**Geometry **

**Let ABC be a triangle such that AB=3, BC=4, and AC=5. Let X be a point in the triangle. Compute the minimal possible value of AX²+BX²+CX².**

**Let ABC be a triangle such that AB=3, BC=4, and AC=5. Let X be a point in the triangle. Compute the minimal possible value of AX²+BX²+CX².**

**Three mathematics classes X,Y and Z take an geometry test. The average of class X is 64, The average of class Y is 80 and the average score of class Z is 76 . The average score of class X and Y is 70 while the average score of class X and Z is 68. What is the ratio of number of students in the class X,Y and Z?**

**(A) 6:10:3
**

**Point O is the center of the regular octagon ABCDEFGH, and X is the midpoint of the side AB. What fraction of the area of the octagon is shaded?
**

(A) 7/16

(B) 11/32

(C)23/32

(D)3/8

(E)15/32

**Let N =123456789…..4344 be the 79-digit number obtained that is formed by writing the integers from 1 to 44 in order, one after the other. What is the remainder when N is divided by 45 ?**

**For how many values of natural number n, both the numbers 48n + 4 and 27n + 19 are perfect square?**

**(1) 0**

**(2) 1**

**(3) 2**

**(4) 3**

**(5) more than 3 **

**In a trapezium PQRS, PQ is parallel to RS, PQ = 20 cm, RS = 3 cm, PQR = 30° and QPS = 60°. What is the length of the line joining the midpoints of PQ and RS? **

**In how many ways a cube can be painted so that each face is painted either red or white?**

**How many digits does the number 1000 ^{1000 } have ?**

**A. 3000**

**B. 3001**

**C. 3010**

**D. 1003**

**E. 100000**

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