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

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3
votes
2answers
90 views

2 is a primitive root mod $3^h$ for any positive integer $h$

It's easy to verify that 2 is a primitive root mod $3^2$. But then why does it follow that 2 is a primitive root mod $3^h$ for any positive integer $h$? This was used in the solution of 2009 Putnam ...
0
votes
2answers
74 views

How to prove $(F,+)$ and $(F\setminus \{0\},\cdot)$ aren't isomorphic, where $(F,+,\cdot)$ is an arbitrary field .

Assume $(F,+,\cdot)$ is an arbitrary field. How to prove $(F,+)$ and $(F\setminus \{0\},\cdot)$ aren't isomorphic? Thanks in advance.
2
votes
0answers
145 views

Korean Math Olympiad 2005 (trapezoid & tangent circles)

In a trapezoid $ABCD$ with $AD||BC$, $O_1$, $O_2$, $O_3$, $O_4$ denote the circles with diameters AB, BC, CD, DA, respectively. Show that there exists a circle with center inside the trapezoid which ...
0
votes
1answer
78 views

Figuring out an angle in an isosceles triangle

A problem from BdMO 2013: Let $ABC$ be an isoscles triangle with $AB=AC$.The bisector of $\angle B$ meets $AC$ at $D$.Given that $BC=BD+AD$,we need to figure out $\angle A$. If we consider ...
10
votes
2answers
107 views

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$ By computing some partial sums, the answers are 0. It seems an inductive argument is possible.
0
votes
0answers
46 views

Find all rational solutions to $x^3 - y^2 = 2$. [duplicate]

Find all rational solutions to $x^3 - y^2 = 2$. The only integers solutions are $(3,\pm5)$: http://mathforum.org/library/drmath/view/51569.html
5
votes
2answers
84 views

Proof that b is not divisible by 6

$$b=\left \lfloor (\sqrt[3]{28}-3)^{-n} \right \rfloor$$ The brackets mean that the number is the largest integer smaller than $(\sqrt[3]{28}-3)^{-n} $ Proof that b is never divisible by 6. I have ...
14
votes
1answer
131 views

Closed form for $\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}}$

Let $$S=\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}},\tag1$$ where $\operatorname{Li}_a(z)$ is the polylogarithm. For $a=1/2$ it can be represented as ...
0
votes
1answer
53 views

Why would the reflections of the orthocentre lie on the circumcircle?

Let ABC be a triangle which it not right-angled. Define a sequence of triangles $A_iB_iC_i$,with $i \geq 0$, as follows: $A_0B_0C_0$ is the triangle $ABC$; and, for $i \geq 0$, $A_{i+1}$, $B_{i+1}$, ...
2
votes
0answers
91 views

Rearranging numbered cards to reverse their order

I have been thinking about this question for a long time, but I can't solve it. Here is the question: We have $9$ cards, with numbers one to nine written on them (in the order $1, 2, \ldots , 9$). ...
0
votes
2answers
113 views

Coordinate Geometry Triangle

ABC is a triangle. BB$_1$ and CC$_1$ are angle bisectors of B and C respectively. E,F are feet of perpendiculars from A on BB$_1$ and CC$_1$ respectively. Suppose D is point at which incircle of ABC ...
1
vote
6answers
218 views

Find $(a,b)$ such that in $x^2+ax+b$, whenever $v$ is a root, then $v^2 - 2$ is also a root

Find $(a,b)$ such that in $x^2+ax+b$, whenever $v$ is a root, then $v^2 - 2$ is also a root $a,b$ are real numbers. Roots may or may not be real. In this question, the aim is to find values of and b ...
1
vote
2answers
68 views

Prove $a = b = c$, given $P_1(x) = ax^2-bx-c$ , $P_2(x) = bx^2-cx-a$, $P_3(x)=cx^2-ax-b$ and $P_1(v)=P_2(v)=P_3(v)$

Prove $a = b = c$, given $P_1(x) = ax^2-bx-c$, $P_2(x) = bx^2-cx-a$, $P_3(x)=cx^2-ax-b$ and $P_1(v)=P_2(v)=P_3(v)$ where $v$ is a real number. $a,b,c$ are non zero real numbers.
7
votes
3answers
374 views

Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$

Given $a + b = 1$, Prove that $a^ab^b + a^bb^a \le 1$; $a$ and $b$ are positive real numbers.
-1
votes
3answers
83 views

Show that $(A',B',C')$ form the vertices of an equilateral triangle.

Let $ABC$ be a triangle with $AB = AC $ and $angle BAC = 30.$ Let $(A')$ be the reflection of A in the line BC $(B')$ be the reflection of $B$ in the line CA $(C')$ be the reflection of C in the line ...
3
votes
1answer
113 views

Korean Math Olympiad 2000 (floor function, quadratic mod) [closed]

Let $p$ be a prime such that $p ≡ 1\ (\mathrm{mod}\ 4)$. Evaluate ...
-2
votes
4answers
142 views

mental ability whiz

I got a difficult question in a national olympiad, and was not able to solve it. I can't wait for answer keys. please solve it for me! If $3a = 4b = 6c$ and $a + b + c = 27 \sqrt{29}$, then what is ...
2
votes
0answers
44 views

Prove $\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$

Let $f\in C^1([0,1])$. Prove the following: $$\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$$ and $$|f(1/2)|\le\int_0^1(|f(t)|+\frac12|f'(t)|)dt$$ Note that ...
1
vote
0answers
63 views

Prove that $0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$

Let $f'$ be integrable. Prove that $$0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$$ Source: ...
2
votes
2answers
85 views

Prove $1<\frac1{e^2(e-1)}\int_e^{e^2}\frac{x}{\ln x}dx<e/2$

Prove the following inequalities: a) $1.43 < \int_0^1e^{x^2}dx < \frac{1+e}2$ b) $2e <\int_0^1 e^{x^2}dx+\int_0^1e^{2-x^2}dx<1+e^2$ c) $1<\frac1{e^2(e-1)}\int_e^{e^2}\frac{x}{\ln ...
2
votes
4answers
78 views

When triangle is divided into four, at least one is not bigger than a quarter of the original proof

The problem: On a triangle $ABC$ the points $M, K, L$ are chosen respectively on the sides $AB, BC, CA$. Prove that the area of at least one of the triangles $AML, BMK, CKL$ will be less than or ...
2
votes
2answers
90 views

Convex Quadrilateral: $ \dfrac {\tan A + \tan B + \tan C + \tan D}{\tan A \tan B \tan C \tan D} = \cot A + \cot B + \cot C + \cot D $

Problem Let $ABCD$ be a convex quadrilateral with no right angles. Show that $$ \dfrac {\tan A + \tan B + \tan C + \tan D}{\tan A \tan B \tan C \tan D} = \cot A + \cot B + \cot C + \cot D. $$ ...
3
votes
1answer
47 views

Neighbors with very different labels

The cells of a square $2011$ by $2011$ array are labelled with the integers $1,2,\dots, 2011^2$ in such a way that every label is used exactly once. We identity the top and bottom edges and the left ...
2
votes
1answer
63 views

How many integral solutions of $a,\ b,\ c$ are there such that $2^a \cdot 3^b + 9 = c^2 $

How many integral solutions of $a,\ b,\ c$ are there such that $$2^a \cdot 3^b + 9 = c^2.$$ we can get that $$2^a \cdot3^b = (c-3)(c+3) $$ we can make cases if $b \ge 2$ then $c=3k$ then ...
2
votes
1answer
118 views

Iran Math Olympiad 2012 (perfect power)

Prove that if $t$ is a natural number then there exists a natural number $n > 1$ such that $(n, t) = 1$ and none of the numbers $n + t, n^2 + t, n^3 + t…$ are perfect powers. There is a solution ...
0
votes
1answer
116 views

Prove $\text{Beta}(x,y) = 2\int_0^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,d\theta, \qquad \mathrm{Re}(x)>0,\ \mathrm{Re}(y)>0$

Prove that $$\int_0^1 x^{k}(1-x)^kdx=\frac{k!k!}{(2k+1)!}.$$ (Edit: Actually the proof can be found here http://en.wikipedia.org/wiki/Beta_function ) How would you show this $\text{Beta}(x,y) = ...
1
vote
2answers
33 views

Locating a point based on a condition

C and D are two points on the same side of straight line AB. Find a point X on AB such that the angles CXA and DXB are equal. How would you go about this?
0
votes
1answer
60 views

Suppose that a cube is inscribed in a sphere of radius one. What is the volume of the cube? my reasoning vs answer

Now my reasoning is that, s^2 + s^2 = 2^2, where s is the side of the cube, giving, s^3 = 2 sqrt 2. But the answer and explanation here is different: http://math.acadiau.ca/aumc/hints4.pdf how is the ...
7
votes
1answer
141 views

Bound on $|f(x)|^2 + |f'(x)|^2$

Let $f\in C^2(\mathbb{R})$ be a twice differentiable function satisfying $$|f(x)|^2\le a$$ and $$|f'(x)|^2 + |f''(x)|^2\le b$$ for all real $x$, where $a$ and $b$ are positive constants. Prove that ...
4
votes
1answer
101 views

What is the minimum number of locks on the cabinet that would satisfy these conditions?

Eleven scientists want to have a cabinet built where they will keep some top secret work. They want multiple locks installed, with keys distributed in such a way that if any six scientists are present ...
4
votes
1answer
68 views

Prove that $2^x < \prod_{p\le x} p < (13/4)^x$ for sufficiently large x

Prove that $2^x < \prod_{p\le x} p < (13/4)^x$ for sufficiently large x. Here $p$ is prime. So if we take logs we need to show for sufficiently large x, $x\log 2 < \sum_{p\le x}\log p < ...
0
votes
0answers
110 views

Iran Math Olympiad 2013 (Perfect Set)

Let $n$ be a natural number and suppose that $w_1,w_2,…,w_n$ are $n$ weights. We call the set of {$w_1,w_2,…,w_n$} to be a Perfect Set if we can achieve all of the 1, 2, …, W weights with sums of ...
1
vote
1answer
67 views

Diophantine Equation.

How many solutions are there in $\mathbb{N} \times \mathbb{N}$ to the equation $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{1995}?$ How would you solve this? I have tried but am not sure how I should ...
2
votes
3answers
88 views

For any real numbers $a,b,c$ show that $\displaystyle \min\{(a-b)^2,(b-c)^2,(c-a)^2\} \leq \frac{a^2+b^2+c^2}{2}$

For any real numbers $a,b,c$ show that: $$ \min\{(a-b)^2,(b-c)^2,(c-a)^2\} \leq \frac{a^2+b^2+c^2}{2}$$ OK. So, here is my attempt to solve the problem: We can assume, Without Loss Of Generality, ...
4
votes
2answers
207 views

Prove that [0,1] is equivalent to (0,1) and give an explicit description of a 1-1 function from [0,1] onto (0,1)

The problem is stated as follows: Show that there is a one-to-one correspondence between the points of the closed interval $[0,1]$ and the points of the open interval $(0,1)$. Give an explicit ...
1
vote
1answer
27 views

Diophantine Approximation and Liouville Theorem

I'm reading the alternative proof of the MO problem: http://koopakoo.wordpress.com/2008/09/03/cgmo-2007-problem-7-and-liouvilles-theorem/ . However, I have a problem, namely that in the alternative ...
5
votes
1answer
118 views

Putnam Series Question

I'm studying for the Putnam Exam and am a bit confused about how to go about solving this problem. Sum the series $$ \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{m^2n}{3^m(n3^m + m3^n)}. $$ ...
0
votes
1answer
81 views

Prime factorization problem

The prime factorizations of $r + 1$ positive integers ($r \geq 1$) together involve only $r$ primes. Prove that there is a subset of these integers whose product is a perfect square. Now, I'm ...
6
votes
3answers
130 views

Nature of roots of quartic equation [closed]

Prove that all roots of a$x^4$ + b$x^3$ + $x^2$ + $x$ + 1 = 0 cannot be real. a,b $\in$ $\Re$. a $\neq$ 0.
19
votes
1answer
235 views

How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?

How can I prove the following identity? $$\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$$
6
votes
5answers
151 views

Show that $\displaystyle \frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} \geq x+y+z $ by considering homogeneity

Well, I'm preparing for an undergrad competition that is held in April and because of that I've been trying to solve the inequalities I find on the internet. I found this problem: $$\displaystyle ...
3
votes
2answers
90 views

Interesting determinant: Let $A$ be an $n$ by $n$ matrix with entries $a_{i,j}$ given that $a_{i,j}=2$ if $i=j$

Let $A$ be an $n$ by $n$ matrix with entries $a_{i,j}$ given that $a_{i,j}=2$ if $i=j$, $a_{i,j}=1$ if $i-j\equiv\pm2\pmod n$, and $a_{i,j}=0$ otherwise. Find $\det A$. It seems that the ...
3
votes
1answer
122 views

Conjecture similar to Fermat's Theorem.

I was wondering about a problem which i could reduce to asking the following Does there exist a set $a,b,c$ of prime numbers such that $$a^a+b^b=c^c$$ Is it really a tough problem or do you think ...
2
votes
1answer
206 views

Korean Math Olympiad 1993 (geometry)

Consider a triangle ABC with BC = $a$, CA = $b$, AB = $c$. Let D be the midpoint of BC and E be the intersection of the bisector of A with BC. The circle through A, D, E meets AC, AB again at F, G ...
1
vote
2answers
55 views

Putnam-Style Sequences Problem

Let $S_1$ denote the sequence of positive integers $1,2,3,4,5,6,\ldots,$ and define the sequence $S_{n+1}$ in terms of $S_n$ by adding $1$ to those integers in $S_n$ which are divisible by $n$. Thus, ...
1
vote
1answer
43 views

Prove that for $n\ge1$, $\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},…,a_2,a_1\rangle\right)^{-1}$

Prove that for $n\ge1$, $$\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},...,a_2,a_1\rangle\right)^{-1}$$ In addition, show that ...
1
vote
2answers
180 views

Find the four digit number?

Find a four digit number which is an exact square such that the first two digits are the same and also its last two digits are also the same.
0
votes
1answer
67 views

Numbers with integer multiples using only digits $2$ and $6$ (Austria Mathematical Olympiad 2006)

Let $N$ be a positive integer. How many non-negative integers $n ≤ N$ are there that have an integer multiple, that only uses the digits $2$ and $6$ in decimal representation? Obviously, $n$ can't be ...
3
votes
1answer
48 views

Prove $\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\geq (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$

Prove that $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\geq (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$ for $a,b,c>0$ Any hints/solutions?
1
vote
1answer
21 views

Interesting continued fraction problem $|r_i-u_0/u_1|\le\frac1{k_ik_{i+1}}$

Let $u_0/u_1$ be a rational number in lowest terms, and write $u_0/u_1=\langle a_0, a_1,...,a_n\rangle$ in standard continued fraction notation. Show that if $0\le i<n$, then ...