# Tagged Questions

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

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### Olympiad Books for Primary Students

I am a teacher of gifted program in primary school and currently I am developing Olympiad Curriculum (topic-wise) for my students. I have those topics that could need some help in terms of questions: ...
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I want to prepare for the maths olympiad and I was wondering if you can recommend me some books about combinatorics, number theory and geometry at a beginner and intermediate level. I would appreciate ...
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### What is the steps for finding a formula [on hold]

I Got a problem then tried to solve it it was mathematical but i failed. Problem statement is given below: Several ages ago Berland was a kingdom. The King of Berland adored math. That's why, when he ...
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### If $x+y=10^{200}$ then prove that 50 divides $x$

Let $x$ be a positive integer and $y$ is another integer obtained after rearranging the digits of $x$. If $x+y=10^{200}$ then prove that $x$ is divisible by 50. My attempt Since $y$ is the digit ...
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### Prove $\sum_{k=1}^{n} 2^n \text{ mod }k > 2n$ where $n > 1000$

This problem is taken from a Russian textbook of past Olympiads. Its statement looks like this : Given a natural number $n > 1000$ prove that $\sum_{k=1}^{n} 2^n \text{ mod }k > 2n$. ...
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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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### Integer solution to the equation below

I wanted to know integer solutions to the equation (1/k1) + (10/k2) + (100/k3) + ..... + (10^18/k19) = 1 (where k1,k2,k3.... are integers) which I believe is ...
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### Economics : Game-theory (Nash equilibrium)

This is a homework question, but resources online are exceedingly complicated, so I was hoping there was a fast, efficient way to solving following question. Question: Six students are going on a ...
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### Sharing and odd pizza

Here is a classical problem, which every mathematician will have seen at least onece in their life: Anne and Ben are sharing a pizza. The pizza is divided into an even number of pieces of unequal ...
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### The problem of congruent areas in a triangle.

A problem was posed in front of me and I couldn't solve it after multiple attempts-- Consider any triangle and 3 concurent cevians are drawn from each of its 3 points . Now the figure formed has 6 ...
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### Independence of radicals: First-principles proof of special case

I've known this problem for a long time: Problem. Show that the number $\alpha=\sqrt{1} + \sqrt{2} + \ldots + \sqrt{n}$ is irrational for $n\geq 2$. but I haven't been able to find a solution from ...
Find a polynomial with integer coefficients which has a global minimum equal to (a)$- \sqrt{2}$, (b)$\sqrt{2}$. It it a high-school math contest problem. The answer is given: $$(a) ~~~~~~~P(x)=N(2x^... 1answer 61 views ### A number which can be factored into a product of k and k+2 consecutive natural numbers (each >1) We say that the number N \in \mathbb{N} has the property P(k) if it can be factored into a product of k consecutive natural numbers (not equal to 1). Find the value of k such that some ... 2answers 100 views ### Favourite problem books at university level As background let me start by stating what I perceive to be the point of problem books, or to put the matter in perhaps more acceptable way, how I define problem books. A large majority of textbooks ... 2answers 49 views ### A little bit confused about the solution for 2016 AIME II #7 Squares ABCD and EFGH have a common center at \overline{AB} || \overline{EF}. The area of ABCD is 2016, and the area of EFGH is a smaller positive integer. Square IJKL is constructed so ... 2answers 68 views ### Chessboard Kings and no check [closed] What is the largest number of kings which can be placed on a chessboard so that no two of them put each other in check? 2answers 107 views ### x^3 +y^2 +z =100z+10y+x What is the largest and smallest integer that satisfies this equation. x^3 + y^2 +z=zyx,where zyx denotes the sequence of the digits. x^3 +y^2 +z =100z+10y+x,where x,y,z>0 The maximum value of x,y,z individually can only be 9. Maximum value: = 9^3 + 9^2 + ... 2answers 43 views ### 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 ... 2answers 57 views ### What is the largest of the five missing numbers? This is Q28 from Australian Maths Competition 2014. A circle is surrounded by 6 other circles,in a hexagonal formation.The leftmost circle is 0,which the rightmost circle is 1000.Each of the five ... 1answer 77 views ### 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$$no ... 2answers 91 views ### 5 numbers add up to 3231.What is the 6th number? This is Q27 from Australian Maths 2013. 3 different non-zero digits are used to form 6 different 3-digit numbers.The sum of 5 of them is 3231.What is the 6 th number? What I tried: Let ... 1answer 37 views ### Prove that a^2 pq + b^2 qr + c^2 rp \leq  given a,b and c are sides of triangle and p+q+r=0 The question is asking to prove that a^2 pq + b^2 qr + c^2 rp \leq 0  given that a,b and c are the sides of a triangle and that p+q+r=0. I have tried AM GM as well as countless pages of ... 2answers 101 views ### What is the 2012th number in this pattern? This is question 30 from Australian Maths 2012 (0,1,2,1,2,3,2,3,4,1,2,3,2,3,4,3,4,5,2,3,4,...) What is the 2012th  number in this list? What I did: I broke up the first few numbers into ... 1answer 85 views ### An Olympiad question on arithmetic progressions. I stuck in the following problem that was one of math Olympiad questions. Can anybody give me some hints please? Suppose that s_1,s_2,s_3,\ldots is a strictly increasing sequence of positive ... 4answers 82 views ### pqrs \cdot 4 =srqp ,then what is the value of qrs? This is question 26 from Australian Maths Competition 2013. pqrs  is a 4-digit number and has the property that pqrs \cdot 4 = srqp.If p=2,what's the value if the 3-digit number qrs? Here's what ... 0answers 20 views ### Special integer system values couting Well I saw this question in a competition: A city uses a special system to represent integers. In the system, there are 5 different numerals A, B, C, D, E, corresponding to the values 1, 6, 36, ... 2answers 44 views ### What can be a good programming algorithm to solve the given equation other than the brute force? [closed] Find all x, y and z for n=100;$$x^2 + y^2 + z^2 = n$$x,\ y, and z should be positive integers. 1answer 82 views ### longest way to rearrange students before returning to original arrangement? [closed] This is Q24 from the 2012 Intermediate Australian Mathematics Competition: "A teacher has a class of twelve students. She thinks it would be a nice idea if they change desks every day, so she has ... 2answers 80 views ### AMC 2012(Senior) Q28 A quadrilateral with sides 15,15,15 and 20 is drawn with each vertex on a circle.Around this circle,a square is drawn,with each side tangent to the circle.What is the area of this square? I know ... 1answer 90 views ### AMC 2012 Junior Question [closed] x^2 +y^2 +z^2 = 100x+10y+z . Find the smallest number and largest number that fit the equation.The numbers are below 1000 I am just baffled at the question.Is there a way to tackle such questions? 3answers 89 views ### Q27 from AMC 2012(Senior) Five consecutive integers p,q,r,s,t,each less than 10000, produce a sum which is a perfect square,while the sum of q,r,s is a perfect cube.What is the value of  \sqrt{p+q+r+s+t} ? What I have ... 5answers 327 views ### 33^{33} is the sum of 33 consecutive odd numbers. Which one is the largest? (Q25 from AMC 2012) The number 33^{33} can be expressed as the sum of 33 consecutive odd numbers. The largest of these odd numbers is \mathrm{A.}\ 33^{32} +32 \mathrm{B.}\ 33^{31} +32 \mathrm{C.}\... 4answers 115 views ### Help in proving an inequality Show that$$a^4 + b^4\ge\frac{1}{8} if $a+b=1.$
The polynomial $Q(x)=x^3-21x+35$ has three distinct real roots $r,s,t$. Find reals $a,b$ so that $P(x)=x^2+ax+b$ satisfies $P(r)=s,P(s)=t,P(t)=r$ or $P(r)=t,P(t)=s,P(s)=r$. I tried using cardano to ...