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

learn more… | top users | synonyms (2)

5
votes
1answer
65 views

Can we improve the constant $6$? $\inf_{0\le x\le 1}\sum_{j=1}^{n}\frac{1}{|x-p_{j}|}\le 6n\left(1+\frac{1}{3}+\cdots+\frac{1}{2n-1}\right)$

Some days ago, when I again read the William Lowell Putnam Mathematical Competition (1979), I found this nice problem: Let $p_{j}\in [0,1],j=1,2,\cdots,n$. Prove, that $$\inf_{0\le x\le ...
1
vote
1answer
88 views

On the solution of a olympics math problem.

The problem in question is The sequence $a_1,a_2,\ldots$ of integers satisfies the following conditions: $1\le a_j\le 2015$ for all $j\ge 1$; $k+a_k\ne\ell+a_\ell$ for all $1\le ...
1
vote
2answers
82 views

Behaviour of $\arctan(x) /x$ when $x\to0$

This is from an MCQ contest. let $f$ be the function defined on $\mathbb{R}$ by $$f(x) =\begin{cases} \dfrac{\arctan(x)}{x} & \text{ if } x \neq 0 \\ \\ 1 & x = 0 \end{cases} $$ ...
0
votes
0answers
65 views

About demonstration of sum of powers.

In the book "250 problems in Elementary Number Theory", the problem 9 asks to prove that $$ 1^3+2^3+\dots + n^3 \big\vert 3(1^5+2^5+\dots + n^5) $$ But in the demonstration (page 25 of Solutions) it ...
1
vote
1answer
83 views

Properties of $ f(x)=|x|^{1/(x-1)} $ (MCQ Contest)

Let $f$ be a function defined by: $$ f(x)=|x|^{\dfrac{1}{x-1}} $$ Then : Choose the correct option. more than one may be correct the domain of $f$ is ...
0
votes
3answers
63 views

An infinite sum in the product of sines

This is an undergrad or lower level question I need help with. Evaluate $$\quad \sum_{n=1}^{\infty} \sin{\left(\frac{a}{3^n}\right)}\sin{\left(\frac{2a}{3^n}\right)}$$ where a is just some real ...
3
votes
1answer
85 views

Algorithm to uniquely determine a number using two adjacent digits

(Russia) Arutyun and Amayak perform a magic trick as follows. A spectator writes down on a board a sequence of $N$ (decimal) digits. Amayak covers two adjacent digits by a black disc. Then ...
1
vote
0answers
28 views

Finding the speed of the faster of two machines.

This is a question from a maths contest and is as follows: Two machines move at constant speeds around a circle of circumference $600$ cm, starting together from the same point. If they travel in ...
0
votes
1answer
62 views

Show that the number or $r$ combinations of $X$ which contain no consecutive integers is given by $\binom{n-r+1}{r}$

Let $X=\{1,2,3,\dots,n\}$ where $n\in N$. Show that the number or $r$ combinations of $X$ which contain no consecutive integers is given by $\binom{n-r+1}{r}$ where $0\le r\le n-r+1$. I am unable ...
-5
votes
2answers
354 views

How many zeros are obtained if we multiply all the natural numbers from 1 to 100? [duplicate]

Options are: 20 21 22 23 24 I recently came across the above question in a competitive exam, where we get about 30 seconds to 1 minute for solving each problem. I want to if there are quick and ...
3
votes
0answers
60 views

Find all pairs of prime numbers $p , q$ for which: $p^2 | q^3 + 1$ and $q^2 | p^6 − 1$.

Find all pairs of prime numbers $p , q$ for which: $$p^2 \mid q^3 + 1 \tag{A}$$ and $$q^2 \mid p^6 − 1 \tag{B}$$ The question is from the Bulgaria National Olympiad 2014. I'm looking for ...
4
votes
5answers
147 views

Solve for $x^{11} + \frac{1}{x^{11}}$ if $x^3 + \frac{1}{x^3} = 18.$

Solve for $x^{11} + \frac{1}{x^{11}}$ if $x^3 + \frac{1}{x^3} = 18.$ I'm familiar with variants of this problem, especially those where the exponent in the given is a factor of the exponent in what ...
6
votes
1answer
117 views

Proving that $\pi(2n) < \pi(n)+\frac{2n}{\log_2(n)}$

Given that $\pi(x)$ is the prime-counting function, prove that, for $n\geq 3$, 1: $\pi(2n) < \pi(n)+\frac{2n}{\log_2(n)}$ 2:$ \pi(2^n) < \frac{2^{n+1}\log_2(n-1)}{n}$ For $x\geq8$ a real ...
0
votes
1answer
62 views

Ratio of Area of Quadrilaterals

This is a question related to the last one I asked yesterday: Ratio of Areas of Quadrilateral Again I tried to make use of the same theorem (*) but have failed. I believe this time the question is ...
6
votes
0answers
100 views

Find the least possible value of $n$ such that there exist $P(x), Q(x) \in \mathbb{Z}[x]$

Find the least possible value of $n, n \geq 2015$ such that there exists polynomial $P(x)$ with degree $n$, integer coefficients, the coefficient of the term $x^n$ is positive and polynomial $Q(x)$ ...
4
votes
2answers
195 views

Combinatorics - sending letters

In a group of 20 people, everybody sends a letter to 10 different people (not to himself). Are there always two people who send each other a letter? So I'm stuck on this neat little problem. I ...
1
vote
1answer
34 views

The minimum cardinal of a geometrical set

Let $S$ be a set of points in a plane $P$, having the following property: for any point $X \in P$ there is at least one point $M \in S$ so that the distance $|XM|$ is rational. Find the minimum ...
2
votes
1answer
54 views

Condition for a polynomial to have root of modulus 1

Prove that the polynomial$$P(X) = X^{n+1} - X^{n} - 1,\text{ }P \in \mathbb {C}[X]$$has a root $z$ with $\left|z\right|=1$ if and only if $6\,|\,(n+2)$. One implication, from left to right, is ...
3
votes
1answer
152 views

Show that there are no integers $x, y$ such that $x^{2015} - y^{2016} = 2115$

Show that there are no integers $x, y$ such that $$x^{2015} - y^{2016} = 2115$$ This is a problem in my school competition. The only thing I can think of is considering two sides of the equation in ...
2
votes
1answer
57 views

Ratio of Areas of Quadrilateral

Given a convex quadrilateral $ABCD$. We trisect $AB, BC, CD ,DA$ by the points $P_1, P_2, R_1, R_2, Q_1,Q_2,S_1,S_2$ respectively as shown below. Show that $$\frac{[KLMN]}{[ABCD]}=\frac{1}{9}.$$ ...
2
votes
1answer
55 views

Set of dominos with ends labeled with integers from 1 to 40 with all possible combinations represented. Longest proper sequence (chain) of dominos?

Consider a set of dominos in which the ends of each domino [are] labeled with distinct integers from 1 to 40 (inclusive) and all possible combinations are represented. A proper sequence of dominos ...
1
vote
2answers
21 views

Nested Sequences and Boundeness

For three sequences $(x_n), (y_n), (z_n)$ with positive starting elements $x_1, y_1, z_1$, we have the following relationship: $$ x_{n+1} = y_n + \frac{1}{z_n} \quad y_{n+1} = z_n + \frac{1}{x_n} ...
5
votes
4answers
527 views

A fair die is rolled four times. What is the probability that each of the final three rolls is at least as large as the roll preceding it?

A fair die is rolled four times. What is the probability that each of the final three rolls is at least as large as the roll preceding it? This question is from the AIME 2001. I am looking for a ...
3
votes
1answer
74 views

$f(x) \ge f(x + \sin x)$, nonconstant functions, infinite number of solutions to $f'(x) = 0$.

Let $\mathcal{F}$ be the set of all the differentiable functions $f: \mathbb{R} \to \mathbb{R}$, which have the property $f(x) \ge f(x + \sin x)$, for all $x \in \mathbb{R}$. Prove that ...
2
votes
0answers
33 views

A polynomial specialized at (1, 1, …, 1)

Consider the polynomial \begin{eqnarray} P_{l, p}(\lambda_1, \cdots, \lambda_{2n+1})=\sum_{j=0}^{2n+1}(-1)^j\sum_{\stackrel{0\leq k_r\leq ...
8
votes
1answer
111 views

Complex Numbers and their relationship with higher Mathematics

Let $z_1, z_2, \cdots, z_n$ be complex numbers satisfying $$|z_1|+|z_2|+\cdots +|z_n|=1.$$ Prove that there is a non-empty subset of $\{z_1,z_2,\cdots,z_n\}$ the sum of whose elements has modulus at ...
1
vote
1answer
33 views

Pairs of cards from 1,2,…,n arranged according to rules on pairwise separation - is it possible for various n?

A deck of $2n+1$ cards consists of a joker and, for each number between $1$ and $n$ inclusive, two cards marked with that number. The $2n + 1$ cards are placed in a row, with the joker in the ...
2
votes
0answers
65 views

An Olympiad problem about the solution of equations

There are positive real numbers $a_1, \dots ,a_{2013}$, that satisfy: equations $x_{k-1}-2x_{k}+x_{k+1}+a_{k}x_{k}=0(1 \le k \le 2013)$ have a solution $(x_0,x_1,\dots ...
4
votes
3answers
122 views

What is the largest possible value from $x_1^{2014}+x_2^{2014}$ from this following problem?

Given quadratic equation $x^2+px+q+1=0$ with two distinct roots $x_1$ and $x_2$. If $p$ and $p^2+q^2$ is prime numbers, what is the largest possible value from $x_1^{2014}+x_2^{2014}$? My ...
33
votes
0answers
1k views

Do $p,q$ exist such $|p-q|+|a_{p}-a_{q}|=2014$

Let $\{a_{1},a_{2},\cdots,a_{2016}\}=\{1,2,3,\cdots,2016\}=A$ be such $$\dfrac{a_{i}-a_{j}}{i-j}\neq 1,\forall i,j\in A\text{ with } i\neq j.$$ Show that there exists $p,q\in A$ such that ...
7
votes
2answers
121 views

How to solve the functional equation $f\left(x^2+f(y)\right)=y+f(x)^2$

How to solve the following functional equation: Find all $f:\mathbb{R}\to\mathbb{R} $ such that: $$ f\left(x^2+f(y)\right)=y+f(x)^2 $$ Holds for every $x,y\in\mathbb{R}$. A friend gave it to me, ...
3
votes
2answers
167 views

Generalization of “BMO2 2001 Question 1 Recurrence Relation”

In this question, a process is proposed. I am going to propose an extension of the process to more players, then ask the same question: $k$ players sit in a circle, with player $i$ starting with ...
0
votes
0answers
174 views

on selling 12 pens for a rupee a seller loses 12%. to earn a profit of 20% how many pens should be sold for a rupee?

On selling 12 pens for a rupee a seller loses 12%. To earn a profit of 20% how many pens should be sold for a rupee? please tell me how to solve this?
3
votes
2answers
81 views

Maximize the area of triangle $ABC$

In triangle $ABC$, $\angle A=\frac{\pi}{3}$, $D$ is a point in the plane, which $DA=DB=8,DC=6$, find the maximum area of triangle $ABC$. It seems no idea to use an Euclidean method. So I tried ...
1
vote
1answer
33 views

Choosing alternating numbers

There are 3000 cards, each labeled with an integer, 1 to 3000. Alex chooses 2000 different cards. Bob then tries to find from these 2000 cards, 1000 cards whose odd or evenness alternates when they ...
2
votes
1answer
36 views

Finite time strategy: search and destroy

A boat, which represented by a point, moves in uniform motion along the real line $\mathbb{R}$. At any moment, the boat's position and speed are not known. The only information available is : (i) its ...
7
votes
7answers
1k views

Finding x in an Olympiad simultaneous equation

I have been practicing for a an upcoming intermediate math olympiad and I came across the following question: Let $x$ and $y$ be positive integers that satisfy the equations $$\begin{cases} xy = ...
2
votes
1answer
80 views

Find the smallest value of $S=\sqrt[5]{\dfrac{abc}{b+c}} + \sqrt[5]{\dfrac{b}{c(1+ab)}} + \sqrt[5]{\dfrac{c}{b(1+ac)}}$

Let $a\ge0$ and $b,c>0$, we need to find the smallest value of the expression $S=\sqrt[5]{\dfrac{abc}{b+c}} + \sqrt[5]{\dfrac{b}{c(1+ab)}} + \sqrt[5]{\dfrac{c}{b(1+ac)}}$. I have no idea for this ...
8
votes
3answers
182 views

Computing $\lim_{A\to\infty} \frac{1}{A} \int\limits_1^A \! A^{\frac{1}{x}} \, \mathrm{d}x.$

On this year's IMC there was this problem: Compute $$ \lim_{A\to\infty} \frac{1}{A} \int\limits_1^A \! A^{\frac{1}{x}} \, \mathrm{d}x. $$ In addition to the two official solutions, I am curious as ...
4
votes
3answers
248 views

BMO2 2001 Question 1 Recurrence Relation

Ahmed and Beth have respectively $p$ and $q$ marbles, with $p > q$. Starting with Ahmed, each in turn gives to the other as many marbles as the other already possesses. It is found that after $2n$ ...
4
votes
5answers
169 views

When the $a^2+b+c+d,b^2+a+c+d,c^2+a+b+d,d^2+a+b+c$ are all perfect squares?

Find all $a,b,c,d\in \mathbb{Z}^+$, which $a^2+b+c+d,b^2+a+c+d,c^2+a+b+d,d^2+a+b+c$ are all perfect squares. I found $(1,1,1,1)$, but I can't find more. Is $a=b=c=d$ true?
1
vote
1answer
55 views

$min\{ \sqrt[4]{\frac{a}{b+c}}+\sqrt[4]{\frac{b}{a+c}}+\sqrt[4]{\frac{c}{a+b}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{a+c}{b}}+\sqrt{\frac{a+b}{c}}\}$

There is an Olympiad problem: $$a,b,c \in \mathbb{R}^+,M=\sqrt[4]{\frac{a}{b+c}}+\sqrt[4]{\frac{b}{a+c}}+\sqrt[4]{\frac{c}{a+b}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{a+c}{b}}+\sqrt{\frac{a+b}{c}},$$ find ...
4
votes
2answers
115 views

Solve $2p+6=n^3$

How can I prove that if $p$ is a prime, there is an unique solution for the equation $$2p+6=n^3$$ where $n$ is an integer? I know that $p=29$ is the solution, but I don't know how to demonstrate ...
5
votes
3answers
94 views

Solve $(x+1)^y-x^z=1$

I want to know if there is another way to solve the equation: $$(x+1)^y-x^z=1$$ for positive integers $x,y,z$ greater than $1$ without using the Catalan identity. Does exist? By now I only know that ...
11
votes
2answers
152 views

The last (and the weirdest) problem from Chen`s “Brief Introduction to Olympiad Inequalities”

Let $a$, $b$, $c$ be positive reals satisfying $a + b + c = \sqrt[\large 7]{a} + \sqrt[\large 7]{b} + \sqrt[\large 7]{c}$. Prove that $a^a b^b c^c ≥ 1$. This is the last problem from this ...
3
votes
3answers
153 views

Show that $(2^n-1)^{1/n}$ is irrational

How to show that $(2^n-1)^{1/n}$ is irrational for all integer $n\ge 2$? If $(2^n-1)^{1/n}=q\in\Bbb Q$ then $q^n=2^n-1$ which doesn't seem right, but I don't get how to prove it.
4
votes
2answers
101 views

Solve the equation $\sqrt{x+5}=5-x^2$

Solve the equation $\sqrt{x+5}=5-x^2$. I have tried to make the substitution $x=\sqrt{5}\tan^2 \theta$ and wanted to make use of the identity $\tan^2\theta+1=\sec^2\theta$ but it didn't work out. I ...
-2
votes
1answer
31 views

Probable winner of last coin game of a series, where winner from one game has disadvantage the next game?

Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner is the first person to obtain a head. They play this game several times, with the stipulation that the loser ...
0
votes
2answers
67 views

Proof if $0\leq a,b<1$ then $a+b<1+ab$

I need to prove that is $0\leq a,b<1$ then $a+b<1+ab$. What I did is see this is equivalent to $a+b-ab<1$. We have: $a+b-ab=a+(1-a)b<a+(1-a)1=1$ as desired. Is there a straightforward ...
3
votes
1answer
60 views

Optimizing number of 6-digit strings differing in at least two places

A certain province issues license plates consisting of six digits (from 0 to 9). The province requires that any two license plates differ in at least two places. (For instance, the numbers ...