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

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3
votes
2answers
146 views

Find the number of series

Find the number of series $(a_1,..., a_{2n})$ that have terms from ${\{0,...9\}}$ so that: $$ 11|\sum_{i=1}^{n}a_i-\sum_{i=n+1}^{2n}a_i $$ (this is not a homework) There is a similar problem ...
1
vote
2answers
143 views

Analysis problem from Romanian Contest - 2 sequences which forms another one

Let $a,b$ be 2 real numbers, and the sequences $(a_n)_{n \geq 1}, (b_n)_{n \geq 1}$ defined by $a_{1}=a$, $b_{1}=b$, $a^2+b^2 <1$ and \begin{cases} ...
2
votes
1answer
87 views

Distance between two points in the plane

my teacher asked in the class today the following question: There exists an infinite set M of points in the plane with the property that any three points are non-collinear and such that the distance ...
1
vote
2answers
64 views

Chinese remainder theorem?

In the 2014 AIME 1, number 8 says: The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit ...
6
votes
2answers
289 views

A generalization of IMO 1977 problem 2

Here is the IMO 1977 problem 2: In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the ...
0
votes
1answer
122 views

Competition math geometry question

The perimeter of triangle ABC is $36$, and its area is $36$. Compute $\tan\frac{A}2 \tan\frac{B}2 \tan\frac{C}2$. I found that the answer is $1/9$, but I was not able to find a reason for this. Could ...
13
votes
4answers
458 views

prove Diophantine equation has no solution $\prod_{i=1}^{2014}(x+i)=\prod_{i=1}^{4028}(y+i)$

show that this equation $$(x+1)(x+2)(x+3)\cdots(x+2014)=(y+1)(y+2)(y+3)\cdots(y+4028)$$ have no positive integer solution. This problem is china TST (2014),I remember a famous result? maybe ...
2
votes
0answers
65 views

An algorithm for solving linear diophantine equations?

I am entering an interesting team based math contest called the purple comet, and quite a lot of questions on this contest involve Diophantine equations. For this contest, you are given a computer, ...
2
votes
2answers
397 views

Putnam 2001 - Problem A-1 (On a 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
131 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
66 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
87 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
46 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
200 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
96 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
46 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
157 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} ...
11
votes
3answers
400 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
104 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
123 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
118 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
77 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
259 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
167 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
60 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
58 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 ...
1
vote
1answer
52 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
254 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
52 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 ...
15
votes
2answers
511 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
33 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 ...
8
votes
2answers
265 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
43 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
184 views

AMC 12 word problem modified to be considerably harder

The original problem is stated as follows: ...
-1
votes
1answer
68 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
74 views

Proving using squeeze principle

This problem sounds very confusing. Please help me solve this problem.
1
vote
1answer
59 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
24 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
220 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
61 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
1answer
401 views

Even or Odd for factorial

Moderator Note: This was a 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 question? ...
0
votes
1answer
151 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 $u=\sin(x)$ ...
0
votes
1answer
61 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 ...
7
votes
2answers
206 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 ...
9
votes
4answers
412 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
25 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 $0.3$. ...
2
votes
1answer
82 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
80 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 ...
6
votes
4answers
529 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
58 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 ...