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

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4
votes
0answers
89 views

Integer solutions of $a^3+2a+1=2^b$

What are the solutions in integers of $a^3+2a+1=2^b$? [Source: Serbian competition problem]
0
votes
0answers
28 views

Game placing numbers in increasing order

Let $k\leq m\leq 100$ be positive integers. Aaron and Britney play a game on a $1\times m$ board, using $100$ paper pieces numbered from $1$ to $100$. The game has $k$ turns. In each turn, Aaron ...
13
votes
0answers
300 views

IMO programs of different nations?

We have a good team in the IMO, and this year I can, and probably will, be part of it. Since we as a country do not have a public training programme, I have to consult the training programms of ...
8
votes
0answers
45 views

Ratio of product from one point and minimum distance

Given points $A_0,A_1,\ldots,A_n$ in the plane, let $m$ denote the minimum distance among any two points. What is the minimum value of $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
7
votes
1answer
91 views

Dividing tournament into “equal” groups

In a tournament of $n$ teams, each team plays all other teams exactly once, with no draw. For which $n$ is it always possible to divide all teams into several groups so that each group of teams won ...
5
votes
1answer
52 views

Selecting cells so that every $2\times 2$ square is odd, then even

Jacob selects some cells from a $12\times9$ table, so that every $2\times 2$ subsquare contains an odd number of selected cells. He then selects some more cells, so that every $2\times 2$ subsquare ...
7
votes
1answer
179 views

Floor function inequality $\lfloor a\sqrt{2}\rfloor\lfloor b\sqrt{7}\rfloor <\lfloor ab\sqrt{14}\rfloor$

Let $a,b$ be positive integers. Show that $\lfloor a\sqrt{2}\rfloor\lfloor b\sqrt{7}\rfloor <\lfloor ab\sqrt{14}\rfloor$. [Source: Russian competition problem]
29
votes
4answers
2k views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
0
votes
2answers
37 views

Percentage Question from GRE

A question in GRE states: In a survey of a town,it was found that 65% of the people surveyed watched the news on television,40% read newspaper, and 25% read a newspaper and watched the news on ...
1
vote
1answer
42 views

Calculus Proof involving exponents.

Prove that $2015^{2013}<2014^{2014}<2013^{2015}$ without the use of a calculator. I don't know where to begin here. Any help or guidance on where to begin would be greatly appreciated.
-1
votes
1answer
48 views

Analytical Question for GRE

In a book prep. MCQ's in analytical portion a question says: "The chairs in the school hall can be set out in 35 equal rows or in 45 equal rows or in 105 equal rows are:" I'm unable to sort out ...
4
votes
1answer
352 views

Need help with GRE question

I encountered a question while preparing for GRE and am stuck. In an examination paper of 5 Questions, 5 percent of the candidates answered all of them and 5 percent none. Of the rest, 25% ...
2
votes
0answers
20 views

Reflection to get within convex polygon

Let $P$ be a convex polygon, and let $A_1$ be a point on the same plane as $P$. Prove that we can find an integer $n$, and points $A_2,A_3,\ldots,A_n$, such that $A_{i+1}$ is a reflection of $A_i$ ...
8
votes
1answer
84 views

Game replacing two numbers by mean

Alicia and Bart plays a game. Alicia first writes $100$ real numbers on the board. After that they move alternately; Bart goes first. In every move, the player chooses two numbers, erases them, and ...
0
votes
0answers
50 views

Finding examples before solving

So I've been solving some contest problems,and most of them require a solution in order to be solved. For example $$S_n=\left\{{n\choose n},{2n\choose n},{3n\choose n},\ldots,{n^2\choose n} \right\}$$ ...
5
votes
2answers
96 views

Integral involving inverse of $x^x$

My brother gave me the following problem: Let $f:[1;\infty)\to[1;\infty)$ be such that for $x≥1$ we have $f(x)=y$ where $y$ is the unique solution of $y^y=x$. Then calculate: $$ \int_0^e f(e^x)dx $$ ...
12
votes
2answers
184 views

Integer solutions of $x^3-x+9=5y^2$

What are the solutions in integers of $x^3-x+9=5y^2$? [Source: Hungarian competition problem]
6
votes
2answers
105 views

How to Solve : $ A =\frac{1}{6}\left((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3\right) $

$ A =\frac{1}{6}\left((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3\right).$ Solve for $2^A.$ (no calculators or graphs are permitted) The way I went about solving this problem was using ...
5
votes
1answer
57 views

Bound on number of breakable sets

Let $\mathcal{S}$ be a finite family of finite sets. A finite set $A$ is called breakable if for every $B\subseteq A$, there exists $S\in \mathcal{S}$ such that $A\cap S=B$. Show that at least ...
1
vote
2answers
30 views

Forming Random Team and Choosing Pair of Friends

n participants of the competition were split into m teams in some manner so that each team has at least one participant. After the competition each pair of participants from the same team became ...
0
votes
1answer
45 views

A problem regarding table decorations

My one friend Alex has r red, g green and b blue balloons. To decorate a single table for the banquet he needs exactly three balloons. Three balloons attached to some table shouldn't have the same ...
1
vote
4answers
107 views

Closed Form for Factorial Sum

I came across this question in some extracurricular problem sets my professor gave me: what is the closed form notation for the following sum: $$S_n = 1\cdot1!+2\cdot2!+ ...+n \cdot n!$$ I tried ...
6
votes
4answers
113 views

If $\sum_{n=1}^\infty \frac{1}{a_n}$ converges, must $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ converge?

Suppose $\sum_{n=1}^\infty \frac{1}{a_n} = A$ is summable, with $a_n > 0,$ $n = 1,2,3,\cdots.$ How can we prove that $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ is also summable? This question ...
0
votes
1answer
45 views

Some Strange Minimum and proof

I read following on Norm chapter in one book. $$\begin{align}\left|\left\|x-y\right\|-\left\|w-z\right\|\right| \leq & \min \{\|x-w\| + \|y-z\| , \|x-z\| + \|y-w\|\}\\\text{ or, }&\min ...
2
votes
1answer
76 views

On the equality $\sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{a_k}= \sqrt[n]{b_1}+\sqrt[n]{b_2}+\cdots+\sqrt[n]{b_m}$

Let $k,m\in \mathbb{N}$. Let $a_1,a_2,\cdots,a_k\ >0$ and $b_1,b_2,\cdots,b_m \ >0$ such that $$\sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{a_k}= ...
6
votes
1answer
73 views

Functional Equation $f(mn)=f(m)f(n)$.

If $f: \mathbb N \mapsto \mathbb N$ is one-to-one and $f(mn) = f(m)f(n)$, what is the smallest possible value of $f(999)$? Easily $f(1)=1$, and I think $f(n)=n$ must be the only map, but not able to ...
6
votes
2answers
112 views

Eating chocolate game on grid

Given is a chocolate of size $m\times n$. Anne and Birgitte plays a game, with Anne starting. In each turn, the player has to divide the chocolate into two rectangular parts along the lines, and eat ...
1
vote
1answer
78 views

For which values of $a$ is the solution for $x^2 - y^2 = a^3$ unique?

For which values of $a$ is the solution unique? $$x^2-y^2=a^3$$ I'm not sure how to do this, so I've been looking at this guy's solution. $x^2 - y^2 = a^3$ is factored into $(x-y)(x+y) = a^3$. ...
-1
votes
1answer
78 views

Check if $N$ is of form $6A + 8B$

Given a number $N$ we need to check if its of form $6A + 8B$ .If its of this form then we need to check if $B$ can be greater than equal to $1$ or not. Like $24$ is of form $6A + 8B$. Also $B$ can ...
0
votes
1answer
60 views

Expected value for Head/Tails

There are $N$ coins placed in a line. A coin may be facing head/tail direction with $0.5$ probability. Now I need to find number of pairs of coins $(i,j)$ such that $i<j$ and on index $i$ , I ...
1
vote
2answers
56 views

Prove that if $abc\ne0$ and $ab+bc+ac=0$ then $a+b+c\ne0$

I tried to do proof by contradiction, but problem is how to get from $ab+bc+ac$ to $a+b+c$ Assuming $a+b+c=0$ my approachs: Adding $ab+ac+bc=0$ and $a+b+c=0$ and try to factor Deriving ...
-1
votes
2answers
158 views

Check if we can turn a string into a palindrome by reversing a substring

Given a string consisting of lower-case characters from English alphabets, we want to reverse a substring from the string such that the string becomes a palindrome. Note : A Palindrome is a string ...
2
votes
1answer
83 views

Find different sequences of game to find winner

Alice and Bob are having a racing competition to see who is the best runner. They don't want to decide this in a single race, so they choose a number N which is the minimum number of points one of ...
2
votes
1answer
115 views

Some challenging Series, maximum value and polynomial factor questions

So I realize that the questions I am gonna ask are going to be a minute's work for some of you but I couldn't do them even after hours of searching for methods or something. They are from a ...
1
vote
1answer
166 views

Count ways to distribute candies

N students sit in a line, and each of them must be given at least one candy. Teacher wants to distribute the candies in such a way that the product of the number of candies any two adjacent students ...
0
votes
2answers
46 views

Adding Combinations - Math Contest

I am studying for a math test, and I'm wondering on an easier way to add combination series. For example, $12 \choose 3$ + $12 \choose 4$ + ... + $12 \choose 8$. Is there an easier way than: $2^{12}$ ...
6
votes
1answer
148 views

permutation and f(n) challenge

Suppose $f(n)$ be the number of permutation from set ${1,2,..,n}$ such that for each $ 1 \leq i \leq n$ we have: $ | \pi(i)-i| \leq 1 $. meaning of $ \pi(i)$ is an elements whose in place $i$ of ...
4
votes
2answers
139 views

I have used Cauchy and Jensen. It is not helping me very much. Advice on solving this problem.

Let $a$, $b$ and $c$ be positive real numbers with $abc=1$. Prove that $$ \frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $$ for each ...
-4
votes
1answer
211 views

Count good numbers in between L and R

Let length(A) denote the count of digits of a number A in its decimal representation. All non-negative numbers of length 1 are Good. Further, a number X with length(X) $≥ 1$ can also be considered ...
0
votes
1answer
58 views

Any thoughts on how to solve this problem? [closed]

How many numbers do there exist having 2013 digits, in which every two-digit number composed of two consecutive digits is a multiple of either 17 or 23? (Taken from Singapore and Asian Schools Math ...
2
votes
2answers
286 views

Interpolation polynomial Challenge

suppose $p(x)=x^k-x^t, k \neq t $ (k,t is a positive integer). function q(x) be a Interpolation polynomial from degree lower or equal n, to data $i=1,...,n+1, (x_i ,p(x_i))$. if ----------- then ...
4
votes
1answer
161 views

Polynomials with coefficients $1$ or $2014$ [closed]

Let $$ P(x)=a_mx^m+a_{m-1}x^{m-1}+ \cdots+a_1x+a_0$$ and $$\quad Q(x) =b_nx^n+b_{n-1}x^{n-1}+ \cdots+b_1x+b_0 $$ be two polynomials where $a_i,b_j \in \{1,2014\}$ for all $i,j$. Suppose that ...
3
votes
1answer
89 views

Functions for non-negative integers [duplicate]

Let $\Bbb{Z}^+$be the set of all non-negative integers where $n$ and $k$ are given natural numbers. We consider the following non-decreasing function, $$f:\Bbb{Z}^+ \to \Bbb{Z}^+$$ such that ...
1
vote
1answer
54 views

Remainder of a summation divided by $2^{12}$

For a positive integer $n$, let $f(n)$ be equal to $n$ if there is an integer $x$ such that $x^2-n$ is divisible by $2^{12}$, and let $f(n)$ be $0$ otherwise. Determine the remainder when ...
0
votes
0answers
13 views

Integer multiplication in 5T(n/3) [duplicate]

x and y has n bits x=x0*(10^2n/3)+x1*10^n/3+x2 y=y0*(10^2n/3)+y1*10^n/3+y2 x*y=x2y2+(x2y1+x1y2)10^n/3+(x2y0+x1y1+x0y2)10^2n/3+(x1y0+x0y1)10^n+x0y0*10^4n/3 now 9 multiplication of n/3 bit numbers ...
2
votes
3answers
95 views

Given $x+y+z=3$ prove that $4 \geq x^2y+y^2z+z^2x$

Given positive reals $x,y,z$, and $x+y+z=3$ prove that $$4 \geq x^2y+y^2z+z^2x$$. This problem was homogenized so I set $x+y+z=3$ to cancel stuff. Now I'm stuck. I have noticed a funny equality ...
1
vote
2answers
106 views

Mind Teasers : Difficult Brain Twister (Today Challenge)

Question can be found in the link below Source: http://gpuzzles.com/mind-teasers/difficult-brain-twister/
8
votes
1answer
203 views

How to prove that $\frac{1}{x_1}+\frac{1}{x_2}+…+\frac{1}{x_n}-\frac{1}{x_1x_2…x_n}\in \mathbb{N}\cup \{0\}$

Question: Show that for every natural number $n$ there exist $n$ natural numbers $ x_1 < x_2 < ... < x_n ,$ such that $$ ...
11
votes
1answer
563 views

A Putnam Integral $\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)} + \sqrt{\ln(x+3)}}.$

This is a Putnam Problem that I have been trying to solve (on and off) for two years, but I have failed. I am in Calculus BC. This problem comes from the book "Calculus Eighth Edition by Larson, ...
2
votes
3answers
155 views

How many strings of $8$ Hs and $8$ Ts are there such that there are at most $2$ consecutive Hs?

How many strings of 8 Hs and 8 Ts are there such that there are at most 2 consecutive Hs? I don't really understand how to approach this question. What would be the quickest way to solve it? Thanks ...