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

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7
votes
7answers
268 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 ...
3
votes
3answers
82 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 ...
0
votes
2answers
43 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 ...
1
vote
1answer
52 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 - ...
1
vote
2answers
40 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 ...
1
vote
0answers
79 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 ...
1
vote
1answer
142 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.
1
vote
2answers
55 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
votes
2answers
222 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?
8
votes
2answers
165 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
votes
4answers
149 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. ...
0
votes
1answer
89 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 $(x^2+y^2)^n$?
6
votes
2answers
79 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 ...
-1
votes
1answer
99 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 ...
5
votes
1answer
82 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 ...
18
votes
2answers
444 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 ...
2
votes
0answers
156 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: $$4x^4-3x^3+3y^4-4y^3=9$$ $$x^3(4x-3)+y^3(3y-4)=9$$
5
votes
1answer
85 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 ...
1
vote
0answers
39 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 ...
1
vote
1answer
59 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 ...
4
votes
2answers
103 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 ...
1
vote
2answers
86 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 $10$)
2
votes
3answers
130 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 ...
7
votes
1answer
108 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, ...
1
vote
0answers
43 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. ...
6
votes
0answers
250 views

IMO 1979 problem

The question is $$If\, p,q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad Prove \,that\, 1979|p.$$ So my solution went like this: ...
1
vote
0answers
74 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
votes
1answer
173 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
votes
0answers
57 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
votes
5answers
109 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
138 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 ...
2
votes
2answers
65 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
72 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 $3n$ ...
39
votes
7answers
9k views

There exists a power of 2 such that the last five digits are all 3's or 6's. Find the last 5 digits of this number

I just took an olympiad and I'm wondering how this problem is solved. Problem: There exists a power of 2 such that the last five digits are all 3's or 6's. Find the last 5 digits of this number. ...
2
votes
2answers
74 views

How to prove the inequality: $\frac{(1+x)^2}{2x^2+(1-x)^2}+\frac{(1+y)^2}{2y^2+(1-y)^2}+\frac{(1+z)^2}{2z^2+(1-z)^2}\leq 8$

Prove that: $$\frac{(1+x)^2}{2x^2+(1-x)^2}+\frac{(1+y)^2}{2y^2+(1-y)^2}+\frac{(1+z)^2}{2z^2+(1-z)^2}\leq 8$$ subject to the constraints: $$x,y,z >0$$ and $$x+y+z=1.$$
0
votes
3answers
134 views

Competition Math Geometry Problem

Note: I am paraphrasing this problem Consider a quadrilateral with 3 sides of equal length, and one longer side. This quadrilateral also has equal diagonals, both of which are equal in length to the ...
4
votes
1answer
222 views

Interesting number theory questions

How many integers less than 1000 can be expressed in the form $$\frac{(x + y + z)^2}{xyz} $$ where $x, y, z$ are positive integers?
2
votes
1answer
67 views

How to prove $\binom{2n}{n}\frac{1}{n+1} = \prod \limits_{i = 2}^n \frac{2i-1}{i+1} $?

How to prove this closed form involving Catalan numbers? $$\binom{2n}{n}\frac{1}{n+1} = \prod \limits_{i = 2}^n \frac{2 \times (2i-1)}{i+1} $$ I have seen this being used here. Not sure how to derive ...
-1
votes
1answer
67 views

Please help me solve this problem [closed]

Every month, a girl gets an allowance. Assume one year ago she had no money, and so saved each month's allowance over the past year. Then, she spends $\frac{1}{2}$ of her money on clothes; then ...
2
votes
2answers
75 views

The floor of a product of fractions

Evaluate: $ \displaystyle \Bigg \lfloor \prod_{n=0}^{248} \frac{33+8n}{29+8n} \Bigg \rfloor= \Bigg \lfloor \frac{33}{29} \times \frac{41}{37} \times \frac{49}{45} \times\ ...\ \times ...
3
votes
2answers
134 views

If $x=123456789101112131415161718$, then $x\equiv 6\pmod{16}$ and $x\equiv 0\pmod 6$

BdMO 2013 Rajshahi Build a number by writing down consecutive natural numbers starting from $1$ which is divisible by $6$ and gives a reminder of $6$ upon division by $16$. Such a number is ...
1
vote
1answer
45 views

Number theory problem, parity

Let $f(x)$ denote the number of (not necessarily distinct) prime factors of $x$. Let $n > 1$ be the smallest positive integer for which there are more $i$ with with $f(i)$ even than $f(i)$ odd in ...
0
votes
1answer
121 views

Maximum area of a rectangle inside a triangle

I recently came across a problem where it gave a triangle with integer side lengths, and it asked you to find the maximum area of a rectangle of a triangle. I solved the problem correctly, but it ...
4
votes
2answers
61 views

Minimum value of: $x^7(yz-1)+y^7(zx-1)+z^7(xy-1)$

$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of: $$x^7(yz-1)+y^7(zx-1)+z^7(xy-1)$$ I put it in the form $x^6y +x^6z+y^6x+y^6z+z^6x +z^6y$. I tried AM-GM but ...
2
votes
3answers
229 views

Book with lots of geometry theorems

I want to study geometry and was looking for some book that has lots of theorems and covers almost all Euclidean geometry that is needed for High School and Maths Olympiads. Thanks.
1
vote
3answers
80 views

There exists an integer with alternating digits $1$ and $2$ which is divisible by $2013$

Could someone give me hints in how to solve the following (rather interesting) problem? Prove that there exists an integer consisting of an alternance of $1$s and $2$s with as many $1$s as $2$s ...
6
votes
2answers
127 views

Prove that $\sqrt[2012]{2012!}<\sqrt[2013]{2013!}$

I need to prove that $\sqrt[2012]{2012!}<\sqrt[2013]{2013!}$ My attempt: Let $a=\sqrt[2012]{2012!}$ and $b=\sqrt[2013]{2013!}$ Then $\displaystyle\frac{b^{2012}}{a^{2012}}=\frac{2013}{b}$ ...
3
votes
1answer
44 views

Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$

Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$ I tried it to do using $AM \ge GM$ but don't know how to proceed. Please help.
8
votes
1answer
119 views

How to fill up $(0,1)$ with disjoint closed intervals all total measure one

This is a problem which was proposed, but not chosen, for a mathematics competition for University students not long ago, and its solution is missing: Let $\sum_{n=1}^\infty a_n=1$, where ...
11
votes
10answers
2k views

Find five consecutive odd integers such that their sum is $55$.

So my professor asked us to do an Olympiad exercise which says that the sum of five consecutive odd integers is $55$, find those integers. But I've never seen such an exercise so it is quite new and ...