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

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0
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1answer
109 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 ...
2
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1answer
161 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 ...
0
votes
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 ...
4
votes
2answers
233 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 ...
4
votes
2answers
229 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 \; ...
1
vote
1answer
132 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 ...
1
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1answer
42 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 ...
2
votes
2answers
129 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, ...
1
vote
1answer
244 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 ...
1
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2answers
50 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 ...
4
votes
2answers
172 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
votes
1answer
191 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
votes
1answer
124 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 ...
3
votes
2answers
154 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 ...
7
votes
7answers
511 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.
4
votes
2answers
146 views

Minimum number of coins to ensure 10 coins of one type are selected

One coin is labelled with the number $1$, two different coins are labelled with the number $2$, three different coins are labelled with the number $3$, $\ldots$ , forty-nine different coins are ...
2
votes
2answers
102 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
votes
2answers
194 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 ...
0
votes
1answer
63 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 $k$ ...
4
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1answer
201 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 ...
7
votes
2answers
137 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 $n=1,2,3$ ...
0
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1answer
79 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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4answers
429 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 ...
0
votes
1answer
56 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 ...
16
votes
1answer
172 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 ...
1
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2answers
133 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 ...
10
votes
2answers
257 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.
1
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2answers
79 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} ...
34
votes
1answer
620 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 ...
1
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2answers
392 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 ...
1
vote
1answer
98 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 ...
1
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2answers
69 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 $0.4$?
1
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1answer
87 views

Find functions such that under the Cartesian coordinate system $F(x, y) = f(x) g(y)$ but under the polar coordinate system $F(x, y) = h(r)$.

Find all non-constant function $F(x, y)\in C^2(\mathbb{R}^2)$ such that under the Cartesian coordinate system $F(x, y) = f(x)  g(y)$ but under the polar coordinate system $F(x, y) = h(r)$. My ...
10
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2answers
375 views

Another math contest problem: $\int_0^{\frac{\ln^22}4}\,\frac{\arccos\frac{\exp\sqrt x}{\sqrt2}}{1-\exp\sqrt{4\,x}}dx$

Prove: $$ {\Large\int_{0}^{\ln^{2}\left(2\right) \over4}}\, \frac{\arccos\left(\vphantom{\huge A} {\exp\left(\vphantom{\large A}\sqrt{x\,}\right) \over \sqrt{\vphantom{\large A}2\,}}\right)} ...
1
vote
1answer
86 views

Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge.

Let $\{a_n\}$ be any sequence of positive real numbers. Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge. For example, ...
4
votes
1answer
83 views

Minimum difference of roots of a polynomial and its derivative

Let $P(x) = (x-x_1)(x-x_2)...(x-x_n)$ where all the n roots are real and distinct. Let $y_1,y_2,...,y_{n-1}$ be the roots of $P'$. Show that $\min_{i\neq j}|x_i-x_j|<\min_{i\neq j}|y_i-y_j|$. My ...
3
votes
1answer
345 views

How find this value of $x,y$

let $x,y\in R$, such $$\begin{cases} \sqrt{1+(x+y)^2}=-y^6+2x^2y^3+4x^4\\ \sqrt{2x^2y^2-x^4y^4}\ge 4x^2y^3+5x^3 \end{cases}$$ find the value of $x,y$. My try: since ...
1
vote
1answer
224 views

Newton's problem of cows and fields

I encountered this problem about Newton's problem of cows and fields: In a field, 17 cows can finish the whole grass in the field for 30 days. 19 cows can finish in 24 days. If a group of cows eat ...
70
votes
3answers
2k views

A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help. Prove: ...
0
votes
2answers
111 views

KVPY Scholarship Exam Problem on finding the area of a rectangle

In a rectangle $ABCD$, the coordinates of $A$ and $B$ are $(1,2)$ and $(3,6)$ respectively and some diameter of the circumscribing circle of $ABCD$ has equation $2x-y+4=0$. Then the area of the ...
0
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0answers
51 views

placing integers on circle with no repeated differences.

Is it possible to place 2008 numbers from 1 to 2009 on a circle such that the absolute values of the differences between numbers and their inmediate neighbors are all different? I think this is from ...
0
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1answer
52 views

The intersections of three polygons in a square with area $=6$

Let three convex polygons with areas equal to $3$, in a square with area equals to $6$. We need to prove that there are two of them which has their intersection with area is at least $1$. I have no ...
3
votes
3answers
152 views

Question from Spring 2012 AMATYC Student Mathematics League

How would you go about solving a problem like this: Let a, b, and c be positive integers which satisfy $a^3+b^3+c^2=2012$. Find $a+b+c.$ It doesn't appear that there's enough information to ...
10
votes
2answers
296 views

How prove this equation have infinite solution?

Let $x,y,z\in Z$, such that $\gcd(x,y)=\gcd(y,z)=\gcd(x,z)=1$. Show that the number of solutions to $$2013x^2+y^3=z^4$$ is infinite. This problem is from the China Mathematical Olympiad ...
1
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2answers
109 views

Determine the least natural number $k$ such that $a(k)>1$

Let $a(n)$ be a sequence with $a(0)=1/2$ and $a(n+1)=a(n)+(a(n)^2)/2013$, $n$ natural number. Determine the least natural number $k$ such that $a(k)>1$. This problem is from Poland proposed to ...
1
vote
1answer
60 views

Let $f,g$ be two distinct functions from $[0,1]$ to $(0, +\infty)$ such that $\int_{0}^{1} g = \int_{0}^{1} f $.

Let $f,g$ be two continuous, distinct functions from $[0,1]$ to $(0, +\infty)$ such that $\int_{0}^{1} g = \int_{0}^{1} f $. Given $n\in \mathbb{N},$ let $y_n = \int_{0}^{1} \frac{f^{(n+1)}}{g^{(n)}} ...
3
votes
5answers
402 views

What is the value of $f(0)+f(8)$?

Suppose $f$ is a polynomial of degree $7$ which satisfies $f(1) =2$, $f(2)=5$, $f(3)=10$, $f(4)=17$, $f(5)=26$, $f(6)=37$ and $f(7)=50$. What is the value of $f(0)+f(8)$?
0
votes
1answer
82 views

Basis and dimensions for quadratic polynomials

How do I find the basis and dimension for the set of all quadratic polynomials p(x)=ax^2+bx+c that satisfy p(1)=0.
0
votes
0answers
40 views

Why is $A^{m} - 1 = (A^{m'} - 1)(A^{m'(a-1)} + A^{m'(a-2)} + … + A^{m'} + 1).$

Show that if $m$ is a multiple of $a^n$, then $(a + 1)^m -1$ is a multiple of $a^{n+1}$. Here is a solution, but I don't understand it: We use induction on $n$. For $n = 0$ we have to show that ...
1
vote
3answers
83 views

Finding the possible location of points

The numbers 1,2,....6 are to be placed in some order at the points A,B,.....F in the figure below. How many ways can the numbers be placed so that each sum of consecutive pairs of points is odd?