complex numbers

complex

13th Friday

Prove that in each year, the 13th day of some month occurs on a Friday.

START 2012, KVPY 2012

A sequence is called tworri c if its first term is 1 and the sum of every pair of
consecutive terms is a positive power of 2. One example of a tworri c sequence is 1, 7, -5,7, 57. (a) Find the shortest possible length of a tworri c sequence that contains the term 2011.
(b) Find the number of tworri c sequences that contain the term 2011 and have this
shortest possible length.
PRACTICE PROBLEMS FOR "KVPY & START 2012"

PASCAL'S Triangle

In Pascal’s triangle the number 1 appears infinitely many times. All other numbers will appear a finite amount of times*.

The number 2 appears just once.
The number 3 appears twice.

...

 

TRIVIA FOR ABR & VEDANSI ON AUSPICIOUS NAVAMI

TRIVIA FOR ABR & VEDANSI ON AUSPICIOUS NAVAMI: Imagine a circular billiard table of unit radius, with centre at (x,y ) coordinates (0,0). It has two billiard balls on it. Ball A is at coordinates (-0.8, -0.1); ball B at (0.6, 0.4). What are the coordinates of the point(s) on the cushion to which one must hit ball A so that it bounces off the cushion once and then hits ball B?

 

Integers

Broken Reciprocal Key

Broken Reciprocal Key

The 1/x key on my calculator is broken. How can I use the trigonometry buttons [[ sin, cos, tan, arc sin, arc cos & arc tan ]] to calculate reciprocals?

Vedansi Chakraborty

In an idle moment, Vedansi Chakraborty picked up two numbers x and y such that 0<x<y<1. She wondered how to combine these simply. She wrote down the numbers: (x + y), (xy), (x/y), (y/x), (x - y), (y - x); Vedansi realized to her surprise that she had written the same number down twice. To her further surprise, noticed that she had written a second number down twice. What were her starting numbers?

ARCH ABR

Mr. Arch Arnab Barman Ray, you are given 10 boxes, each large enough to contain exactly 10 wooden building blocks, and a total of 100 blocks in 10 different colours, but not necessarily the same number of each colour. Prove that the blocks can be arranged so that at least one box contains blocks of the same colour and no box contains blocks with more than 2 colours.[[[[This problem is based on a interview question at the University of Oxford.]]]]]

 

mirror problems

[[[Priyanka Laskar (Puja)]]]] 

Keen to make a good impression at the start of term, you stand in front of the mirror to check you look OK. How long does the mirror need to be so that you can see yourself top to toe? How far from the mirror should you stand? If you wanted most people to be able to use the same mirror, how long should it be and how high up on the wall should you hang it?

 

 

Chords in polygons

souradeep bhattacharya:Chords in polygons

A regular polygon is inscribed in a unit circle and all the different length chords connecting pairs of vertices are drawn.
(a) 3 chords connecting 4 vertices (b) 4 chords connecting 5 vertices

What is the link between the number of sides of the polygon and the sum of the squares of the lengths of these chords?

souradeep bhattacharya:Chords in polygons

souradeep bhattacharya:Chords in polygons

A regular polygon is inscribed in a unit circle and all the different length chords connecting pairs of vertices are drawn.
(a) 3 chords connecting 4 vertices (b) 4 chords connecting 5 vertices

What is the link between the number of sides of the polygon and the sum of the squares of the lengths of these chords?

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Sagnik Bhattacharya's Safe combinations

Sagnik Bhattacharya's Safe combinations

A group of 11 scientists are working on a secret project, the materials of which are kept in a safe. They want to be able to open the safe only when a majority of the group is present. Therefore the safe is provided with a number of different locks, and each scientist is given the keys to certain of these locks. How many locks are required, and how many keys must each scientist have?

[[ PROBLEMS FOR RMO 2011 ]]

Determine with proof the number of positive integers "m" such that a convex regular
polygon with "m" sides has interior angles whose measures, in degrees, are integers.[[ PROBLEMS FOR RMO 2011 ]]

{{ PROBLEM 2 FOR RMO 2011}}

Find three isosceles triangles, no two of which are congruent, with integer sides, such
that each triangle’s area is numerically equal to 6 times its perimeter.
{{ PROBLEM 2 FOR RMO 2011}}

PROBLEM 3 FOR RMO 2011

PROBLEM 3 FOR RMO 2011
A set is reciprocally whole if its elements are distinct integers greater than 1 and
the sum of the reciprocals of all those elements is exactly 1. Find a set S, as small
as possible, that contains two reciprocally whole subsets, I and J, which are distinct
but not necessarily disjoint (meaning they may share elements, but they may not be
the same subset). Prove that no set with fewer elements than S can contain two
reciprocally whole subsets.

more problems for RMO

writes a sequence of integers starting with the number 12. Each subsequent integer she writes is chosen randomly with equal chance from among the positive divisors of the previous integer (including the possibility of the integer itself). She keeps writing integers until she writes the integer 1 for the first time, and then she stops. One such sequence is 12, 6, 6, 3, 3, 3, 1.
What is the expected value of the number of terms in Anna’s sequence?

more problems for RMO 2011

An increasing arithmetic sequence with infinitely many terms is determined as follows.
A single die is thrown and the number that appears is taken as the first term. The die is thrown again and the second number that appears is taken as the common difference between each pair of consecutive terms. Determine with proof how many of the 36 possible sequences formed in this way contain at least one perfect square.

PRACTICE PROBLEMS FOR RMO 2011

Two trains are on the same track a distance 100 km apart heading towards one another, each at a speed of 50 km/h. A fly starting out at the front of one train, flies towards the other at a speed of 75 km/h. Upon reaching the other train, the fly turns around and continues towards the first train. How many kilometers does the fly travel before getting squashed in the collision of the two trains?

In Ron Howard's 2001 film A Beautiful Mind, John Nash (played by Russell Crowe) can be overheard discussing this problem with a group of students in the library.

KVPY 2011 SA/SB/SX ANSWERKEY

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