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

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1answer
140 views

Rigorous Statements: “It suffices to show that […]” and Variations

Mathematical proofs display a variety of proof styles and language used. One of the common statements I have seen are "It suffices to show that [...]" and "We want to show that [...]". Of course, ...
3
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2answers
118 views

The number 3211000 is 7-special

Define a positive integer $k$ to be $n$-special if it satisfies the following properties: It has $n$ digits (0, 1, ..., 9) The 1st digit is equal to the number of 0's in the decimal representation ...
5
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2answers
104 views

$a_1=1,a_{n+1}=\frac{n}{a_n}+\frac{a_n}{n}$. Prove that for $n\ge4$, $\lfloor{a_n^2}\rfloor=n$

Define a sequence $\left\lbrace a_{n}\right\rbrace$ by $\displaystyle{a_{1} = 1\,,\ a_{n + 1} = {n \over a_n} + {a_n \over n}.\quad}$ Prove that for $n \geq 4,\,\,\left\lfloor ...
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3answers
127 views

Find all polynomials $P(x)$ such that $2xP(x)=(x+1)P(x-1)+(x-1)P(x+1)$.

Find all polynomials $P(x)$ such that $2xP(x)=(x+1)P(x-1)+(x-1)P(x+1)$. Well, if $\deg P\le 3$ this is easy since we can deduce $P(0)=P(1)=P(-1)$ by letting $x=0,1,-1$
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1answer
181 views

No primes in this sequence

Here's a fun little problem: Prove that the sequence $$10001, 100010001, 1000100010001, \cdots$$ contains no prime numbers.
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1answer
81 views

In a party with 2000 persons, determine # of people who know everyone

In a party with 2000 persons, among any set of four there is at least one person who knows each of the other three. There are three people who are not mutually acquainted with each other. How many ...
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1answer
75 views

Czech Republic Math Olympiad 2008 Problem

In decimal representation, we call an integer k-carboxylic if and only if it can be represented as a sum of k distinct integers, all of them greater than 9, whose digits are the same. For instance, ...
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2answers
137 views

Continuous functions on $\mathbb{R}^2$ with special property

The following problem is from Miklos Schweitzer competition (Year 1983, Problem 7): Prove that if the function $f: \mathbb{R}^{2}\to [0, 1]$ is continuous, and its average on every circle of ...
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2answers
49 views

Recursive formular and closed-form questions

Follow the question the $f(n)=4n-1$ and $F(n)=\sum_{k=0}^nf(k)$. And it ask you to write the recursive of $F(n)$. But I only know the recursive of $f(n)$ is $$f(n)=\begin{cases} -1,&\text{if ...
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1answer
56 views

Conditions for convergence of a geometric series [duplicate]

This question concerns the infinite geometric series formula. It turns out there is a nice formula for the sum of an infinite geometric series. Consider the infinite geometric series ...
5
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1answer
78 views

Macedonia National Olympiad 2010

Problem The point O is the center of the circumscribed circle of the acute-angled triangle ABC. The line AO cuts the side BC in point N, and the line BO cuts the side AC at point M. Prove that if ...
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2answers
51 views

Contest Math Possible Triangles

In the xy-plane, how many triangles have each of their vertices at points (a,b) where a,b are integers satisfying 1 ≤ a ≤ 5 and 1≤b≤5? I got twenty-five, but something tells me this isn't right. I ...
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1answer
54 views

Calculating number of pages from sum of page numbers

A novel has 6 chapters. As usual, starting from the first page of the first chapter, the pages of the novel are numbered $1, 2, 3, 4, \ldots$. Also, each chapter begins on a new page. The last ...
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1answer
48 views

Young Tableaux Generalizing

The entries in a array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? (2010 AMC12 B) The ...
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2answers
167 views

France Olympiad Team Selection Test 2005

In an international meeting of n ≥ 3 participants, 14 languages are spoken. We know that: - Any 3 participants speak a common language. - No language is spoken by more than half of the participants. ...
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0answers
57 views

Who made now part of the problem?

Who came up with the meme of putting the current year as a four digit number into exercise problems? Is there a known first historical account?
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1answer
96 views

AMC Problem Help 12B 2010

A geometric sequence $(a_n)$ has $a_1=\sin x$, $a_2=\cos x$ , and $a_3=\tan x$ for some real number $x$. For what value of $n$ does $a_n=1+\cos x$? The AMC website has a solution to this, and ...
3
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2answers
116 views

Combinatorial Proof Of A Number Theory Theorem--Confusion

I came across a combinatorial proof of the Fermat's Little Theorem which states that If $p$ is a prime number then the number ($a$$p$-$a$) is a multiple of $p$ for any natural number $a$. Let me ...
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4answers
817 views

How to do well on Math Olympiads

I'm a high school student who really likes maths and I'm quite good at school. I want to start training maths by myself but I think I need some guidelines. I want to do well on IMO but I don't know ...
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1answer
100 views

Polynomials as sum of squares

Sometimes I have seen some math's competition problem solutions made by completing the expression as sum of squares. What is the intuition/computer program behind these solutions? For example, Prove ...
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1answer
147 views

Number of ways of expressing $n$ as a sum of positive integers

a) Let $s_n$ denote the number of ways of expressing $n$ as a sum of positive integers. Thus $s_1=1$, $s_2=2$, and $s_3=4$ (the four ways are $3$, $2+1$, $1+2$, and $1+1+1$). Prove that ...
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0answers
21 views

Number Of Triangles of All Sizes in an Equilateral Triangle [duplicate]

https://mail.google.com/mail/u/0/?ui=2&ik=4622e6803e&view=att&th=1422d3806080ed0d&attid=0.1&disp=emb&realattid=ii_1422415e014f71c5&zw&atsh=1 Consider an ...
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2answers
182 views

Logic Puzzle of Diamonds and sons

I came across a math problem and I need a solution for this. An old man has 49 diamonds. Each one has a different worth as $1, $2, $3, ….. $49. He has 7 sons and he ...
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2answers
186 views

Algebra question from Australia national olympiad 2013

Find all positive integers $n$ for which there are real numbers $x_1, \; x_2, \cdots,\; x_n$ satisfying $$(i) \; \; -1<x_i<1 \; for \; i=1,2, \cdots n$$ $$(ii) \; \; x_1+x_2+ \cdots +x_n=0 \; ...
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1answer
97 views

China Girls Math Olympiad (CGMO) 2002

There are 3n girl students who took part in a summer camp. There were three girl students to be on duty every day. When the summer camp ended, it was found that any two of the 3n students had been on ...
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1answer
40 views

Prove that number of $(A, B, C)$ with $A ∩ B ∩ C = \emptyset$, $A ∩ B \ne \emptyset$, $B ∩ C \ne \emptyset$ is $7^n − 2\cdot6^n + 5^n$

Prove that the number of triples $(A, B, C)$ where $A, B, C$ are subsets of $\{1,2,\cdots,n\}$ such that $A ∩ B ∩ C = \emptyset$, $A ∩ B \ne \emptyset$, $B ∩ C \ne \emptyset$ is $7^n − 2\cdot6^n ...
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2answers
120 views

Checkers on a Chessboard

Given 2k pieces on a k by k chessboard, prove that there is always a sequence of pieces $K_1, K_2 \ldots K_{2n}$ such that $K_1$ and $K_2$ are in the same row, $K_2$ and $K_3$ are in the same column, ...
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1answer
144 views

What books to refer while preparing for rmo?

I am preparing for Regional Mathematics Olympiad and would like to know the books I should refer to prepare for the same. I basically would prefer the ones which enhance the ability to strike and ...
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2answers
47 views

Number theory recursion congruence problem.

this is a problem a friend of mine asked me: for any integer $n: a_1=n $ and for $a_k$ and $k$ an integer such that $k>1$ we have a_k the only integer such that $0\leq a_k<k$ and ...
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2answers
163 views

Exercise sum equal to 1 using only the digits 1,2, 3,…,9

Give a method to write the number one as the sum of three fractions, where each fraction the numerator is a one-digit number the denominator is a two-digit number and numbers that can be used are from ...
5
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1answer
167 views

Tricky Puzzle!! Please help.

I stumbled upon a puzzle I can't crack. It goes like this: In a certain Code language: 7321=6 5342=3 8645=15 Then 9312=? The Answer is 9. But I can't seem to find the logic behind it??
4
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1answer
118 views

Number Theory Contest Problem

Given that $x, y$ are positive integers with $x(x + 1)\mid y(y + 1)$, but neither $x$ nor $x + 1$ divides either of $y$ or $y + 1$, and $x^2+ y^2$ as small as possible, find $x^2+ y^2$. I have tried ...
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2answers
134 views

Number Theory Contest Math

Find the smallest positive integer $n$ such that $n^4 + (n + 1)^4$ is composite. Find the sum of the first $5$ positive integers $n$ such that $n^2 - 1$ is the product of 3 distinct primes. Answer to ...
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7answers
369 views

If $a^3 + b^3 +3ab = 1$, find $a+b$

Given that the real numbers $a,b$ satisfy $a^3 + b^3 +3ab = 1$, find $a+b$. I tried to factorize it but unable to do it.
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2answers
81 views

Finding the index such that all partial sums are nonnegative

Given an array a[] of integers of arbitrary size N that sum to 0 (for example, a[] = {-1, 0, 5, 3, -9, 2}), does there always exists an index i ($0\le i\le N-1$) such that each partial sum $S_j = ...
2
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2answers
175 views

Solving two simultaneous equations

Suppose that $x$, $y$ and $z$ are three integers (positive,negative or zero) such that we get the following relationships simultaneously $x + y = 1 - z$ and $x^3 + y^3= 1 - z^2$ Find all such ...
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1answer
60 views

Partition of circumference into $3k$ arcs

The following problem is from 1982 Russian Mathematical Olympiad. If you go to this link, and scroll down to the section Russian Math Olympiad, then this is Problem 333 in that text-file. Let $k$ ...
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1answer
187 views

Online Math Open Contest 2 Problem 50

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be ...
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2answers
122 views

Compute $\int_0^1\int_0^1…\int_0^1\lfloor{x_1+x_2+…+x_n}\rfloor dx_1dx_2…dx_n$

Compute $\int_0^1\int_0^1...\int_0^1\lfloor{x_1+x_2+...+x_n}\rfloor dx_1dx_2...dx_n$ where the integrand consists of the floor (or greatest integer less than or equal) function. The case $n=1,2,3$ ...
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1answer
63 views

Choosing a Set of r elements from a set having n elements.

Define a set $X$={$1$,$2$,$...$,$n$} . Determine the number of ways of selecting a subset of $X$ such that it contains no consecutive integers .
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331 views

“If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer”

If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer. I found this question in RMO 1992 paper ! Can anyone help me to prove ...
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1answer
55 views

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$.

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and either of the following is true: $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$. We see that if $(a,b,c)$ is a solution, then so is ...
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1answer
164 views

$|3^a-2^b|\neq p$, from a contest

I recently came across an old contest problem: (I did not find the solution anywhere) Find the least prime number which cannot be written in the form $|3^a-2^b|$ where $a$ and $b$ are ...
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2answers
98 views

Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.

Let $f$ be a continuous, nonnegative, real-valued function and $$\int_0^{\infty}f(x)dx<\infty.$$ Prove that $$\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0.$$ A start: If ...
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1answer
197 views

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$, $f(x)f(yf(x))=f(x+y)$

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$$$f(x)f(yf(x))=f(x+y)$$ A start: set y=0 to get $f(x)f(0)=f(x)$. So $f(0)=1$ unless $f$ is identically zero.
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2answers
75 views

Determine the value of $p>0$ for which $\sum_{n=1}^{\infty}(-1)^{\lfloor{\sqrt{n}}\rfloor}/n^p$ converges.

Determine the value of $p>0$ for which $$\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor{\sqrt{n}}\rfloor}}{n^p}$$ converges. By considering $\lfloor{\sqrt{n}}\rfloor$, we see the series is $$\sum_{k\ge1} ...
33
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1answer
549 views

Integral $\int_0^1\frac{x^9\left(x^4+x^2-x-1-5\ln x\right)}{\left(x^{10}-1\right)\ln x}dx$

A friend of mine sent me an integral that she had not been able to crack, and me neither. It comes with a result, but without a proof (I suppose it originated in some math contest). Could you please ...
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2answers
324 views

A truth teller and liar puzzle of Ramanujan mathematical olympiad 2013

On an island each person always tells the truth or each person always tells a lie. Three people say $A$ , $B$ and $C$ have a conversation. $A$ says that $B$ is lying , $B$ says that $C$ is lying and ...
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1answer
96 views

How find this maximum of $P_{1}+P_{n}$

Question $n$ students attend a test of $m$ problems where $m, n \ge 2$. The scoring rule for each problem is: If $x$ students answer a problem incorrectly, then a correct answer worth $x$ points ...
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2answers
54 views

Square root of decimal Places

I'm searching to find how to get the sqaure root of a number having decimal places. How to find the Square root of say $0.4$?