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

learn more… | top users | synonyms (2)

7
votes
4answers
421 views

Given that $f(1)= 2013,$ find the value of $f(2013)$?

Suppose that $f$ is a function defined on the set of natural numbers such that $$f(1)+ 2^2f(2)+ 3^2f(3)+...+n^2f(n) = n^3f(n)$$ for all positive integers $n$. Given that $f(1)= 2013$, find the value ...
1
vote
1answer
54 views

how to prove this question about limit and derivative

Suppose $f:(a,b)\to\mathbb R$ that $ f $ satisfies: $$f\in C^1$$ $$\lim_{x\to a ^ +}f^2(x)=0$$ $$\lim_{x\to b ^ -}f^2(x)=e-1$$ if $\forall x \in(a,b) : 2f(x)f '(x)-f^2(x)\ge1 $, then how to ...
0
votes
1answer
59 views

Equation with fractional parts

How many solutions has the equation: $\{20 \cdot \{ 13 \cdot\{20 \cdot\{ 13\cdot x\}\} \}\}=x^{2013}$ -? Here $\{z\}=z-[z]$, where $[z]=m \in \mathbb{Z}, \ m \le z < m+1$.
10
votes
2answers
129 views

how prove $n! \mid \prod_{k=0}^{n-1}(2^n-2^k) :\forall n \in \mathbb N$?

how prove $$n! \mid \prod_{k=0}^{n-1}(2^n-2^k) :\forall n\in \mathbb N $$ Thanks in advance
2
votes
3answers
78 views

Help with inequality please

Once again I have come across an olympiad-type problem which probably requires some sort of insight even though it looks simple. The question is as follows: Let $a$, $b$ and $c$ be positive real ...
11
votes
2answers
246 views

Functions satisfying $f(m+f(n)) = f(m) + n$

I am a real newbie when it comes to funtions, and I don't understand what is supposed to happen or what I'm supposed to find when I get given an olympiad type question concerning functions. Could you ...
6
votes
1answer
182 views

how to prove $\sum _{|k|\lt\sqrt m}\binom{2m}{m+k}\ge2^{2m-1}$

how to prove $$\sum _{|k|\lt\sqrt m}\binom{2m}{m+k}\ge2^{2m-1},\forall m\ge1$$ Thanks in advance .
8
votes
1answer
355 views

Probem proposed for IMO 26

I have come across this problem and I really don't know how to construct this. Any ideas would be very much apreciated. Given 3 concentric circles, construct an equilateral triangle with a vertex on ...
1
vote
3answers
92 views

proportion probability problem

I am just doing some olympiad exercises to practice my probability skills, but I have difficulties with this one: Lets consider a village where each resident gets off work at a random time. ...
3
votes
1answer
204 views

Uniformly distributed probability problem

May you have an idea for the following exercise I found from some olympiad. Each day you have to bring home one full can of water. To do so you go to the next well and make the can full. On the way ...
6
votes
2answers
403 views

Divisibility - Math Olympiad

Show that for any positive integer $m$, there is an infinite number of pairs of integers $(x,y)$ satisfying the conditions: i) $\gcd(x,y)=1 $; ii) $y \mid x^2+m$; iii) $x \mid y^2+m$.
10
votes
4answers
890 views

How to prove this inequality? $ab+ac+ad+bc+bd+cd\le a+b+c+d+2abcd$

let $a,b,c,d\ge 0$,and $a^2+b^2+c^2+d^2=3$,prove that $ab+ac+ad+bc+bd+cd\le a+b+c+d+2abcd$ I find this inequality are same as Crux 3059 Problem.
2
votes
1answer
53 views

Covering of a $2\times 3$ rectangle using smaller rectangles

We have a $2 \times 3$ rectangle, that we wish to cover using $1 \times 2$ rectangles and $1 \times 1$ squares. How many possible ways are there to do so? Using $0$ and $3$ rectangles, it appears ...
11
votes
1answer
295 views

Math Olympiad - pre-periodic function

Let $c \in \mathbb{Q}$, $f(x)=x^2+c$. Define $$f^{0}(x)=x, \ \ f^{n+1}(x)=f(f^{n}(x)), \ \forall n \in \mathbb{N}$$ We say that $x \in \mathbb{R}$ is pre-periodic if $\{f^{n}(x), n \in \mathbb{N}\}$ ...
2
votes
1answer
270 views

Trigonometry / Geometry Puzzle with a Circle Inscribed within a Square

Point P is any point on the inscribed circle. You must prove that (tan(a))^2 + (tan(B))^2 = 8 I first moved point P down to the point where the square would be tangent to the curve to make the ...
6
votes
1answer
69 views

question about limit and series

consider following hypotheses $ m\in\mathbb N$ $c\in \mathbb C\,$ ,$\, \; a_j\in \mathbb C$ $a_j\in \mathbb C\;$ , $\;|a_j|=1,\;\forall\;1\le j\le m$ if ...
3
votes
0answers
118 views

Number of collinear subsets in a set

Call a set of points $(x,y)$ good if all the points in the set are collinear (i.e. they all lie on a line).Let S be the set of points $(x,y)$ such that $0\leq x,y \leq n$ ( $ x,y $ are restricted to ...
1
vote
1answer
440 views

Polynomial of polynomials (from Brilliant.org)

Moderator Note: This is a current question on brilliant.org if f(x) is a polynomial satisfying $$27 f(x^3) -4f(x^2) - x^6 f(3x) + 46 = 0 ,$$ what is $f(10)$? We can get $$f(0)=-2,\quad ...
-1
votes
1answer
140 views

Finding sum of all integral parts

Given two numbers $M$ and $N$, Let $q_i$ be the integer part of $\frac{iN}{M}$. What is $$ \sum_{i=0}^{M-1} q_i? $$ The Sum is obviously can be calculated in $O(M)$. Can this be done in less time, ...
5
votes
2answers
207 views

The roots of the derivative $P'(z)$ of the polynomial $P(z)\in\mathbb C[x]$ lie in the convex hull of the set of roots of $P(z)$.

Assume $S=\{z_1,z_2,...,z_k\}, z_i\in \mathbb C$$, C(S)$ and define $$C(S):=\{z=a_1z_1+a_2z_2+...+a_kz_k | a_i\ge0 ,a_1+a_2+...+a_k=1\}$$ where $$A:=\{z\in \mathbb C:f(z)=0 ...
0
votes
1answer
15 views

Representing expression through a Summatory

I need represent this expression $L_{u_3}u(k-1)+L_{u_2}u(k-2)+L_{u_1}u(k-3)$ by using a summatory, $L_{u}$ is a vector that contains d elements $L_u=\begin{pmatrix}L_{u_1} L_{u_2} ...
6
votes
2answers
138 views

If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $[a,b]\subset f([c,d])$, how to prove there is some $[r,s]$ such that $f([r,s])=[a,b]$?

Let $f:\mathbb R\to\mathbb R$ satisfy the following: $f$ is continuous there exist closed intervals $[a,b]$ and $[c,d]$ such that $[a,b]\subset f([c,d])$ How to prove that there ...
2
votes
3answers
228 views

Integers with 15 divisors (from brilliant.org)

How many integers from 1 to 19999 have exactly 15 divisors? Note: This is a past question from brilliant.org.
0
votes
3answers
213 views

Find the function that satisfies the following

Let $f: \mathbb{R} \to \mathbb{R}$ inconstant so that $\exists \lim_{x \to +\infty} f(x) $ and for any arithmetical progression $(a_n)$ the sequence $(f(a_n))$ is an arithmetical progression. ...
4
votes
2answers
206 views

Prove that function is bijective

Let $n \in \mathbb{N} \setminus \{ 0 \} $ and $A \in M_n(\mathbb{R})$ with $m \in \mathbb{N} \setminus \{ 0 \}$ as $A^m= \alpha \times I_n$, with $ \alpha \in \mathbb{R} \setminus \{ -1,1 \}$. ...
5
votes
3answers
213 views

Show that $\operatorname{rank}(A^2+A+I_3)=1$

If $A \in M_3(\mathbb{R}), A \ne I_3 $ and $A^3=I_3$ Show that $\operatorname{rank}(A^2+A+I_3)=1$. What I have reached so far is that $\operatorname{rank}(A-I_3)+\operatorname{rank}(A^2+A+I_3)\le ...
6
votes
3answers
472 views

Expected value uniform decreasing function

We are given a function $f(n,k)$ as for(i=0;i < k;i++) n = rand(n); return n; rand is defined as a random number generator that uniformly generates values ...
7
votes
1answer
1k views

Can three distinct points in the plane always be separated into bounded regions by four lines?

How can I show that for any three points in the plane, four lines can be drawn that separate the three points into distinct enclosed regions? Can any six points be enclosed in distinct regions formed ...
1
vote
1answer
258 views

Existence of Gergonne point, without Ceva theorem

The intersection at one point (called Gergonne point) of the lines from vertices of a triangle to contact points of the inscribed circle can be proved immediately using Ceva's theorem. Is there a ...
3
votes
0answers
386 views

An easy question on geometry.

The question says that I have to derive a formula to find the maximum number of enclosed regions formed by $n$ lines. This is how I proceeded: Let $f(n)$ be the maximum number of enclosed regions ...
3
votes
2answers
390 views

A number theory problem

Let $S$ be a set of real numbers satisfying the following conditions: i. $0$ is in $S$. ii. Whenever $x$ is in $S$ then $2^x+3^x$ is in S. iii. Whenever $x^2+x^3$ is in $S$ then $x$ is in $S$. ...
3
votes
3answers
68 views

If $f:[a,b]\to[a,b]$ is increasing, continuous, and $f(a)=a$, how to prove $f(E)=E$ where $E=\{x:a\le x\le b,f(x)\ge x\}$?

Let $f:[a,b]\to[a,b]$ satisfy: $f$ is increasing $f$ is continuous $f(a)=a$ If $E=\{x:a\le x\le b,f(x)\ge x\}$, then how can we prove that $f(E)=E$?
12
votes
4answers
383 views

Proving the inequality $\arctan\frac{\pi}{2}\ge1$

Do you see any nice way to prove that $$\arctan\frac{\pi}{2}\ge1 ?$$ Thanks! Sis.
1
vote
1answer
68 views

how find all function $f:(0,+\infty)\to(0,+\infty)$ that satisfy in following conditions?

how find all function $f:(0,+\infty)\to(0,+\infty)$ such that $\forall w,x,y,z\in \mathbb R^+ ,wx=yz$$$\frac{f(w)^2+f(x)^2}{f(y^2)+f(z^2)}=\frac{w^2+x^2}{y^2+z^2}.$$Thanks for any hint .
3
votes
1answer
385 views

Minimizing the integral of a function

For what real values of the argument $a$ the area which is limited by the function $$f:\Bbb R\to \Bbb R;\ f(x)=\frac{x^3}{3}-x^2+a;\ x=0; x=2; y=0$$ is minimum. (National Mathematical ...
9
votes
1answer
117 views

Geometric inequality with a triangle

The positive real numbers $x,y,z$ are the side lengths of a triangle iff $$x^2 + y^2 + z^2 < 2\sqrt{x^2y^2 + y^2z^2 + z^2x^2}$$
8
votes
3answers
96 views

How to prove there is no positive and continuous function satisfying some conditions

Let $\alpha \in \mathbb R $. How could I prove there isn't any positive and continuous function $f$ such that the following conditions hold? $\int_{0}^1 f(x)dx=1$ $\int_{0}^1 xf(x)dx=\alpha $ ...
17
votes
10answers
1k views

Find the integer closest to $\ln(2013)$

I encounter such a problem, in a Maths contest, to find out the closest integer to $\ln(2013)$, without using a calculator. I really get stuck. I tried to turn $\ln(2013)$ into ...
12
votes
2answers
416 views

Evaluate Integral (Romanian Olympiad)

$$ \int\cos x\cdot\cos^2(2x)\cdot\cos^3(3x)\cdot\cos^4(4x)\cdot\ldots\cdot\cos^{2002}(2002x)dx $$ Taken from the 2002 Romanian olympiad
4
votes
1answer
626 views

Pennies on a checkerboard.

Here is a question on pennies on checkerboard. It isnt a homework question. I saw it in a book. ...
3
votes
3answers
79 views

how prove $\sum_{m=0}^{n}\left(\frac{n!}{m!(n-m)!}\right)^2=\frac{(2n)!}{(n!)^2}$?

How to prove $\forall n \in \mathbb N$ $$\sum_{m = 0}^{n} \left(\frac{n!}{m!(n-m)!}\right)^2=\frac{(2n)!}{(n!)^2}$$
8
votes
3answers
254 views

Proving that $x(1-e^{-1/x})$ is strictly increasing

Prove that the function below is strictly increasing $$f(x)=x(1-e^{-1/x}), \quad x>0$$
0
votes
2answers
245 views

Solving $\frac{1}{a}+\frac{1}{b}=\frac{1}{200}\;$

How many ordered pairs of integers $(a,b)$ are there such that $$\frac{1}{a}+\frac{1}{b}=\frac{1}{200}\;?$$
2
votes
0answers
73 views

How to make a string of numbers not to become a square number? [duplicate]

Moderator Note: this is a question from the Federal Mathematics Competition 2013. Anja and Bernd are playing the following game: They take turns write down digits on the blackboard. Anja starts. ...
4
votes
7answers
836 views

Integrals from MIT integration bee

$\int\frac{dx}{2+2\sin x+\cos x}$ $\int_0^{\infty}\frac{\ln x}{1+x^2}dx$ $\int\frac{dx}{x(1+x^3)}$ In general what is $\int \frac{dx}{a+b\sin x}$?
4
votes
2answers
71 views

let $A\in M_n\mathbb R$ .how prove these statements with following condition?

Assume $A\in M_n(\mathbb{R})$, $A\neq 0$ such that: \begin{align*} A=(a_{ij}),\ 1\le i,j\le n,\\ a_{ik}a_{jk}=a_{kk}a_{ij},\ \forall\, i,j \end{align*} How to prove that: ...
2
votes
1answer
100 views

Prove that sum of 2010 vectors is $\neq 0$ if these vector create a set with lengths numbers $\{1,2,\ldots,2010\}$

A set $V$ has 2010-vectors: $V=\{v_{1}, \ldots,v_{2010}\}$ and these vectors create another set with the lengths of these vectors: $B=\{1,2,\ldots,2010\}$. Each vector is parallel to one of $2$ given ...
0
votes
1answer
58 views

Geometric inequality regarding a tetrahedron

The circumradius of a tetrahedron $ ABCD$ is $ R$, and the lenghts of the segments connecting the vertices $ A,B,C,D$ with the centroids of the opposite faces are equal to $ m_a,m_b,m_c$ and $ m_d$, ...
5
votes
3answers
138 views

Find $\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}.$

Find $$\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}\;.$$
13
votes
4answers
600 views

Showing that $ |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$

For every nonnegative integer $n$ and every real number $ x$ prove the inequality: $$\sum_{k=0}^n|\cos(2^kx)|= |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$$