Problems of the Week

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2017-10-23 Basic

Can you form 24 using all the numbers 3, 3, 9, and 11, along with any of the 4 arithmetic operations (addition, subtraction, multiplication, and division) and parentheses?

A concave polygon can be equilateral. For example, the concave octagon to the right is equilateral.

What is the least number of sides a concave equilateral polygon can have?

One day, Alice, Betty, Cathy, David, and Edward were at home playing hide and seek. Unfortunately, one of them accidentally collided into a flower vase. When their mother came back home and saw the broken vase, she asked them who did it.

Alice: "Betty broke it!"
Betty: "Alice is not telling the truth."
Cathy: "I did it!"
David: "I did not break it and Betty is lying!"
Edward: "David is telling the truth!"

Given that at least three children are telling the truth, who broke the vase?

A catalyst is a molecule that can increase the rate of a chemical reaction.

Suppose we have a reaction that releases energy $E.$ If we introduce a catalyst into the reaction and assume that no other conditions change, does the amount of energy released during any given reaction increase, decrease, or stay the same?

You are given a set of cards arranged in a line. Each card is either $\color{#333333}{\text{black}}$ or $\color{#D61F06}{\text{red}}$. You may divide the cards into two parts by picking a line before or after the whole cards (so that a part can contain 0 cards), or somewhere between two cards.

No matter how many cards are arranged in whatever manner, can you always ensure the number of $\color{#333333}{\text{black}}$ cards in the left part is exactly the same as the number of $\color{#D61F06}{\text{red}}$ cards in the right part?

An example split that fulfills the condition is shown below.

There is 1 $\color{#333333}{\text{black}}$ card in the left part and 1 $\color{#D61F06}{\text{red}}$ card in the right part

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