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

learn more… | top users | synonyms (2)

4
votes
0answers
21 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 ...
4
votes
0answers
41 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]
0
votes
0answers
18 views

Newton-Raphson with Exponentials

I'm having trouble getting initial values for x and y to be thrown into the Newton Raphson formulae, aka Xv1 and Yv1 respectively. Question; Show that the equation: 10e^-2x = 2x + 3x^2 has a root ...
0
votes
0answers
233 views

Integral Contest [on hold]

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
24 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
37 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
30 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
317 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
14 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 $A_i$ ...
6
votes
0answers
49 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
43 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
89 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
156 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
90 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 ...
3
votes
0answers
22 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
21 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
22 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 ...
2
votes
4answers
50 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
98 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 ...
4
votes
1answer
40 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
74 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}= ...
5
votes
1answer
55 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
79 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
77 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
0answers
43 views

Count edges that can be removed

Given are N nodes and M edges, each edge connects two nodes. The edges are bidirectional , i.e., substance can flow in either direction through the edge. We start from node 1 and end up at node N. ...
-1
votes
1answer
74 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 ...
4
votes
0answers
130 views

Can I use any theorem I know at an IMO? [closed]

What if I happen to know a (fairly well-known) theorem that trivializes a given problem set at a math contest? Could my answer be rejected (unless I provide proof)? For example, see this question on ...
0
votes
1answer
46 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
49 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
90 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 ...
-4
votes
3answers
83 views

Expected number of good pair of coins [closed]

N coins are being put in a line, each of them is either facing Heads or Tail with equal probability.A pair of indices (i,j) is called good coin pair if coin at index i is facing Heads, and coin at ...
2
votes
1answer
78 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
83 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
141 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
42 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
145 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
115 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
185 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
55 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 ...
8
votes
2answers
247 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
146 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
80 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
45 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
78 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 ...
2
votes
2answers
81 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/
-1
votes
0answers
109 views

Minimum number of points that should be added to obtain a square?

Given a set of N points. How to find the minimum number of points that should be added to the given set in order to obtain at least one square whose vertices belong to the points in the given set ? ...
8
votes
1answer
192 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
457 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
145 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 ...