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High School
By TrueShelf Inc. on Aug. 28, 2013 | Updated Jan. 16, 2017

7 points inside a hexagon

Consider a hexagon \(H\) with side length 1. Given any 7 points inside \(H\), show that at least two points are separated by a distance of at most 1.
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High School
By TrueShelf Inc. on Nov. 14, 2016 | Updated Feb. 24, 2017

IMO 2016 Problem 5

The equation \((x-1)(x-2)(x-3)...(x-2016) = (x-1)(x-2)(x-3)...(x-2016)\) is written on a board, with 2016 linear factors on each side. What is the least possible value of \(k\) for which it is possi…
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High School
By Shiva Kintali on June 24, 2012 | Updated Jan. 16, 2017

Toggling 100 doors

There are 100 doors numbered 1 to 100 in a row. There are 100 people. The first person opens all the doors. The second person closes all the even-numbered doors. The third person changes the state of…
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High School
By TrueShelf Inc. on July 21, 2016 | Updated Jan. 16, 2017

IMO 2016 Problem 3

Let \(P = A_1, A_2 \dots A_k\) be a convex polygon on the plane. The vertices \(P = A_1, A_2 \dots A_k\) have integral coordinates and lie on a circle. Let \(S\) be the area of \(P\). An odd positive …
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Undergraduate
By TrueShelf Inc. on May 21, 2014 | Updated Jan. 16, 2017

99 fair coins

Person \(A\) flips 99 fair coins and obtains \(a\) heads. Person \(B\) flips 100 fair coins and obtains \(b\) heads. What is the probability that \(a < b\) ?
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High School
By TrueShelf Inc. on June 6, 2014 | Updated Jan. 16, 2017

Constructible polygons

A constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular hepta…
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High School
By TrueShelf Inc. on Oct. 21, 2014 | Updated Jan. 16, 2017

Farmers and Chickens

Three farmers were selling chickens at the local market. One farmer had 10 chickens to sell, another had 16 chickens to sell, and the last had 26 chickens to sell. In order not to compete with each …
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High School
By Shiva Kintali on April 1, 2014 | Updated Jan. 16, 2017

Designing two dice

You are given two blank dice i.e., all the sides of the dice are blank. You are allowed to write any integers on the sides of the dice. Only one integer on each side. Write the integers such that …
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High School
By TrueShelf Inc. on May 7, 2014 | Updated Jan. 16, 2017

Sock Drawer puzzle

There are 10 socks of each of the following colors in a drawer: red blue green black white i.e., there are 50 socks. The socks are arbitrarily distributed in the drawer. You are blind-folded. …
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Undergraduate
By 123forman on May 8, 2014 | Updated Jan. 16, 2017

Democratic distribution of wealth

A crew of 100 pirates have captured 100 laptops. The pirates are ranked 1 through 100, with pirate 100 being the pirate king. Captured booty is distributed as follows. The pirate king proposes a dis…
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