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

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3
votes
0answers
21 views

Maximize $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$

Given three real numbers $x+y+z = 3$, find the maximum value of $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$. (Source : Singapore Math Olympiad 2012, Senior section, Round 1, question 29). I ...
0
votes
0answers
14 views

Euclidean Geometry problem: prove that $C'$ is the midpoint of $A'B'$.

The tangents to a circumference centered at $O$, passing through an exterior point $C$, meet the circumference at the points $A$ and $B$. Let $S$ be an arbitrary point on the circumference. The ...
1
vote
0answers
23 views

Problem solution by model theory

Sorry if that's not the right place for asking this, but didn't have anywhere else to go. I was cheking out some math problems in the Mathematical Olympiad site, and I found this one: Let $Q_{>0}$ ...
12
votes
5answers
306 views

Integral $\lim_{n\to \infty} \int_0^1 \int_0^1…\int_0^1 \cos^2\big(\frac{\pi}{2n}(x_1+x_2+…x_n)\big)dx_1 dx_2…dx_n$

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 ...
4
votes
1answer
38 views

Lines covering points on napkin

Suppose we place a $100\times 100$ napkin on an infinite lattice plane. What is the minimum number of lines that can always cover all the lattice points lying inside or on the border of the napkin, no ...
1
vote
0answers
22 views

Almost perfect numbers

A positive integer $n$ is called almost perfect if the sum of its divisors smaller than $n$ is $n-1$. What are all almost perfect numbers $n$ such that some power $n^k$ is also almost perfect for at ...
5
votes
2answers
85 views

A different type binomial expansion problem

Suppose we have $$(1+x+x^2)^n = a_0 + a_1 x + a_2 x^2 + \cdots + a_{2n} x^{2n}.$$ What will be the value of $a_0^2 - a_1^2 + a_2^2 - \cdots + a_{2n}^2$? The answer is $a_n$, but I can't solve it. ...
1
vote
2answers
46 views

Let $n$ be a positive integer. Show that if $2^n -1$ is a prime number, then $n$ is a prime number.

Let $n$ be a positive integer. Show that if $2^n -1$ is a prime number, then $n$ is a prime number. This is how I started to tackle this question: Assume that instead of $n$ being a prime number, it ...
7
votes
2answers
186 views

How prove this inequality generalized from 1969 IMO problem 6

Let $x_{1},x_{2},y_{1},y_{2},z_{1},z_{2},w_{1},w_{2} $ are all positive numbers, and such $$x_{1}y_{1}z_{1}-w^3_{1}>0,\; \text{ and }\;x_{2}y_{2}z_{2}-w^3_{2}>0.$$ show that ...
8
votes
4answers
108 views

If $a,b,c,d>0$ and $a+b+c+d=4$, then $\frac{1}{11+a^2}+\frac{1}{11+b^2}+\frac{1}{11+c^2}+\frac{1}{11+d^2} \leq \frac {1}{3}$

Prove if $a,b,c,d>0$ and $a+b+c+d=4$, then $$\dfrac{1}{11+a^2}+\dfrac{1}{11+b^2}+\dfrac{1}{11+c^2}+\dfrac{1}{11+d^2} \leq \dfrac {1}{3}$$ This was an Inequality Olympiad Problem1. I proved by ...
1
vote
2answers
100 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 ...
2
votes
2answers
93 views

Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$ $$10^{-3}x^{\log_{10}x} + x(\log_{10}^2x - 2\log_{10} x) = x^2 + 3x$$ To be fair I wasn't able to make any progress. I tried using substitution for the ...
6
votes
2answers
163 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 ...
10
votes
2answers
360 views

Another math contest problem: $\int_0^{\frac{\ln^22}4}\,\frac{\arccos\frac{\exp\sqrt x}{\sqrt2}}{1-\exp\sqrt{4\,x}}dx$

Prove: $$ {\Large\int_{0}^{\ln^{2}\left(2\right) \over4}}\, \frac{\arccos\left(\vphantom{\huge A} {\exp\left(\vphantom{\large A}\sqrt{x\,}\right) \over \sqrt{\vphantom{\large A}2\,}}\right)} ...
11
votes
1answer
125 views

Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''\,dx=\int_0^ty'y''\,dx$.

Let $y(x)$ be a solution to $y''+e^xy=0$. Prove that there exists $t$ such that $0\le t\le T$ and $$\int_0^Te^{-x} y'y'' \, dx=\int_0^ty'y''\,dx.$$
2
votes
1answer
50 views

Finding all primes $(p,q)$ for perfect squares.

Find all prime pairs $(p,q)$ such that $2p-1, 2q-1, 2pq-1$ are all perfect squares. Source: St.Petersburg Olympiad 2011 I have only found the pair $(5,5)$ so I am thinking that maybe a modulo $5$ ...
3
votes
3answers
99 views

How to solve $\ln x+x=1$

How can I solve this equation: $$\ln x+x=1$$ We had it on a local Olympiad math contest problem.
1
vote
1answer
37 views

question from russian math olympiad

Let $f(x,y)=\frac{1}{2}(x+y-1)(x+y-2)$ be a function of two positive integers. Prove that for any positive integer $z$ there exists a single pair $x,y$ such that $f(x,y)=z$. Isn't this clearly ...
7
votes
1answer
283 views

IMO 1979 problem

The question is $$\text{If }\, p, \ q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad \text{Prove that } 1979\mid p.$$ So my solution went like ...
7
votes
1answer
353 views

Putnam PigeonHole

This is from page 12 of Putnam and Beyond. Problem: Prove that for every set $X ={x_1,x_2, \ldots ,x_n}$ of $n$ real numbers, there exists a nonempty subset $S$ of $X$ and an integer $m$ such that ...
1
vote
4answers
74 views

Let $p(x) = 1+a_1x+a_2x^2+\cdots+a_nx^n$ be a polynomial where $a_1,\ldots,a_n$ are integers, and $a_1 + … + a_n$ is even.

Let $p(x) = 1+a_1x+a_2x^2+\cdots+a_nx^n$ be a polynomial where $a_1,\ldots,a_n$ are integers, and $a_1 + \cdots + a_n$ is even. Prove that there is no integer x such that $p(x) =0$ I have started ...
1
vote
4answers
56 views

A function $f$ has the property that $f(x+y)=f(x)+f(y)+3xy$. If $f(1)=2$, what is $f(8)$?

A function $f$ has the property that $f(x+y)=f(x)+f(y)+3xy$. If f(1)=2, what is f(8)? I would like to try to tackle this problem but I need somewhere to start as I really have no idea at all on how ...
1
vote
3answers
57 views

If $x^2+bx+a=0$ and $x^2+ax+b=0$ have a common root $c$, Then what values of $(a,b)$ would work?

Let $a$ and $b$ be distinct integers. If $x^2+bx+a=0$ and $x^2+ax+b=0$ have a common root $c$, Then which of the following statements are true? 1) $c*(a+c)=-b$ 2) $a+b=-1$ 3) $a+b+c=0$ 4) $c=0$ ...
7
votes
1answer
128 views

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $a_{n+2} = a_{n+1} - a_na_{n+1}/2$ for $n$ a positive integer.

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $$a_{n+2} = a_{n+1} - a_na_{n+1}/2$$ for $n$ a positive integer. Find $$\lim_{n\to\infty}na_n$$ if it exists. Well, we can deduce that $\lim a_n=0$ by ...
7
votes
0answers
92 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
-2
votes
0answers
25 views

Convergence of series (putnam training) [closed]

Does the series $\sum_{n=1}^{\infty} \frac {|\sin n|}{n} $ converge?
2
votes
2answers
37 views

Maximum of a function from integers to integers

Suppose $f$ is a function form positive integers to positive integers satisfying $f(1)=1$, $f(2n)=f(n)$, and $f(2n+1)=f(2n)+1$ for all positive integers $n$. Job: Find the maximum of $f(n)$ when $n$ ...
10
votes
0answers
110 views

Curious number theory problem

$k,m,n\in\mathbb{N}$ satisfy $k^{m+n}=nm^n$. How can I show that $m=k$ and $n=k^k?$
3
votes
1answer
45 views

Knot Theory: Mutations

Show that if we have three tangles as in Figure 2.33a, we can mutate several times in order to permute the tangles. Note that we can then permute n tangles in a row. This is from Colin Adams; The ...
0
votes
1answer
19 views

1995 MathCounts State Team #8

In professional football, it is possible to score 2 points (for a safety), 3 points (for a field goal), or 6 points (for a touchdown). If a touchdown is scored, it is possible to score 1 additional ...
7
votes
2answers
105 views

$a^3+3a^2+a$ is never a perfect square.

Prove that no number of the form $ a^3+3a^2+a $, for a positive integer $a$, is a perfect square. This problem was published in the Italian national competition (Cesenatico 1991). I've been ...
2
votes
0answers
26 views

Triangles packed into a unit circle

2014 triangles have non-overlapping interiors contained in a unit circle.What is the largest possible value of the sum of their areas? What are some ideas that might help me start this?
1
vote
1answer
42 views

Knot theory: pretzel knot

Prove that pretzel knot $K(p_1,p_2,p_3,\dots,p_n)$ with all $p_i >0$ is an alternating knot or link? I think since all $p_i$'s are positive, the sign has a lot to do with it but how to prove it is ...
2
votes
2answers
76 views

For any positive integers $a$ and $b$, the number $(36a+b)(a+36b)$ can never be a power of $2$.

APMO 1998: Show that for any positive integers $a$ and $b$, the number $(36a+b)(a+36b)$ can never be a power of $2$. The solution I've read substitutes $a=2^Ap,b=2^Bq$ where $p$ and $q$ are ...
4
votes
1answer
82 views

Math Olympiads: GCD of terms in a sequence equals GCD of terms in other sequence

Recently, someone asked for a proof of a problem from the Russian Mathematical Olympiad, 1995. Math Olympiads: GCD of terms in a sequence equals GCD of their indices. The problem was to show that if ...
0
votes
1answer
80 views

If $x + \frac{1}{x} = k$, what's the value of this sum?

Friends, if $x + \frac{1}{x} = k$, with $k$ positive real number, what's the value of $$1+ x^6 + x^{12} + x^{18} + x^{24}+x^{30}$$ I tried with the substitution $u= x^6$: $$a= 1+ u + u^2 + u^3 + u^4 ...
1
vote
3answers
40 views

Ratio Math Problem

This is from a competition math problem I had recently that I just couldn't figure out. If $ (x+y):(y+z):(x+z) = 1:2:4$ and $x+y+z=35$ compute the value of x. I can tell that $7*(x+y)=2x+2y+2z$ ...
1
vote
2answers
60 views

Maths brain teaser. Fifty minutes ago it was four times as many minutes past three o'clock

Fifty minutes ago it was four times as many minutes past three o'clock. How many minutes is it to six o'clock..? I have got the solution online but have doubts in it : ...
5
votes
2answers
138 views

Math Olympiads: GCD of terms in a sequence equals GCD of their indices.

The sequence $a_1 ,a_2 ,a_3 ,...$ of positive integers satisfies $\text{gcd}(a_i ,a_j ) = \text{gcd} (i, j)$ for $i \neq j$. Prove that $a_i = i$ for all $i$. Source: Russian Mathematical Olympiad, ...
3
votes
2answers
186 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 ...
2
votes
2answers
125 views

Cool Integral $\int_0^{\pi/2}dx\ln \sinh x$

$$ I_1=\int_0^{\pi/2}dx\ln \sinh x,\quad I_2=\int_0^{\pi/2}dx\ln \cosh x, \quad I_1\neq I_2. $$ I am trying to calculate these integrals. We know the similar looking integrals $$ \int_0^{\pi/2}dx\ln ...
1
vote
2answers
36 views

Maximize the inradius given the base and the area of the triangle

BdMO 2013 Secondary: A triangle has base of length 8 and area 12. What is the radius of the largest circle that can be inscribed in this triangle? Let $A,r,s$ denote the area,inradius and ...
2
votes
3answers
248 views

2014 AMC 12 B problem 25

What is the sum of all positive real solutions $x$ to the following equation? $$2\cos(2x)\left( \cos(2x) - \cos{\left(\frac{2014\pi^2}{x^2}\right)} \right) = \cos(4x) - 1 $$
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
3answers
46 views

Prove that $(2m+1)^2 - 4(2n+1)$ can never be a perfect square where m, n are integers

I could prove it hit and trial method. But I was thinking to come up with a general and a more 'mathematically' correct method, but I did not reach anywhere. Thanks a lot for any help.
3
votes
1answer
55 views

Locus of the centres of equilateral triangles (contest problem)

Given a triangle $A_0A_1A_2$ determine the locus of the centres of the equilateral triangles $X_0X_1X_2$ satisfying the condition that each of the lines $X_kX_{k+1}$, $k=0,1,2$ passes through ...
1
vote
3answers
41 views

What's the probability that $y\ge x+1$?

"Two numbers, $x$ and $y$ are selected at random from the interval $(0,3)$. What is the probability that $y\ge x+1$?" The answer key says $\frac 29$, but I can't figure out how to get to that answer. ...
0
votes
1answer
19 views

Bound the Number of Acute-angled Triangles

I encounter the following problem with solution. But I do not quite understand the argument for 5, 10 points and eventually 100 points. Can someone elucidate the details? Problem In a plane there ...
2
votes
1answer
38 views

Find the sum of the maximum and minimum

For a real number $x$ find the sum of the maximum and minimum. $$y=\frac{x^2-2x-3}{2x^2+2x+1}$$ This is a sample question for a math competition. It seems like calculus would be used to solve this, ...
4
votes
3answers
137 views

How find this function $f(x)$ such $f(a+f(b))=f(a+b)+f(b)$

let function $f:R_{+}\to R_{+}$,and such $$f(a+f(b))=f(a+b)+f(b),\forall a,b\in R_{+}$$ Find $f(x)$. my try: let $a=b=1$,then $$f(1+f(1))=f(2)+f(1)$$ $a=1,b=2$,then $$f(1+f(2))=f(3)+f(2)$$ then I ...