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

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0
votes
1answer
21 views

Help with distance question points A and B

Ok. I had no idea how to do the question but I tried fiddling with the triangles to see if I can get any value but only managed to get $MN$. I read the solution to this question, and it said that I ...
1
vote
1answer
35 views

Explain the following part of the exercise…

I've been looking at this IMO 2006 contest exercise and found a thing I cannot understand: The exercise is: Now what I do not understand is , why ??? Can you explain me this? Thank you in ...
12
votes
5answers
404 views

Putnam Exam Integral

I am trying to evaluate$$ \lim_{n\to \infty} \int_0^1 \int_0^1...\int_0^1 \cos^2\big(\frac{\pi}{2n}(x_1+x_2+...x_n)\big)dx_1 dx_2...dx_n. $$ This is from an old Putnam mathematics competition. Either ...
6
votes
2answers
129 views

Integrating $ \int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx. $

Compute $$ \int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx. $$ I am not sure how to start this one...I am thinking of a substitution to get started.
4
votes
1answer
94 views

determine all polynomials $P(x)$ such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial. clearly we have to show $(x+1)P(x-1)-(x-1)P(x)=c$ for all values of $x$ ($c$ is a ...
3
votes
2answers
144 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
141 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
68 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
56 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
273 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
73 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
429 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
55 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, ...
3
votes
2answers
340 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
103 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
63 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
78 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
45 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
193 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
89 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
35 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
127 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
239 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
91 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
111 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
109 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
68 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
216 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
147 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
55 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
49 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
43 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
215 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
47 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 ...
11
votes
2answers
385 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
31 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
239 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
38 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
152 views

AMC 12 word problem modified to be considerably harder

The original problem is stated as follows: ...
-1
votes
1answer
48 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
65 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
22 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
205 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
59 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
155 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
114 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
54 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 ...
6
votes
2answers
185 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 ...
6
votes
3answers
235 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 ...