Problem Of The Day

Some rules we all encouraged to abide by in the channel

This is efnet-math.org's Problem of the Day section. New problems are added every few days. Problems are archived at the end of the month. This and last month is always kept on this page. To view past problems, browse through the Problem of the Day Archive

Table of contents

1.1 Saturday, August 30, 2008

2.1 Thursday, May 19, 2008

3.1 Monday, April 7, 2008

4.1 Friday, March 29, 2008
4.2 Sunday, March 23, 2008

5 February 2008

6.1 Friday, 18th of January
6.2 Sunday, 13th of January

7 December 2007

7.1 Monday, 17th of December
7.2 Monday, 10th of December
7.3 Tuesday, 4th of December

8 November 2007

8.1 Thursday, 29th of November
8.2 Wednesday, 28th of November
8.3 Sunday, 18th of November
8.4 Monday, 12th of November
8.5 Sunday, 11th of November

9.1 Monday, 22nd of October
9.2 Monday, 15th of October
9.3 Saturday, 13th of October
9.4 Friday, 12th of October
9.5 Friday, 5th of October
9.6 Wednesday, 3rd of October

10 September 2007

10.1 Friday, 28th of September
10.2 Thursday, 27th of September
10.3 Wednesday, 26th of September
10.4 Tuesday, 25th of Septemeber
10.5 Saturday, 22nd of September
10.6 Thursday, 20th of Setember
10.7 Wednesday, 19th of September
10.8 Friday, 14th of September
10.9 Tuesday, 11th of September
10.10 Saturday, 1st of September

11.1 Wednesday, 29th of August
11.2 Saturday, 11th of August

12.1 Thursday, 26th of July
12.2 Saturday, 21st of July
12.3 Tuesday, 3rd of July

13.1 Friday, 29th of June
13.2 Thursday, 28th of June
13.3 Monday, 25th of June
13.4 Sunday, 24th of June
13.5 Friday, 22nd of June
13.6 Tuesday, 14th of June
13.7 Sunday, 10th of June
13.8 Tuesday, 5th of June
13.9 Sunday, 3rd of June
13.10 Saturday, 2nd of June

14.1 Wednesday, 30th of May
14.2 Tuesday, 29th of May
14.3 Friday, 25th of May
14.4 Tuesday, 22nd of May
14.5 Friday, 18th of May
14.6 Friday, 11th of May

August 2008

Saturday, August 30, 2008

Let $f$ be a holomorphic function on the open unit disc $Δ$ such that $| f (z) | < 1$ for all $z\in\Delta$ . Suppose that $f\Big(\frac{1}{2}\Big) = f\Big(-\frac{1}{2}\Big) = 0$ . Show that $|f(0)| \leq \frac{1}{3}$ .

May 2008

Thursday, May 19, 2008

from Zabrien

Consider a completely filled Sudoku, written as a 9x9 matrix. Show that the determinant of this matrix is divisible by 405.

April 2008

Monday, April 7, 2008

from beigebox

Given a set of points $S = \{x_1,x_2,\ldots,x_n\}$ in the plane so that for each two points $d(x_i,x_j) \geq 1$ , show that there are at most 3n pairs of points of distance exactly 1.

from Crito

The sequence ${x_n} n\geq 1$ is defined by $x_{1} = 2, x_{n + 1} = \frac {2 + x_{n}}{1 - 2x_{n}}\;\; (n \in \mathbb{N})$ . Prove that

a) for all $n\in\mathbb{N}$ ,

b) ${x_n} n\geq 1$ is not periodic.

March 2008

Friday, March 29, 2008

from Polytope

Solve the following system of equations:

$x^2-yz=4, \quad y^2-xz=37, \quad z^2-xy=-29$

Sunday, March 23, 2008

from dedekind

Does there exist a continuous function $f$ such that $f(f(x)) = x 2 - 2.$

Solution (http://www.artofproblemsolving.com/Forum/viewtopic.php?search_id=778734284&t=103001)

February 2008

January 2008

Friday, 18th of January

Find the number of solutions to $y 2 = x (x 2 + B)$ over the finite field with $p$ elements where $p \equiv 3 \pmod 4$ and $p$ does not divide $B$ .

Sunday, 13th of January

posted by Crito

1. A hyper-primitive root is a k-tuple $(a_{1},a_{2},\dots,a_{k})$ and $(m_{1},m_{2},\dots,m_{k})$ with the following property:

For each $a\in\mathbb N$ , that $(a, m) = 1$ , has a unique representation in the following form:

$a\equiv a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}\dots a_{k}^{\alpha_{k}}\pmod{m}\qquad 1\leq\alpha_{i}\leq m_{i}$

Prove that for each m we have a hyper-primitive root.

2. Let p be prime number and n be non negative integers.

(1) Let m be integer such that $0\leq m\leq n$ . How many numbers are there among integers 1 through $p n + 1$ which can be divided by $p m$ not but by $p m + 1$ ?

(2) For two integers $x,\ y$ 1 through $p n + 1$ , how many pairs of (x, y) such that the product xy can be divided by $p n + 1$ ?

December 2007

Monday, 17th of December

posted by crito

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that $\,f(xf(y)+f(x)) = 2f(x)+xy$ for every reals x,y.

Monday, 10th of December

Let $S$ be a simple random walk on $\mathbb{Z}^2$ starting at the origin. Show that the probability that $S$ returns to the origin before hitting $(1,0)$ is $\frac{1}{2}$ .

Tuesday, 4th of December

posted by Crito

Let $f (x) = x 2 + 2007 x + 1$ . Prove that for every positive integer n, the equation $\underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) = 0$ has at least one real solution.

November 2007

Thursday, 29th of November

posted by dioid

Let $f (x) = - x 2 + 2 x sin(x) + cos(2 x)$ prove that $f (x) < 0$ for $\pi/2 \leq x \leq \pi$

Experienced solvers, please don't spoil it in channel, HS calc level

Wednesday, 28th of November

posted by nerdy2

Classify GL(V) orbits of pairs of nondegenerate symmetric bilinear forms on a vector space V over an algebraically closed field.

Sunday, 18th of November

posted by Crito

Let n be a natural number, such that $(n,2(2 1386 - 1)) = 1$ . Let $\{a_{1},a_{2},\dots,a_{\varphi(n)}\}$ be a reduced residue system for $n$ . Prove that:

$n|a_{1}^{1386}+a_{2}^{1386}+\dots+a_{\varphi(n)}^{1386}$

Solution by int-e

Monday, 12th of November

posted by Crito

Let $p$ be a prime number and $n$ be a non-negative integer.

(1) Let $m$ be an integer, $0\leq m\leq n$ . How many integers $k$ exist, $1\leq k\leq p^{n+1}$ , such that $p^m\mid k$ but $p^{m+1}\nmid k$ ?

(2) How many pairs of integers $(x, y)$ exist such that $x,y\in\left[1,p^{n+1}\right]$ and $p^{n+1}\mid xy$ ?

Bonus by nerdy2

Given a function $f : \mathbb R \mapsto \mathbb R$ , suppose $g(x) = \lim_{t \to x} f(t)$ exists everywhere. Show that g is continuous.

Sunday, 11th of November

Find the number of integers c such that $-2007 \leq c \leq 2007$ and there exists an integer $x$ such that $x 2 + c$ is a multiple of $22007$ .

October 2007

Monday, 22nd of October

Posted by HiLander

Let $n \ge 3$ and let $F$ be a collection of subsets of $\{1,2,\ldots,n\}$ with $| F | = 2 n - 1 + n + 1$ . Show that there are $A,B,C \in F$ with $A \cap B \ne \emptyset, A \cap C \ne \emptyset, B \cap C \ne \emptyset$ , but $A \cap B \cap C = \emptyset$ .

Monday, 15th of October

Posted by Galois

(A challenge from Fermat to the English mathematicians.)

Find all integer solutions $x, y$ of the equation $y 2 = x 3 - 2$ .

Saturday, 13th of October

Posted by Crito

Prove that for a set $S\subset\mathbb N$ , there exists a sequence $\{a_{i}\}_{i = 0}^{\infty}$ in S such that for each n, $\sum_{i = 0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$ if and only if $|S|\geq2$ .

Extra: Posted by Crito

1. Positive integers x>1 and y satisfy an equation $2 x 2 - 1 = y 15$ . Prove that 5 divides x.

2. Find integral solutions to the equation $(m 2 - n 2) 2 = 16 n + 1$ .

Friday, 12th of October

posted by Crito

a) Let $n_{1},n_{2},\dots$ be a sequence of natural number such that $n_{i}\geq2$ and $\epsilon_{1},\epsilon_{2},\dots$ be a sequence such that $\epsilon_{i}\in\{1,2\}$ . Prove that the sequence:

$\sqrt[n_{1}]{\epsilon_{1}+\sqrt[n_{2}]{\epsilon_{2}+\dots+\sqrt[n_{k}]{\epsilon_{k}}}}$ is convergent and its limit is in $(1,2]$ .

Define $\sqrt[n_{1}]{\epsilon_{1}+\sqrt[n_{2}]{\epsilon_{2}+\dots}}$ to be this limit.

b) Prove that for each $x\in(1,2]$ there exist sequences $n_{1},n_{2},\dots\in\mathbb N$ and $n_{i}\geq2$ and $\epsilon_{1},\epsilon_{2},\dots$ , such that $n_{i}\geq2$ and $\epsilon_{i}\in\{1,2\}$ , and $x=\sqrt[n_{1}]{\epsilon_{1}+\sqrt[n_{2}]{\epsilon_{2}+\dots}}$

Friday, 5th of October

posted by Crito

Let n be a natural number, and $n = 2 2007 k + 1$ , such that k is an odd number. Prove that $n\not|\,2^{n-1}+1$

Wednesday, 3rd of October

posted by Crito

Sequence ${a n}$ is defined by $a_{0}= 2007,\, a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}$ for $n \ge 1$ . Prove that $\lfloor a_{n}\rfloor =2007-n$ for $0 \le n \le 1004$ , where $\lfloor x\rfloor$ denotes the largest integer no larger than x.

Solution by int-e.

September 2007

Friday, 28th of September

posted by Crito

Find all positive integers $k$ with the following property: There exists an integer $a$ so that $(a + k) 3 - a 3$ is a multiple of 2007.

Thursday, 27th of September

posted by Crito

Let a,b,c,d be real numbers which satisfy $\frac{1}{2}\leq a,b,c,d\leq 2$ and abcd=1. Find the maximum value of $\left(a+\frac{1}{b}\right)\left(b+\frac{1}{c}\right)\left(c+\frac{1}{d}\right)\left(d+\frac{1}{a}\right)$ .

Wednesday, 26th of September

posted by Crito

Let a,b,c,d be positive real numbers with a+b+c+d = 4. Prove that $a^{2}bc+b^{2}cd+c^{2}da+d^{2}ab\leq 4$ .

Tuesday, 25th of Septemeber

posted by Crito

Determine all pairs (x,y) of positive integers satisfying the equation $x! + y! = x y$ .

Saturday, 22nd of September

posted by Crito

Prove that for two non-zero polynomials f(x,y),g(x,y) with real coefficients the system:

f(x,y)=0

g(x,y)=0

has finitely many solutions in $\mathbb C^{2}$ if and only if f(x,y) and g(x,y) are coprime.

Thursday, 20th of Setember

posted by Crito

Does there exist a sequence $\{b_{i}\}_{i=1}^\infty$ of positive real numbers such that for each natural m: $b_{m}+b_{2m}+b_{3m}+\dots=\frac1m$

Solution by int-e.

Wednesday, 19th of September

posted by Crito

Given an integer m, define the sequence $\left\{a_{n}\right\}$ as follows: $a_{1}=\frac{m}{2},\ a_{n+1}=a_{n}\left\lceil a_{n}\right\rceil$ if $\geq 1$ Find all values of m for which $a 2007$ is the first integer appearing in the sequence.

Friday, 14th of September

from \\Steve

If $H, G$ are groups, let $H\diamondsuit G$ denote that $H < G$ such that for every automorphism $\phi:G\to G$ and $h\in H$ , $\phi(h)\in H$ . Let $H\triangleleft G$ denote that $H$ is a normal subgroup of $G$ . If $H\diamondsuit N\triangleleft G$ , show that $H\triangleleft G$ .

Tuesday, 11th of September

posted by Crito

Let a and b be positive integers. Show that if $4 a b - 1$ divides $(4 a 2 - 1) 2$ , then $a = b$ .

Solution by int-e.

Saturday, 1st of September

posted by Crito

Let p > 5 be a prime number.

For any integer x, define ${f_p}(x) = \sum_{k=1}^{p-1} \frac{1}{(px+k)^2}$

Prove that for any pair of positive integers x, y, the numerator of $f p (x) - f p (y)$ , when written as a fraction in lowest terms, is divisible by p^3.

August 2007

Wednesday, 29th of August

posted by Crito

Let $p \geq 5$ be a prime.

(a) Show that exists a prime $q \neq p$ such that $q | (p - 1) p + 1$

(b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: $\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2$

Saturday, 11th of August

Prove that the set of strict local maximum points of a real function is countable.

Solution (http://encyclomaniacs.sound-club.org/~fs/math/POTD-2007-08-11.pdf) by flamingspinach

July 2007

Thursday, 26th of July

edited by landen, not in Easy Series

Find the following limit in terms of $n\,.$

$\lim_{m\rightarrow \infty} \frac{1^n+2^n+\dots+m^n}{m^{n+1}},\ n\in\mathbb{N}$

For a gold star try it for $n \in \mathbb{R}$ with $n \in (-1,\infty).\,$

Solution from landen

Saturday, 21st of July

Problem 6 from landen's Easy Series
Might be a little harder than usual.

What is the largest positive integer $n\,$ such that $n^3+100\,$ is divisible by $n+10.\,$

Solution by int-e.

Tuesday, 3rd of July

Problem 5 from landen's Easy Series
from John of Palermo about 1224 CE

Three men own a share in a heap of coins; the first owns 1/2, the second 1/3, and the third 1/6. The money is divided by having each man take an amount arbitrarily. The first man returns 1/2 of the coins he has taken, the second 1/3, and the third 1/6. The money thus returned is divided into three equal shares, which are given to each man, and it turns out that now everyone has his proper part. How much money was there, and how much money did each obtain the first time?

Problem 6 Easy Series

Six positive integers form a strictly increasing series. Each number except the first is a multiple of the preceding number. Their sum is 79. Find out all you can about the numbers.

June 2007

Friday, 29th of June

2 more from landen's Easy Series

Show that the rational number

${{12\,m+1}\over{30\,m+2}}$

is in lowest terms for any positive integer $m$ .

A cube has all sides labeled with a positive integer. Then at each corner of the cube the corner is labeled with the product of the numbers on the three sides that come together at the corner. The sum of the numbers from all 8 corners is 1001. What is the sum of the 6 numbers on the sides?

Solutions by int-e.

Thursday, 28th of June

NEW!! landen's Easy Series. Formal training not required.

Find all positive integers $n\,$ such that $3n-4,4n-5,\,$ and $5n-3\,$ are all prime numbers.

$p\,$ and $q\,$ are prime numbers. $x^2\, -\,p\,x\, +\,q\ =\ 0$ has distinct rational roots. Find all $p\,$ and $q\,$ which work.

Solutions by int-e.

Monday, 25th of June

Posted by Crito

Let $x, y, z \ge 0$ be real numbers. Prove that: $\frac{x^{3}+y^{3}+z^{3}}{3}\ge xyz+\frac{3}{4}|(x-y)(y-z)(z-x)|$ .

Find the maximal real constant $α$ that can replace $\frac{3}{4}$ such that the inequality is still true for any non-negative x,y,z.

Sunday, 24th of June

Posted by Crito

If $F$ is a finite set of at least three positive integers each dividing the sum $\sigma = \sum\nolimits_F$ , where gcd $(F) = 1$ , show that the product $\prod\nolimits_F$ divides $σ | F | - 2$ .

Friday, 22nd of June

Posted by Crito

Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$ , such that for each $i\neq j, (a_{i},a_{j})=1$ , and for each n, the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$ ?

Solution by int-e.

Tuesday, 14th of June

Posted by Crito

Let k be a given natural number. Prove that for any positive numbers x; y; z with the sum 1 the following inequality holds: $\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\frac{y^{k+2}}{y^{k+1}+z^{k}+x^{k}}+\frac{z^{k+2}}{z^{k+1}+x^{k}+y^{k}}\geq \frac{1}{7}$ . When does equality occur?

Sunday, 10th of June

Posted by Crito

Determine all pairs of natural numbers (x; n) that satisfy the equation $x 3 + 2 x + 1 = 2 n$ .

Tuesday, 5th of June

Posted by Crito

Let $f: \mathbb{Q}\rightarrow \mathbb{R}$ be a function such that $|f(x)-f(y)|\leq (x-y)^{2}$ for all $x,y \in\mathbb{Q}$ . Prove that f is constant.

Sunday, 3rd of June

Posted by Crito

i) Find all infinite arithmetic progressions of positive integers $(d_n)_{n\in\mathbb{N}}$ such that $d p$ is prime for all sufficiently large primes $p$ .

ii) Find all polynomials $f(X) \in \mathbb{Z}[X]$ such that $\left|f(p)\right|$ is prime for all sufficiently large primes $p$ .

Solution by int-e.

Saturday, 2nd of June

Posted by Crito

Find all polynomials of degree 3, such that for each $x,y\geq 0: p(x+y)\geq p(x)+p(y)$

Solution by int-e.

May 2007

Wednesday, 30th of May

Posted by Crito

Let a,b,c,d be positive reals such that $a + b + c + d = 1$ .

Prove that: $6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}$ .

Solution by int-e

Tuesday, 29th of May

Posted by Crito

Does there exist two unfair 6-sided dice labeled with numbers 1..6 each such that probability of their sum being j is a number in $\left(\frac2{33},\frac4{33}\right)$ for each $2\leq j\leq 12$ ?

Solution by int-e

Bonus posted by Crito

Prove that the function $f : \mathbb{N}\longrightarrow \mathbb{Z}$ defined by $f (n) = n 2007 - n!$ , is injective.

Friday, 25th of May

Posted by Crito

Natural numbers a, b and c are pairwise distinct and satisfy $a | b + c + b c, b | c + a + c a, c | a + b + a b$ .

Prove that at least one of the numbers a, b, c is not prime.

Solution by int-e.

Tuesday, 22nd of May

Suggested by feydrauth

Prove or disprove:

For any positive integer $n\,$ there is a positive integer $m\,$ such that $n\,m$ has only 0's and 7's as decimal digits.

Solution by int-e.

Friday, 18th of May

poted by Crito

Find all real $α,β$ such that the following limit exists and is finite:

$\lim_{x,y\rightarrow 0^{+}}\frac{x^{2\alpha}y^{2\beta}}{x^{2\alpha}+y^{3\beta}}$

Friday, 11th of May

from Magnuss

Determine whether there exists a function $f: {\mathbb N} \to {\mathbb N}\,$ such that $f(f(n)) = n^2 - 19n + 99\,$ for all positive integers $n.\,$

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