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

learn more… | top users | synonyms (2)

2
votes
2answers
295 views

Putnam $\bf 2001$ problem A$\bf 1$ (Binary operation)

Let $*$ be a binary operation acting on a non-empty set $S$ such that $(a*b)*a=b$ for all $a,b\in S$. Prove that $a*(b*a)=b$ for all $a,b \in S$.
4
votes
2answers
74 views

Finding the value of $(bc-ad)(ac-bd)(ab-cd)$

Let $a,b,c,d$ be $4$ distinct non-zero integers such that $a+b+c+d = 0$. It is know that the number $$M = (bc - ad)(ac - bd)(ab-cd)$$ lies strictly between $96100$ and $98000$. Determine the value ...
2
votes
1answer
59 views

Solve without convergance?

Two days ago I recalled a problem I was given a long time ago. The problem is: Four ants are placed on the vertices of a square with side 1. The ants start moving, each directed towards its left ...
2
votes
1answer
67 views

Putnam inspired problem

The following is a beautiful problem from Putnam 2003 minimize $|\sin x + \cos x + \tan x + \csc x + \sec x + \cot x|$ I was thinking about a small variation of the above problem minimize $|\sin ...
1
vote
0answers
42 views

Prove that for every prime $p > 100$ and every integer $r$ $\exists a, b$ such that $p \mid a^2 + b^5 - r$

Prove that for every prime $p > 100$ and every integer $r$ $\exists a, b \in \mathbb{Z}$ such that $p \mid a^2 + b^5 - r$ Preferably without using Jacobi sums, I have already seen a solution using ...
5
votes
1answer
177 views

Limits of the solutions to $x\sin x = 1$

Let $x_n$ be the sequence of increasing solutions to $x\sin{x} = 1$. Define $$a = \lim_{n \to \infty} n(x_{2n+1} - 2\pi n) $$ and $$b = \lim_{n \to \infty} n^3 \left( x_{2n+1} - 2\pi n - \frac{a}{n} ...
4
votes
3answers
84 views

Factoring $x^5 + x^4 + x^3 + x^2 + x + 1$ without using $\frac{x^n - 1}{x-1}$?

I was at a math team meet today and one of the problems was to factor $x^5 + x^4 + x^3 + x^2 + x + 1$. It also gave the hint that it decomposes into two trinomials and a binomial. The solution they ...
2
votes
1answer
34 views

Prove $f$ not continuous at SEEMOUS Contest

Let $n$ be a nonzero natural number and $f:\mathbb{R}\to\mathbb{R}\setminus\{0\}$ be a function such that $f(2014) = 1 − f(2013)$. Let $x_1,x_2,x_3,...,x_n$ be real numbers not equal to each other. ...
10
votes
1answer
107 views

An IMO inspired problem

This problem from IMO 1988 is said to be one of the most elegant ones in functional equations. Problem : The function $f$ is defined on the set of all positive integers as follows: \begin{align} ...
6
votes
2answers
163 views

Integral $ \int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx$

Hello there I am trying to calculate $$ \int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx $$ NOT using mathematica, matlab, etc. We are given that $\sigma, \omega$ are complex. Note, the ...
2
votes
0answers
81 views

quadrilateral geometry question

I recently took the AIME, and the following question was one I was not able to answer: On square $ABCD$, points $E,F,G,$ and $H$ lie on sides $\overline{AB}$,$\overline{BC}$,$\overline{CD}$, and ...
2
votes
2answers
83 views

$f(x)=\sin(43x)+\cos(2x)$ is periodic function?

$f(x)=\sin(43x)+\cos(2x)$ is periodic function. I got the period of $\sin(43x)$ is $\frac{2\pi/}{43}$ and period of $\cos(2x)$ is $\pi$. Then the period of $f(x)$ is $2\pi$. Am I right? Any comment? ...
10
votes
1answer
103 views

Question concerning finite intersecting sets

Let $\{X_i\}_{i=1}^{\infty}$, $\{Y_j\}_{j=1}^{\infty}$ be finite sets of cardinality at most $n$. If for any finite $F$, there are $i,j \in \mathbb{N}$ such that $F \cap X_i \cap Y_j = \emptyset$, ...
2
votes
4answers
65 views

determining the amount of total questions needed in a game given the probabilty

I'm creating a game and can't seem to quite figure this out - driving me crazy. There are 8 questions in my game You can play the game an unlimited amount of times the test bank doesn't change. so ...
0
votes
3answers
186 views

Sets of integers in the form $a^2 + 4ab + b^2$

Let $A$ be the set of all integers of the form $ a^2 + 4ab + b^2$ where $a, b$ are integers if $x, y$ are in $A$, prove that $xy$ is in $A$ (I have tried opening everything, it gets nowhere) Show ...
3
votes
2answers
124 views

The square of a number's last few digits remain the same.

The number $9376$ has a property that the last four digits of $9376^2$ remain the same. How many $4$ digit numbers have this property? Are there values of $n>4$ such that a $n$-digit number has ...
6
votes
2answers
51 views

Given $p(x)$ is a polynomial with integer coefficients and that $p(a)=1$ for some integer $a$ prove that $p(x)$ has no more than two integral roots. [duplicate]

Given $p(x)$ is a polynomial with integer coefficients and that $p(a)=1$ for some integer $a$ prove that $p(x)$ has no more than two integral roots. I've attempted a proof by contradiction assuming ...
1
vote
2answers
28 views

Connecting square vertexes with minimal road

I have four cities in $A=(0,0),B=(1,0),C=(1,1),D=(0,1)$. I am asked to build the shortest motorway to connect the cities. How can I do that? I was thinking that first I need some compactness argument ...
0
votes
0answers
140 views

Query Preprocessing

Moderator Note: This is a current contest question on codechef.com. I have array of x integers and i need to answer y queries. Each query have 3 integers ( Number, Left index, Right Index). I ...
1
vote
1answer
38 views

chain of divisibility relation

Let $a$ and $b$ be positive integers such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5, \cdots $ Prove that $a = b$. My way is as follows: Let $A=v_p(a), B=v_p(b)$ be the exact power of a prime $p$ ...
4
votes
1answer
178 views

Inequality in triangle involving side lenghs, medians and area

A, B and C are the vertices of a triangle. Denote $m_a$, $m_b$ and $m_c$ the medians from A, B and C. Prove the inequality: $$\sum_{cyc}{a^2bcm_a}\geq\sum_{cyc}{cS(a^2+b^2)}$$where a, b and c are the ...
0
votes
0answers
45 views

Solving the recurrence $F(0) = X$ and $F(i)=A \cdot (F(i-1))^2 + B \cdot F(i-1) + C$

Moderator Note: This is a current contest question on codechef.com. I am given $F(0)=X$ and $F(i)=A \cdot (F(i-1))^2 + B \cdot F(i-1) + C$ for $1 \leq i \leq N$. Now given $N,A,B,C$ and $X$, how ...
8
votes
2answers
220 views

Integral…$\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$.

I am trying to calculate $$ I=\frac{1}{\pi}\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta=\frac{11\pi^4}{180}=\frac{11\zeta(4)}{2}. $$ Note, we can expand the log in the integral to ...
1
vote
1answer
26 views

Number of moves to switch all tiles from black to red?

Four tiles are arranged as per the diagram and all start off black. On each move, two connected tiles may be interchanged, and upon doing so each of the two tiles switches color from red to black ...
1
vote
0answers
44 views

Computing the last non-zero digit of ${1027 \choose 41}$?

I am working on the following problem: Let $x_n$ be a sequence of positive odd numbers. If $N$ is the number of ordered pairs $(x_1, x_2, x_3, \dots, x_{42})$ such that $$x_1 + x_2 + x_3 + \dots + ...
1
vote
1answer
34 views

Number theory problem - powers

Find the smallest prime $p$ such that for any $1 \leq k \leq 10$ relatively prime to $p$, one of $k, k^2,\ldots k^{p - 2}$ is congruent to $1$ modulo $p$. I am honestly not sure how to approach this ...
3
votes
2answers
132 views

AMC 12 word problem modified to be considerably harder

The original problem is stated as follows: ...
-1
votes
1answer
37 views

Three Bags Marble Probability

I found this problem in a maths test, and although I am sure there is a method to solve it, I don't know how. I have three bags. Two bags have identical contents- 1 black marble and 2 white ones. The ...
0
votes
1answer
60 views

Proving using squeeze principle

This problem sounds very confusing. Please help me solve this problem.
1
vote
1answer
57 views

Prove the derivative

Let $f(x) = (x^2-1)^{\frac{1}{2}}, x>1$. How do I prove that the $n$th derivative of $f(x) > 0$ for odd $n$, and the $n$th derivative of $f(x) < 0$ for even $n$?
1
vote
0answers
20 views

Convex quadrilateral

In a convex quadrilateral (the two diagonals are interior to the quadrilateral) prove that the sum lengths of the diagonals is less than the perimeter but great than one-half the perimeter.
3
votes
2answers
182 views

Integral, Definite Integral $ \int_{-\infty}^\infty \exp{\big(\alpha x^4+\beta x^3+\gamma x^2 +\delta x+\epsilon}\big)dx, \ \alpha <0. $

Calculate the integral $$ I=\int_{-\infty}^\infty \exp{\big(\alpha x^4+\beta x^3+\gamma x^2 +\delta x+\epsilon}\big)dx, \ \alpha <0. $$ The answer can be expressed analytically in terms of a ...
0
votes
1answer
52 views

Trigonometric eq.

The equation $3\sin(x)+4\cos(x)=5$ is well-known. The equation $3\sin^m(x)+4\cos^n(x)=5$ where $m$ and $n$ are non-negative integers is much more interesting.. I would like to see a nice, elementary ...
-2
votes
2answers
98 views

Even or Odd for factorial

Moderator Note: This is a current contest question on codechef.com. Given $N$ and $M$ I need to tell whether $\left\lfloor \large\frac{N!}{M} \right\rfloor$ is even or odd.How to do this ...
0
votes
1answer
76 views

Integral $\int_0^{\pi/2} x\cot(x)dx$, Differntiation wrt parameter only.

Integrate using differentiation wrt parameter only. $$\int_0^{\pi/2} x\cot(x)dx$$ We can express this as $$\int_0^{\pi/2} x\cdot\frac{\cos(x)}{\sin(x)}dx$$ Notice we can write $u=\sin(x)$ ...
-1
votes
1answer
47 views

different wrt parameter $I=\int_0^\infty \frac{1}{(x^2+p)^{n+1}}dx$

Integrate using differentiation with respect to parameter only: $$ I=\int_0^\infty \frac{1}{(x^2+p)^{n+1}}dx, \ n\geq 0, \ p\geq1 $$ No complex methods allowed. This is a rather useful integral to ...
5
votes
1answer
146 views

Computing the integral $ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi. $

Integrate $$ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2) \, d\phi. $$ Something that may help $(1-2x\cos\phi+x^2)=(1-xe^{i\phi})(1-xe^{-i\phi})$. And using the series ...
0
votes
2answers
81 views

Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve $$ \int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx $$ No complex variables, only this approach. Interesting ...
1
vote
1answer
18 views

Probability that one buys bread Exactly Three times in the next Five minutes

The problem states that a typical customer buys the bread $60\%$ of the time and fruit $50\%$ of the time on each visit. Also the probability that the customers buy both bread and fruit is $0.3$. ...
1
vote
1answer
38 views

finding the Prime numbers easily

I was doing some of the previous math contests and faced a question that asked me "the number of two digit primes that are still primes when the digits are reversed". I actually wrote down every two ...
1
vote
1answer
68 views

definiteinteggral

The integral is given by $$\int_0^1 \frac{\ln (1-x)\ln x}{1+x} dx = \frac{1}{8}\big(-\pi^2\ln(4) +13\zeta(3)\big).$$ Any ideas how to prove? We cannot solve the integral so easily because we cannot ...
5
votes
4answers
119 views

Some Questions regarding preparing for Math Olympiads (searched but didn't get answers)

Many questions have been asked on this site regarding preparation for olympiads like the Putnam. I've read those questions and accordingly decided to start with Engel's "Problem Solving" but I have a ...
1
vote
1answer
41 views

Average Train Speed

I'm repeating this question since they don't seem to like it over there: http://stackoverflow.com/questions/21972403/average-train-speed This is the question I have: If a train is traveling at 50 ...
0
votes
3answers
72 views

Finding the Rate of distance between hands of clock

First, I think I don't understand the problem which asks about the greatest rate of change in distance between the tips of the hands of clocks. Does it mean where the increasing of distance is the ...
2
votes
1answer
37 views

Relabelling players in a tournament

BdMO 2014 $n$ players take part in a chess tournament where each player plays with all others only once and the only outcomes of the games are win and loss.Prove that it is possible,after the ...
2
votes
1answer
140 views

2012 USAJMO Problem 5

For distinct positive integers $a, b < 2012$, define $f(a, b)$ to be the number of integers $k$ with $1 \le k<2012$ such that the remainder when $ak$ divided by 2012 is greater than that of $bk$ ...
0
votes
1answer
116 views

Do degenerate triangles count? (2014 AMC 12B #12)

The problem is this: A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can ...
2
votes
3answers
244 views

2014 AMC 12 B problem 25

What is the sum of all positive real solutions $x$ to the following equation? $$2\cos(2x)\left( \cos(2x) - \cos{\left(\frac{2014\pi^2}{x^2}\right)} \right) = \cos(4x) - 1 $$
1
vote
2answers
90 views

A tricky question from the AMC test (American Mathematics competitions)

A man walks into a store with just enough money to buy exactly 30 balloons, he then he discovers the store has a buy 1 get, one 1/3 off, sale. (a rather ridiculous sale if I do say so myself) how many ...
2
votes
1answer
41 views

Functional equation of non-negative function

Find all $ f:[0,\infty)\rightarrow [0,\infty) $ such that $ f (2)=0 $, $ f (x)\not= 0 $ for $ x\in [0, 2) $ and $$ f (xf (y)) f (y)=f (x+y) $$ for all $ x, y\ge 0 $. I tried plugging in values ...