Problem:
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space?
Answer Choices:
A. 9
B. 1221​
C. 15
D. 1521​
E. 17
Solution:
Let EQ=c and TQ=s as indicated in the figure. Triangles QUV and FEQ are similar since ∠FQE and ∠QVU are congruent because both are complementary to ∠VQU. So
UVQU​=EQFE​
and thus QU=c1​. Then AB=1+c+1/c+1=5 and so c+1/c=3. Since the area of square ABCD equals the sum of areas of square QRST, four unit squares, four 1×c triangles, and four c1​×1 triangles, it follows that
25​===​s2+4(1+2c​+2c1​)s2+4+2(c+c1​)s2+4+2⋅3​
Therefore, the area of square QRST=s2=15.
OR
Square FVMP has area 9, the four triangles FQV,VRM,MSP, and PTF each have area 23​. So the area of square STQR is 9+6=15.
Answer: C​.
The problems on this page are the property of the MAA's American Mathematics Competitions
