is due to the fact the lines B L and C K are perpendicular to the corresponding internal angle bisectors. 5. Since the excenter D lies on the internal angle bisector A I, we conclude that A I must bisect the segment K L. Specifically, in a parallelogram, the diagonals bisect each other. Therefore, since D K A L is a parallelogram and A I is one of the diagonals, it bisects the other diagonal K L. # Conclusion: [ boxed{text{A} , I , text{bisects} , K L} ] This completes the proof.

answer:**Analysis** This question examines the method of proof by contradiction, which can be solved directly from the definition of proof by contradiction. **Answer** In the process of deriving a contradiction using the method of proof by contradiction, the conditions typically used include: (①) the judgment opposite to the conclusion, i.e., the assumption; (②) the conditions of the original proposition; (③) axioms, theorems, definitions, etc. Therefore, the correct choice is boxed{text{C}}.

answer:Let x be the number of hours with the tailwind (at 15 mph) and 12-x be the hours against the wind (at 10 mph). The total distance traveled is given by the equation: [ 15x + 10(12 - x) = 150 ] Expanding and simplifying the equation: [ 15x + 120 - 10x = 150 ] [ 5x + 120 = 150 ] [ 5x = 30 ] [ x = frac{30}{5} = boxed{6} ]