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

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2
votes
2answers
78 views

Prove the root is less than $2^n$

A polynomial $f(x)$ of degree $n$ such that coefficient of $x^k$ is $a_k$. Another constructed polynomial $g(x)$ of degree $n$ is present such that the coefficeint of $x^k$ is $\frac{a_k}{2^k-1}$. ...
-1
votes
1answer
33 views

Combinatoric meaning of multinomial coefficients

$$\binom{n}{k}$$ means how many ways there are to choose $k$ objects from $n$ total objects. What is the combinatoric meaning of: $$\binom{n}{k_1, k_2, ... , k_n}$$ ??
1
vote
1answer
39 views

Distinguishability in Round Table Combinatorics

I have stumbled upon many questions, and one of the weaknesses is the ability to test if the concept is distinguishable or not. For example this: Nine delegates, three each from three different ...
3
votes
1answer
87 views

Determine all functions $f:\mathbb{Q}\to\mathbb{Q}$ satisfying the functional equation $f(2f(x) + f(y)) = 2x + y$

Determine all functions $f$ defined on the set of rational numbers that take rational values for which $$f(2f(x) + f(y)) = 2x + y \tag{1}$$ for each x and y. This question is from the 2008 ...
3
votes
1answer
86 views

Solving the number theoretic equation $ \sum_{d|n}{d^4}=n^4+n^3+n^2+n+1 $

I found an interesting problem: Find all $n\in\mathbb N$ such that $$ \sum_{d|n}{d^4}=n^4+n^3+n^2+n+1 $$ If we define $s(n)=\sum_{d|n}{d^4}$, we can show, that $s(mn)=s(m)s(n)$ if $\gcd(m,n)=1$. ...
12
votes
2answers
137 views

Exist complex $z_{0}$ ,such $|z_{0}|=1$,and $|f(z_{0})|\le|f(z)|,\forall |z|\ge 1$

Let $a\in (0,1), f(z)=z^2-z+a, z\in \mathbb C$. Does there exist a complex number $z_{0}$ such that $|z_{0}|=1$, and $$|f(z_{0})|\le|f(z)|,\forall |z|\ge 1$$ I just have no idea where to even begin ...
0
votes
2answers
72 views

Can anybody solve this geometry? [closed]

Triangle ABC has CA < CB. The bisector of angle C meets the perpendicular bisector of side AB in G. The perpendicular from G to BC meets BC in F. Prove that CF = 1/ 2 (CA + CB).
9
votes
1answer
60 views

inequality $\max\{a_1,a_2,\cdots,a_n \}\leq {n^2}^{n-1}.$with Egyptian fraction

Let $a_1,a_2,\cdots,a_n $ be positive integer such that$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.$ Prove that$$\max\{a_1,a_2,\cdots,a_n \}\leq {n^2}^{n-1}.$$ This Problem from:1
1
vote
2answers
111 views

Determine all positive integers $n$ which have a divisor $d$ with the property that $dn+1$ is a divisor of $d^2 + n^2$

Determine all positive integers $n$ which have a divisor $d$ with the property that $dn+1$ is a divisor of $d^2 + n^2$. So i formed the equation that $$\frac{n}{d} = \frac{d^2 + n^2}{dn + 1}$$ And ...
0
votes
1answer
30 views

Smoothing Inequalities

Can anyone explain to me how the "smoothing argument for inequalities" works? I know that basically it can be used to prove an inequality $f(a_1,a_2,\cdots,a_n)\geq C$ subject to the constraint ...
2
votes
2answers
147 views

2014 iberoamerican olympiad Problem 3

2014 points are placed on a circumference. On each of the segments with end points on two of the $2014$ points is written a non-negative real number. For any convex polygon with vertices on some of ...
0
votes
2answers
89 views

The probability that each delegate sits next to at least one delegate from another country

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate ...
14
votes
3answers
281 views

Prove that there exists infinitely many positive integers $n$ such that $\sin^2{(na)}+\sin^2{(nb)}\le \frac{2\pi^2}{n}$

Can anyone please help me with the following proof: Prove that there exists infinitely many positive integers, $n$, such that $$\sin^2{(na)}+\sin^2{(nb)}\le \dfrac{2\pi^2}{n}\quad a,b\in \Bbb R$$
7
votes
1answer
63 views

What is maximum a number of to form right-triangles from in n straight lines

I am interested what is maximum a number of to form right-triangles from in $n=100$ straight lines such $n=3$,then maximum number of is $1$,see fig:$\Delta ABC$ is right-triangles. $n=4$ then ...
-1
votes
2answers
135 views

Finding the sum of this series: $1+\frac 12 + \frac 13 + \cdots + \frac 1{50}$ [duplicate]

I need to find the sum of this series: $1+\frac 12 + \frac 13 + \cdots + \frac 1{50}$ Please help me find the sum of this series.
3
votes
2answers
47 views

A plane contains a set of marked points, such that any three can be covered by a unit disk. Prove that the entire set can be covered by a unit disk.

A set of points is marked on the plane, with the property that any three marked points can be covered with a disk of radius 1. Prove that the set of all marked points can be covered with a disk ...
4
votes
2answers
90 views

Let $a,b,c>0$ so that $a+b+c=1$…

Let $a,b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c}+6\geq 2\sqrt{2}\left ( ...
5
votes
3answers
85 views

Solving a Complicated Trig Problem

I am preparing for AIME and I came across this problem which I need help solving: $$\begin{eqnarray} 10^{10^{10}} \sin\left( \frac{109}{10^{10^{10}}} \right) - 9^{9^{9}} \sin\left( ...
19
votes
2answers
170 views

For all $n$ there exists $x$ such that $\varphi(x)<\varphi(x+1)<\ldots<\varphi(x+n)$

Let $\varphi$ be the Euler's function, i.e. $\varphi(n)$ stands for the number of integers $m \in \{1,\ldots,n\}$ such that $\text{gcd}(m,n)=1$. Let $n\ge 2$ be a positive integer. Show that there ...
3
votes
1answer
107 views

Undergraduate mathematics competitions

I am a freshman (math undergraduate) here in Argentina and I am deeply interested in mathematical olympiads but I really need some advice. Right now, my problem solving skills are good but not that ...
1
vote
1answer
73 views

Olympiad-like Inequality Problem

Let $$ A := \frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \times\cdots\times \frac{2013}{2014};$$ let $$ B := \frac{2}{3} \times \frac{4}{5} \times \frac{6}{7} \times\cdots \times ...
0
votes
1answer
34 views

Find the number of sets satisfying the conditions

Let $ N$ be the number of ordered pairs of nonempty sets $ \mathcal{A}$ and $ \mathcal{B}$ that have the following properties: • $ \mathcal{A} \cup \mathcal{B} = ...
1
vote
1answer
48 views

Why this polynomial can't be expressed as a sum of squares of some polynomials (the real coefficients)

Before we has prove this polynomials is postive semi-definite,Now I want to show that this postive semi-definite polynomials $$x^2(x^2-1)^2+y^2(y^2-1)^2+(x^2-1)(y^2-1)(x^2+y^2-1)$$ can't be ...
16
votes
1answer
147 views

Prove this deg inequality $\deg{P(x)}\cdot \deg{Q(x)}\cdot \deg{R(x)}\ge 656$

Let three non-constant polynomails $P(x),Q(x),R(x)\in \mathbb Z[x]$,and if this equation $P(x)Q(x)R(x)=2015$ has $49$ distrinct integer roots. Prove that $$\deg{P(x)}\cdot \deg{Q(x)}\cdot ...
3
votes
1answer
37 views

Prove that $\left (\sum_{k=1}^{n}\frac{1+x^{2k}}{1+x^{4k}} \right )\left ( \sum_{k=1}^{n}\frac{1+y^{2k}}{1+y^{4k}} \right )< \frac{1}{(1-x)(1-y)}.$

Let $n$ be a positive integer, and let $x$ and $y$ be positive real numbers such that $x^{n}+y^{n}=1$ Prove that $$\left (\sum_{k=1}^{n}\frac{1+x^{2k}}{1+x^{4k}} \right )\left ( ...
2
votes
0answers
42 views

Find the smallest number $n$ such that there exist polynomials $f_{1}, f_{2},…,f_{n}$ with rational coefficients

Find the smallest number $n$ such that there exist polynomials $f_{1}, f_{2},...,f_{n}$ with rational coefficients satisfying $$x^{2}+7=f_{1}(x)^{2}+f_{2}(x)^{2}+...+f_{n}(x)^{2}.$$ It's Olympiad ...
1
vote
1answer
34 views

Permutations where no partial sum is divisible by 3 (contest question)

A permutation of the integers $1901,1902\dots 2000$ is a sequence in which each of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums $$s_1 = ...
3
votes
1answer
56 views

Maximize the Cyclic sum

Let $x_1,x_2,\dots ,x_6$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5+x_6=1$, and $x_1x_3x_5+x_2x_4x_6 \geq \frac{1}{540}$. Let $p$ and $q$ be positive relatively prime integers such ...
4
votes
0answers
64 views

Mathematical Olympiad Problem

Let $\Bbb{R}$ be the set of real numbers. Determine all functions $f:\Bbb{R}\longrightarrow \Bbb{R}$ satisfying the equation $$f(x+f(x+y))+f(xy) = x + f(x+y)+yf(x)$$ for all real numbers $x$ and $y$.
11
votes
3answers
170 views

Determine all functions satisfying $f\left ( f(x)^{2}y \right )=x^{3}f(xy)$

Denote by $\mathbb{Q}^{+} $ the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the following equation for all $x,y \in ...
1
vote
2answers
68 views

Ghosts closing and opening doors [duplicate]

There are $1000$ doors $D_1,D_2,D_3,\dots,D_{1000}$ and $1000$ persons $P_1,P_2,\dots,P_{1000}$. Initially all the doors were closed. Person $P_1$ goes and opens all the doors. Then person $P_2$ ...
3
votes
2answers
71 views

Prove that $\frac{1}{1999} < \prod_{i=1}^{999}{\frac{2i−1}{2i}} < \frac{1}{44}$

Prove that $$\dfrac{1}{1999} < \prod_{i=1}^{999}{\dfrac{2i−1}{2i}} < \dfrac{1}{44}$$ from the 1997 Canada National Olympiad. I have been able to prove the left half of the inequality using ...
0
votes
0answers
45 views

IMO 2003 #6 Question

Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.
1
vote
0answers
36 views

divisibility of $a^m+a-1$ by $a^n+a^2-1$.

Find all integers $m,n\geq 3$ such as the polynomial $a^m+a-1$ is divisible by $a^n+a^2-1$. It is clear that $m=n+k$ for some integer $k\geq 0$, we find that ...
7
votes
3answers
60 views

At least one of $|f(x)|$ and $|g(x)|$ not less than $a+1$

Let $a\in(0,1),f(x)=ax^3+(1-4a)x^2+(5a-1)x-5a+3$,$g(x)=(1-a)x^3-x^2+(2-a)x-3a-1 $. Prove that: For any real number $x$ ,at least one of $|f(x)|$ and $|g(x)|$ not less than $a+1$ since ...
3
votes
2answers
61 views

a matrix of rank $r$ satisfies a polynomial of degree $r+1$.

Let $M$ be an $n\times n$ matrix with coefficients in $\mathbb C$. Suppose $M$ has rank $r$ with $r<n$. Prove there is a polynomial $P(x)$ with degree $r+1$ and coefficients in $\mathbb C$ such ...
15
votes
1answer
214 views

Proof a Rng cannot have exactly five non-zero divisors.

Let $R$ be a Rng (a ring which does not necessarily have a $1$). We call an element $a$ regular if $xa=0$ implies $x=0$ and $ax=0$ implies $x=0$. Prove $R$ cannot have exactly five regular elements. ...
0
votes
1answer
19 views

Find the maximum sum of real part roots

Let $z_1,z_2,z_3,\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of ...
-1
votes
2answers
33 views

How many ways to arrange colors (constraints)

Ed has five identical green marbles and a large supply of identical red marbles. He arranges the green marbles and some of the red marbles in a row and finds that the number of marbles whose right ...
0
votes
0answers
55 views

Where did I go wrong in this putnam question? [duplicate]

I won't be writing each step explicitly because it takes me forever to write this up in math format. $$ \int_2^4 \frac{\sqrt{\ln{(9-x)}}}{{\sqrt{\ln{(9-x)}}}+{\sqrt{\ln{(x+3)}}}}\,dx $$ My solving ...
2
votes
4answers
67 views

Number of Interesting Quadruples

Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \le a<b<c<d \le 10$, and a+d>b+c. How many interesting ordered quadruples are there? This is a bit of trouble ...
2
votes
1answer
50 views

Prove that $4$ divides $n$ [duplicate]

Let $a_1$,$a_2$,$a_3$,.......,$a_n$ be $n$ such that each $a_i$ either $1$ or $-1$.If $a_1 a_2 a_3 a_4+a_2 a_3 a_4 a_5+......+a_n a_1 a_2 a_3=0$, then prove that $4$ divides $n$. I tried this for ...
3
votes
1answer
33 views

Prove for primes p $>2$ that $\sum_{k=1}^{p−1}{k^{2p−1}}\equiv\frac{1}{2}p(p+1)\pmod {p^2}$

Let $p$ be an odd prime. Prove that: $$\sum_{k=1}^{p−1}{k^{2p−1}}\equiv\dfrac{p(p + 1)}{2}\pmod {p^2}$$ The problem is taken from the 2004 Canada National Olympiad. I am only able to show ...
-2
votes
1answer
74 views

find minimum number of card which is needed to perform trick

some one ask you to choose number between $1$ and $n$ both inclusively then he shows you some card having one of more element and asks whether the card contains your chosen number or not after showing ...
2
votes
1answer
44 views

Help with trig identities to solve an AIME geometry question

Quadrilateral $ABCD$ has side lengths $AB = 20$, $BC = 15$, $CD = 7$, and $AD = 24$, with diagonal length $AC = 25$. If we write $\angle ACB = \alpha$ and $\angle ABD = \beta$, then $\tan (\alpha + ...
3
votes
3answers
68 views

Let $(a, b, c)$ be a Pythagorean triple. Prove that $\left(\dfrac{􀀀c}{a}+\dfrac{􀀀c}{b}\right)^2$ is greater than 8 and never an integer.

Let $(a, b, c)$ be a Pythagorean triple, i.e. a triplet of positive integers with $a^2 + b^2 = c^2$. a) Prove that $$\left(\dfrac{􀀀c}{a}+\dfrac{􀀀c}{b}\right)^2 > 8$$ b) Prove that ...
6
votes
3answers
162 views

Proving that $\sum_{i=1}^n\frac{a_i}{1-a_i}\leq\sum_{i=1}^n\frac{b_i}{1-b_i}$

Suppose $a_1,a_2,...,a_n>0$ and $\sum_{i=1}^na_i=1$. Define $b_1,b_2,...,b_n$ by $b_i=\frac{a_i^2}{\sum_{j=1}^n(a_j^2)}$. Show that ...
0
votes
2answers
50 views

Distinguishable Objects in a Circular Arrangement

I asked a question, AOPS Math Jam If you look at #9: **Please CTRL:F -> ** this: *"Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least ...
1
vote
1answer
98 views

A difficult Limit problem consisting $n^{1/n}$ in a high school level exam .

I was asked in an exam to solve the following question-- If $\lim_{n \to \infty}(a \sqrt[n]{n} + b)^{n/\ln n}$ has the value equal to $e^{-3}$ ,then find the value of $(4b+3a)$. I tried with ...
-1
votes
1answer
35 views

Prove that $A=60$, if, and only if, $BC'+CB'=BC$ [closed]

In triangle $ABC$ the bisector of angle $B$ meets the opposite side $AC$ at $B'$ Similarly, the bisector of angle $C$ meets the opposite side $AB$ at $C'$. Prove that $A=60$, if, and only if, ...