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

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Diophantine Equation (2014 AMC 12A)

There are exactly $N$ distinct rational numbers $k$ such that $|k| < 200$ and $$5x^2 + kx + 12 = 0 $$ has at least one integer solution for $x$. What is $N$? (My idea was to consider the equation ...
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1answer
144 views

how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\le 1000$

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$,For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\le 1000$ $A:38 , B:40 C:42 ...
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83 views

Contest geometry problem

$|AM|=|CM|$ $\angle BCA = 15^{\circ}$ $\angle CBM = \angle ABH$ $\angle BHC = 90^\circ$ Find $|AC|$ The solution states that $\overline{BM}$ is the isogonal conjugate of $\overline{BH}$ but I ...
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1answer
140 views

AMC 12 2010B Problem Help #18

Can someone explain this solution? A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final ...
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2answers
87 views

Choosing people around a circular table

There are 20 people around a circular table.We have to choose $3$ of them such that at least $2$ of them are sitting together.In how many ways can this be done? Number of ways of choosing 3 people ...
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1answer
192 views

Sequence where the sum of digits of all numbers is 7

BdMO 2014 We define a sequence starting with $a_1=7,a_2=16,\ldots,\,$ such that the sum of digits of all numbers of the sequence is $7$ and if $m>n$,then $a_m>a_n$ i.e. all such numbers are ...
3
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1answer
192 views

The library with 999 books.

In the town of Capibara there is a library with books in 999 themes. Since Capibara is an international town they have books in various languages. We know that for every language we can find exactly ...
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1answer
306 views

putnam mathematics

The tail of a giant kangaroo is attached by a giant rubber band to a stake in the ground. A flea is sitting on top of the stake eyeing the kangaroo (hungrily). The kangaroo sees the flea leaps into ...
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1answer
31 views

Compute number of points having same property

I have been given a cuboid which has either green or red color for each point (integer coordinates) in it. I am also given another cuboid whose lower left corner is (x1, y1, z1) and upper right corner ...
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2answers
137 views

$7$ points inside a circle at equal distances

BdMO 2014 There are $7$ points on a circle.Any 2 consecutive points are at equal distance from one another.How many acute angled triangles can you form taking any 3 of these points? I believe ...
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3answers
109 views

Show that the number of 5-tuples (a, b, c, d, e) such that abcde = 5(bcde + acde + abde + abce + abcd) is odd

Show that the number of 5-tuples $(a,b,c,d,e)$ such that$$abcde=5(bcde+acde+abde+abce+abcd)$$ is odd.
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1answer
77 views

Eliminating numbers from the sequence $1,2,3,4,5,6,7…400$

BdMO 2014 Let us take the sequence $1,2,3,4,5,6,7....400$ .We are going to remove numbers from the sequence such that the sum of any 2 numbers of the remaining sequence is not divisible by 7.What ...
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1answer
62 views

Brazil 2002 first problem neater result?

Brazil's 2002 first problem basically asks to prove that for any positive integer n, there are n integers $m_1,m_2\dots m_n$ where $1\leq m_i\leq9$ such that $m_1^2+m_2^2+\ldots+m_n^2=a^2$ for some ...
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0answers
48 views

Separating points on a plane

BdMO 2011 There are $25$ points on a plane, no three of which lie on a line. Find the minimum number of lines needed to separate them from one another. Can we assume that the points lie on a ...
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1answer
85 views

Arranging red and blue tiles in a line with at least 1 blue tile between any 2 red tiles

BdMO 2010 Nationals: Tom and Jerry have $8$ blue tiles and $6$ red tiles.They want to arrange them in a straight line so that between any $2$ red tiles there is always at least $1$ blue tile.In ...
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4answers
516 views

Find all bijections $\,\,f:[0,1]\rightarrow[0,1],\,$ which satisfy $\,\,f\big(2x-f(x)\big)=x$.

A friend of mine gave me the following problem: Find all functions $f:[0,1]\to[0,1]$, which are one-to-one and onto and satisfy the following functional relation: $$ f\big(2x-f(x)\big)=x, \tag{1} $$ ...
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4answers
137 views

Sum $\sum_{k=0}^{2013}2^ka_{k}$

let real sequence $a_{0},a_{1},a_{2},\cdots,a_{n}$,such $$a_{0}=2013,a_{n}=-\dfrac{2013}{n}\sum_{k=0}^{n-1}a_{k},n\ge 1$$ How find this sum $$\sum_{k=0}^{2013}2^ka_{k}$$ My idea: since ...
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7answers
291 views

$211!$ or $106^{211}$:Which is greater?

A BdMO question: Let $a=211!$ and $b=106^{211}$. Show which is greater with proper logic. By matching term by term,it is pretty easy to note that $106!<106^{106}$ $106^{105}<107\cdot ...
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3answers
129 views

Prove $4(x + y + z)^3 > 27(x^2y + y^2z + z^2x)$

Prove that, for all positive real numbers $x$, $y$ and $z$, $$4(x + y + z)^3 > 27(x^2y + y^2z + z^2x)$$ I've tried expressing it as a sum of squares, but haven't got anywhere. Hints are also ...
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2answers
54 views

What is the minimal number of weighings required to find an odd (lighter) coin out of 80?

I have $80$ coins. Among them, exactly one coin is lighter compared to all the others. I was given a physical balance, suddenly. What is the minimal number of weighings required to find the lighter ...
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1answer
95 views

How to calculate time in this condition?

A starts at 11:00AM and travels at a speed of 4km/hr. B starts at 1:00PM and travels at 1km/hr for the first 1hr and 2km/hr for the next hr and so on. At what time they will meet each other ? Note - ...
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2answers
47 views

Show that there exists a convex hexagon in the plane such that (a) all its interior angles are equal,

Show that there exists a convex hexagon in the plane such that (a) all its interior angles are equal, (b) all its sides are 1, 2, 3, 4, 5, 6 in some order. it is the 9th question inmo 1993. i cant ...
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0answers
191 views

International undergraduate mathematics Olympiad preparation

I have decided to try out and compete at the next international undergraduate mathematics Olympiad but first I need to get selected for the team that is sent for the competition from my country. I ...
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1answer
162 views

Proving there exist no positive integers $m,n$ such that $ m/n +(n+1)/m = 4$

Prove that there exist no positive integers $m,n$ such that $$ \frac{m}{n} + \frac{n+1}{m} = 4.$$ I worked on cases considering $m$ and $n$ are even or odd, but I couldn't get anything.
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2answers
60 views

Given $x^2 + y^2 + z^2 = 3$ prove that $x/\sqrt{x^2+y+z} + y/\sqrt{y^2+x+z} + z/\sqrt{z^2+x+z} \le \sqrt3$

Given $x^2 + y^2 + z^2 = 3$ Then prove that $${x\over\sqrt{x^2+y+z}} + {y\over\sqrt{y^2+x+z}} + {z\over\sqrt{z^2+x+y}} \le \sqrt 3$$ I tried using Cauchy-Schwartz inequality but the inequality is ...
3
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2answers
361 views

Finding all positive integers $x,y,z$ that satisfy $3^x - 5^y = z^2$

Find all positive integers $x,y,z$ that satisfy: $$3^x - 5^y = z^2.$$ I think that $(x,y,z)= (2,1,2)$ will be the only solution. But how to prove that?
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2answers
211 views

No cont function $f\colon\mathbb{R}\to\mathbb{R}$ with $f(x)$ rational $\iff f(x+1)$ irrational.

Prove that there are no continuous functions $f\colon \mathbb{R} \to \mathbb{R}$ with the property: For any $x \in \mathbb{R}$, $f(x)$ is a rational number if and only if $f(x+1)$ is an irrational ...
5
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4answers
156 views

$x$ is equal to at least $51$ of $a_1,\frac{a_1+a_2}{2},\ldots,\frac{a_1+a_2+\ldots+a_{100}}{100}$. Prove that $2$ of $a_1,\ldots,a_{100}$ are equal.

If $x$ is equal to at least $51$ number of the array $a_1, \frac{a_1+a_2}{2},\ldots,\frac{a_1+a_2+\ldots+a_{100}}{100}$, prove that $2$ numbers of the array $a_1,a_2\ldots,a_{100}$ are equal. ...
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1answer
98 views

What is the meaning of $(x^2+y^2)^n$? Is this an already known geometric object?

We all know that $x^2+y^2=r^2$ is a circle. What does $(x^2+y^2)^2$ signify? In general, what is $(x^2+y^2)^n$?
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4answers
118 views

If $a,b,c,d>0$ and $a+b+c+d=4$, then $\frac{1}{11+a^2}+\frac{1}{11+b^2}+\frac{1}{11+c^2}+\frac{1}{11+d^2} \leq \frac {1}{3}$

Prove if $a,b,c,d>0$ and $a+b+c+d=4$, then $$\dfrac{1}{11+a^2}+\dfrac{1}{11+b^2}+\dfrac{1}{11+c^2}+\dfrac{1}{11+d^2} \leq \dfrac {1}{3}$$ This was an Inequality Olympiad Problem1. I proved by ...
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1answer
104 views

$a^3+b^3+c^3 + 21abc \geq 3$ for $(a+b)(a+c)(b + c) = 1$ and $a,b,c>0$

$a, b, c \gt 0$ and $(a+b)(a+c)(b + c) = 1$ Prove that $a^3+b^3+c^3 + 21abc \geq 3$ In this problem I spotted one trick $(a+b)(a+c)(b + c) = 1 \Leftrightarrow \\(a \sqrt{b+c})^2+(b\sqrt{a+c})^2+(c ...
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1answer
93 views

Maximum value of the lowest sum in a set of numbers

Last year in a maths contest held in Catalonia called Cangur it was posed the following qüestion: We write numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10, in a certain order around a circumference. Then ...
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2answers
524 views

An upper bound on certain finite trigonometric series given a lower bound

Let $f$ be the function $f(x)=1+a\sin{x}+b\cos x+c\sin{(2x)}+d\cos{(2x)}$, where $a,b,c,d$ are arbitrary real numbers. Prove that if $f(x)>0$ for all $x\in \mathbb R$, then $f(x)<3$ for all ...
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0answers
177 views

How Find $3x^3+4y^3=7,4x^4+3y^4=16$

if postive real number $x,y$ such $$\begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases}$$ Find $x+y=?$ My try: $$4x^4-3x^3+3y^4-4y^3=9$$ $$x^3(4x-3)+y^3(3y-4)=9$$
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1answer
91 views

P0lyn0mial questi0n

Suppose $P(x)$ is a polynomial of degree $2012$ and $P(x) = 1/x$ when $x$ takes the integer values $1\cdots2013$ (inclusive). What is the value of $P(2014)$? I get $1/1007$ but I'm not sure if ...
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0answers
47 views

Generalized inequality with parameters $\alpha, \beta$

Let $d$ be a positive integer, and let $\alpha, \beta$ be positive real numbers such that $\alpha+\beta=1$. Consider the inequality in $k$ variables $x_1, x_2, …, x_k$, $$ \alpha \cdot \sum_{i=1}^k ...
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1answer
67 views

math contest ranking problem?

A math contest is held among 4 middle schools. Each school enters a team of 3 students. The 12 contestants are ranked from 1 (best performance) to 12 (worst performance). The team that has the overall ...
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2answers
132 views

Cute coloring problem on a board

Suppose we color an $n\times n$ square board using $n$ colors exactly $n$ times each. Prove that there is either a column or a row containing at least $\lceil \sqrt n \rceil$ different colors. A ...
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2answers
91 views

If $7$ is the first digit of $2^n$, what is the first digit of $5^n$?

Let $2^n = 7\cdot 10^x + p$ and $5^n = a\cdot 10^y + r$ And now what? (We're in base $10$)
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3answers
160 views

Moscow Math Olympiad 1973

In every polyhedron there is at least one pair of faces with the same number of sides. Solution: Let $N$ be the greatest number of sides in a face of a given polyhedron. Then the number of ...
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1answer
145 views

South Africa National Olympiad 2000 (Tile 4xn rectangle using 2x1 tiles)

Let $A_n$ be the number of ways to tile a $4×n$ rectangle using $2×1$ tiles. Prove that $A_n$ is divisible by 2 if and only if $A_n$ is divisible by 3. My attempt: Define basic shapes A, B and C, ...
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0answers
44 views

Two Perfect Squares--$(3n+1) \& (4n+1)$. [duplicate]

Assume $n$ is a Natural Number which satisfies the following 2 properties simultaneously: $01$ . $(3n+1)$=$a$12 for some Natural Number $a$1. $02$ . $(4n+1)$=$a$22 for some Natural Number $a$2. ...
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1answer
418 views

IMO 1979 problem

The question is $$\text{If }\, p, \ q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad \text{Prove that } 1979\mid p.$$ So my solution went like ...
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0answers
76 views

(AIME) number theory question [duplicate]

How many integers less than 1000 can be expressed in the form $$\frac{(x + y + z)^2}{xyz} $$ where $x, y, z$ are integers? So far, I've attempted substituting certain values of $x, y, z$. For ...
4
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1answer
187 views

Which methods different than the natural $\lim_{n\to\infty}\frac{|\cos{1}|+|\cos{2}|+|\cos{3}|+\cdots+|\cos{n}|}{n}$ [closed]

Compute this limit $$\lim_{n\to\infty}\dfrac{|\cos{1}|+|\cos{2}|+|\cos{3}|+\cdots+|\cos{n}|}{n}$$ This problem is from $13^{th}$ Annual Harvard-MIT Mathematics Tournament problem 4. and this Answer ...
0
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0answers
63 views

Ratios in a rhombus

NOTE: I am NOT looking for a full answer,just a hint. Last problem on this question. BdMO 2013 Chittagong: Let $ABCD$ be a rhombus.Let $G$ be a point outside the rhombus such that GE is ...
2
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5answers
126 views

Prove that there exist infinitely many pythagorean integers $a²+b²=c²$

Prove that there exist infinitely many Pythagorean integers $a²+b²=c²$ My key idea is to show that there exists infinitely many integers that can be the length of the sides of a right triangle, but ...
5
votes
2answers
150 views

How find this $\sum_{i=0}^{5}\frac{1}{2+\cos{\left(x+\frac{i\pi}{3}\right)}}\cdot \frac{1}{2+\cos{\left(x+\frac{(i+1)\pi}{3}\right)}}$

Find this follow function $f(x)$ range ,where $x\in R$, $$f(x)=\sum_{i=0}^{5}\dfrac{1}{2+\cos{\left(x+\dfrac{i\pi}{3}\right)}}\cdot \dfrac{1}{2+\cos{\left(x+\dfrac{(i+1)\pi}{3}\right)}}$$ or ...
3
votes
2answers
68 views

Divisibility Of Positve Integers [closed]

Suppose a,b and c are three positive integers which satisfy the condition that ($a$2+$b$2+$c$2) is divisible by $(a+b+c)$. Prove that there exists infinitely many positive integers $n$ for which ...
2
votes
3answers
87 views

Separating $3n$ points on the plane by a line

I am trying to solve a problem in geometry (a contest-type question), and I wondering if the following result is true. (If it is true, then it makes life much easier!) Suppose there are $3n$ ...