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British Math Olympiad 2001 R1

The document presents 5 problems for the 2001 British Mathematical Olympiad: 1) Find all two-digit integers N such that the sum of the digits of 10N - N is divisible by 170. 2) Given circles S and T touching at point A, show that the angle subtended by two chords from a point P on T is equal to 26 degrees. 3) Prove there are 7 distinct tetrominoes under rotations and determine if they can pack into a 4x7 rectangle without overlapping. 4) Find the smallest k such that the terms of the defined integer sequence are 2001 consecutive integers. 5) Prove a triangle

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Ajay Negi
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404 views1 page

British Math Olympiad 2001 R1

The document presents 5 problems for the 2001 British Mathematical Olympiad: 1) Find all two-digit integers N such that the sum of the digits of 10N - N is divisible by 170. 2) Given circles S and T touching at point A, show that the angle subtended by two chords from a point P on T is equal to 26 degrees. 3) Prove there are 7 distinct tetrominoes under rotations and determine if they can pack into a 4x7 rectangle without overlapping. 4) Find the smallest k such that the terms of the defined integer sequence are 2001 consecutive integers. 5) Prove a triangle

Uploaded by

Ajay Negi
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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2001 British Mathematical Olympiad

BRITISH MATHEMATICAL OLYMPIAD Round 1


Round 1 : Wednesday, 17 January 2001
1. Find all two-digit integers N for which the sum of the digits
Time allowed Three and a half hours. of 10N − N is divisible by 170.
Instructions • Full written solutions - not just answers - are
required, with complete proofs of any assertions 2. Circle S lies inside circle T and touches it at A. From a
you may make. Marks awarded will depend on the point P (distinct from A) on T , chords P Q and P R of T
clarity of your mathematical presentation. Work are drawn touching S at X and Y respectively. Show that
in rough first, and then draft your final version 6 QAR = 26 XAY .
carefully before writing up your best attempt.
Do not hand in rough work.
3. A tetromino is a figure made up of four unit squares connected
• One complete solution will gain far more credit by common edges.
than several unfinished attempts. It is more (i) If we do not distinguish between the possible rotations of
important to complete a small number of questions a tetromino within its plane, prove that there are seven
than to try all five problems. distinct tetrominoes.
• Each question carries 10 marks. (ii) Prove or disprove the statement: It is possible to pack all
• The use of rulers and compasses is allowed, but seven distinct tetrominoes into a 4×7 rectangle without
calculators and protractors are forbidden. overlapping.
• Start each question on a fresh sheet of paper. Write
on one side of the paper only. On each sheet of 4. Define the sequence (an ) by
working write the number of the question in the √
an = n + { n },
top left hand corner and your name, initials and
where n is a positive integer and {x} denotes the nearest
school in the top right hand corner.
integer to x, where halves are rounded up if necessary.
• Complete the cover sheet provided and attach it to Determine the smallest integer k for which the terms
the front of your script, followed by the questions ak , ak+1 , . . . , ak+2000 form a sequence of 2001 consecutive
1,2,3,4,5 in order. integers.
• Staple all the pages neatly together in the top left
hand corner.
5. A triangle has sides of length a, b, c and its circumcircle has
radius R. Prove that the triangle is right-angled if and only
if a2 + b2 + c2 = 8R2 .

Do not turn over until told to do so.

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