pigeonhole principal

Prove that any sequence of nm + 1 distinct real numbers either contains an
increasing subsequence of length n + 1 or a decreasing subsequence of length m + 1.

Twenty children are queueing for ice cream that is sold at R5 per cone. Ten of the children have a R5 coin, the others want to pay with R10 bill.At the beginning , the ice cream man doesn't have any change. How many possible arrangements of the twenty kids in a queue are there so that the ice cream man will never run out of change

Q. FIND ALL INTEGERS x AND y SUCH THAT (x,y)=8 AND [x,y]=64.  [NUMBER THEORY].

Suppose n≥0 is an integer and all the roots of x³+bx+4-(2×2016ⁿ)=0 are integers . Find all possible b.

Let ABC be a ∆ with circumcircle £ and in centre I. Let internal angle bisectors of ∆ ABC meet £ at A',B',C'. respectively. Let B'C' intersect AA'in P and AC in Q,and let BB' intersect AC in R.
Suppose the quadrilateral PIRQ is a kite I.e. IP=IR AND QP=QR.
PROVE that ∆ ABC is equilateral.

Q.)  The present ages of two brothers A AND B and their father are 3 distinct positive integers a,b,c respectively.
Suppose (b-1)/a-1 and (b+1)/a+1 are two consecutive integers and (c-1)/b-1 and (c+1)/b+1 are two consecutive integers . If  a+b+c≤150 determine a,b,c.

Let ABC be an isosceles ∆ with AB=AC. Let ¶ represent the circumcirle of ∆ABC and O be the centre of circle .Let CO meet ¶ in D.Draw a line parallel to AC through D .
Let it intersect AB at E. Suppose AE:EB=2:1.
PROVE THAT ABC IS EQUILATERAL ∆.

(Hint: you may join BD and DA. And extend DE to BC at F.

Hello friends..   Here's one more interesting question on GEOMETRY.. .         
Please put your solution in comments.
Thanks

https://m.youtube.com/watch?v=iiwu9_MC-SI

(geometry) Q. 2].         Consider a ∆ PQR with a point A within the ∆ .Lines PA, QA, and RA intersect the opposite side s in points DEF respectively.
PROVE THAT out of the numbers PA/AD, QA/AE, RA/AF, at least one is ≤2 and at least one is ≥2.


post your solution in the comments .
Thank you.

Link for previous year CMI papers:https://www.cmi.ac.in//admissions/syllabus.php