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shl-20102
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- Nursing 220 Final Exam Review
Exam of 10 pages for the course Nursing 220 Final Exam Review at Nursing 220 Final Exam Review (shl-20102)
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Chapter 12010 Shortlist JBMO - Problems
1.1 Algebra
A1 The real numbers a, b, c, d satisfy simultaneously the equations
abc − d = 1, bcd − a = 2, cda − b = 3, dab − c = −6.
Prove that a + b + c + d 6= 0.
A2 Determine all four digit numbers abcd such that
a(a + b + c + d)(a2 + b2 + c2 + d2 )(a6 + 2b6 + 3c6 + 4d6 ) = abcd.
A3 Find all pairs (x, y) of real numbers such that |x| + |y| = 1340 and x3 + y 3 + 2010xy =
6703 .
A4 Let a, b, c be positive real numbers such that abc(a + b + c) = 3. Prove the inequality
(a + b)(b + c)(c + a) ≥ 8,
and determine all cases when equality holds.
A5 The real positive numbers x, y, z satisfy the relations x ≤ 2, y ≤ 3, x + y + z = 11.
√
Prove that xyz ≤ 6.
1.2 Combinatorics
C1 There are two piles of coins, each containing 2010 pieces. Two players A and B play
a game taking turns (A plays first). At each turn, the player on play has to take one
or more coins from one pile or exactly one coin from each pile. Whoever takes the last
coin is the winner. Which player will win if they both play in the best possible way?
C2 A 9 × 7 rectangle is tiled with pieces of 1two types, shown in the picture below.
, 2 CHAPTER 1. 2010 SHORTLIST JBMO - PROBLEMS
Find all possible values of the number of the 2 × 2 pieces which can be used in such a
tiling.
1.3 Geometry
G1 Consider a triangle ABC with ∠ACB = 90◦ . Let F be the foot of the altitude from
C. Circle ω touches the line segment F B at point P , the altitude CF at point Q and the
circumcircle of ABC at point R. Prove that points A, Q, R are collinear and AP = AC.
G2 Consider a triangle ABC and let M be the midpoint of the side BC. Suppose
∠M AC = ∠ABC and ∠BAM = 105◦ . Find the measure of ∠ABC.
G3 Let ABC be an acute-angled triangle. A circle ω1 (O1 , R1 ) passes through points B
and C and meets the sides AB and AC at points D and E, respectively. Let ω2 (O2 , R2 )
be the circumcircle of the triangle ADE. Prove that O1 O2 is equal to the circumradius
of the triangle ABC.
G4 Let AL and BK be angle bisectors in the non-isosceles triangle ABC (L ∈ BC, K ∈
AC). The perpendicular bisector of BK intesects the line AL at point M . Point N lies
on the line BK such that LN k M K. Prove that LN = N A.
1.4 Number Theory
NT1 Find all positive integers n such that n2n+1 + 1 is a perfect square.
NT2 Find all positive integers n such that 36n − 6 is a product of two or more consecu-
tive positive integers.
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