# Tagged Questions

Problems from or inspired by mathematics competitions. Questions regarding mathematics competitions.

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### Chicken nuggets come in boxes of $389$ and $691$. What is the largest number of nuggests one cannot order? [duplicate]

Chicken nuggets come in boxes of $389$ and $691$. What is the largest number of nuggets one cannot order?
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### Number of People Required (Arena Survival Question)

Let's say that in an arena slaves are fighting 1v1. Only slaves with the identical number of wins can fight with each other. If a slave has 3 losses, the slave will be kicked out of the arena. If a ...
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### a particular linear combination

Fix $a_1,\ldots,a_n\in\mathbb{N}$. I'd like to know if one can characterize the natural numbers that belong to the set $$\{b_1a_1+\ldots+b_na_n:\,b_j\in\{-1,0,1\}\}.$$ EDIT: Maybe this question doesn'...
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### Discriminant of Cubics and Math Olympiad

Let $a,b,c$ be distinct nonzero real numbers. If the equations $E_1: ax^3+bx+c=0, E_2: bx^3+cx+a=0$ and $E_3: cx^3+ax+b=0$ have a common root, prove that at least one of these equations has three real ...
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### Proving circumcenter lies on altitude

Problem: In $\triangle ABC$, let $D$ be the intersection of the tangents to the circumcircle at $B$ and $C$, let $B'$ be the reflection of $B$ across $AC$, let $C'$ be the reflection of $C$ across $AB$...
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### IMO 2016 P3, number theory with the area of a polygon

Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \cdots A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is ...
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### Determine all functions $f$ on $\mathbb R$ such that $f(x^2+yf(x))=f(x)f(x+y)$ for all $x,y$

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that $$f(x^2+yf(x))=f(x)f(x+y).$$ for all $x,y$ real numbers. I think that the only three solutions are: $f(x)=0$, $f(x)=1$ and $f(x)=x$...
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### Maths Puzzle!!! [duplicate]

I am planning on taking an interview in the near future and was practicing on some previously asked aptitude questions. During my prep I came across a problem for which I couldn't find an answer. ...
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### Who becomes king? [closed]

5 earls argue which becomes king and which becomes treasurer. A will be happy only if D or E is treasurer. B will be happy only if C is treasurer. C will be happy only if D is either king or ...
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### How to improve mathematics for Programming Contests?

You might close this question or downvote it, but I can't stop myself from asking the experts of mathematics who solve thousands of math problems. I'm a C++/C programmer who wants to improve his ...
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### PAMO G Qualification Exam Question

ABCD is rectangular court with AB = 50m and BC = 30m. Four girls stand at different positions in that court so that the distance between the two girls next to each other is maximised. What is this ...
### Find all odd positive integers $n$, which there exists odd positive integers $x_1,x_2,..,x_n$, such that $x_1^2+x_2^2+\cdots+x_n^2=n^4$
Find all odd positive integers $n$, which there exists odd positive integers $x_1,x_2,..,x_n$, such that $$x_1^2+x_2^2+\cdots+x_n^2=n^4$$ My work so far 1) $n=3$ $$x_1^2+x_2^2+x_3^2=81$$ no ...