# Problems of the Week

Contribute a problem

# 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 $$\color{black}{\text{black}}$$ or $$\color{red}{\text{red}}$$. 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 $$\color{black}{\text{black}}$$ cards in the left part is exactly the same as the number of $$\color{red}{\text{red}}$$ cards in the right part?

An example split that fulfills the condition is shown below.

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

×