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Sorry Boys: She is ranked No.1 in the International Mathematical Olympiad Hall of Fame

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Lisa Sauermann (born September 25, 1992) is a German schoolgirl who became the most successful participant in the International Mathematical Olympiad. She is ranked No.1 in the International Mathematical Olympiad Hall of Fame, having won four gold medals (2008–2011) and one silver medal (2007) at this event.

In all of those occasions she represented Germany. She was the only student to achieve a perfect score at IMO 2011.

Sauermann attended Martin-Andersen-Nexö-Gymnasium Dresden when she was in 12th grade. She won the Franz Ludwig Gehe Prize in 2011 and the gold medal in the age group III, the 11th–12th grade competition. As a result she won a trip to the Royal Academy of Sciences in Stockholm. To achieve this, she presented a new mathematical theorem with a proof in an outstanding work entitled “Forests with Hypergraphs”.

She is currently studying at the University of Bonn.

Source: http://www.wikipedia.com

How smart are you: Then solve these tough Mathematical’s

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1 In a 10 by 10 grid of dots, what is the maximum number of lines that can be drawn connecting two dots on the grid so that no two lines are parallel?

2 If r_1, r_2, and r_3 are the solutions to the equation x^3 – 5x^2 + 6x – 1 = 0, then what is the value of r_1^2 + r_2^2 + r_3^2?

3 The expression \circ \ 1\ \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012 is written on a blackboard. Catherine places a + sign or a – sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?

4 Parallel lines \ell_1 and \ell_2 are drawn in a plane. Points A_1, A_2, \dots, A_n are chosen on \ell_1, and points B_1, B_2, \dots, B_{n+1} are chosen on \ell_2. All segments A_iB_j are drawn, such that 1 \le i \le n and 1 \le j \le n+1. Let the number of total intersections between these segments (not including endpoints) be denoted by Q. Given that no three segments are concurrent, besides at endpoints, prove that Q is divisible by 3.

5 In convex hexagon ABCDEF, \angle A \cong \angle B, \angle C \cong \angle D, and \angle E \cong \angle F. Prove that the perpendicular bisectors of \overline{AB}, \overline{CD}, and \overline{EF} pass through a common point.

6 The positive numbers a, b, c satisfy 4abc(a+b+c) = (a+b)^2(a+c)^2. Prove that a(a+b+c)=bc.

7 For how many positive integers n \le 500 is n! divisible by 2^{n-2}?

8 A convex 2012-gon A_1A_2A_3 \dots A_{2012} has the property that for every integer 1 \le i \le 1006, \overline{A_iA_{i+1006}} partitions the polygon into two congruent regions. Show that for every pair of integers 1 \le j < k \le 1006, quadrilateral A_jA_kA_{j+1006}A_{k+1006} is a parallelogram.

Source: http://www.artofproblemsolving.com