Problem Of The Day

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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 July 2012

1.1 Monday, July 30, 2012

2 June 2012

2.1 Saturday, June 30, 2012
2.2 Wednesday, June 27, 2012
2.3 Thursday, June 14, 2012
2.4 Tuesday, June 12, 2012

3 May 2012

3.1 Tuesday, May 29, 2012
3.2 Wednesday, May 23, 2012
3.3 Monday, May 21, 2012

4 April 2012

4.1 Thursday, April 19, 2012
4.2 Tuesday, April 17, 2012
4.3 Sunday, April 15, 2012
4.4 Tuesday, April 10, 2012
4.5 Friday, April 6, 2012
4.6 Wednesday, April 4, 2012
4.7 Monday, April 2, 2012
4.8 Sunday, April 1, 2012

5 March 2012

5.1 Saturday, March 31, 2012
5.2 Monday, March 26, 2012
5.3 Saturday, March 24, 2012
5.4 Sunday, March 18, 2012
5.5 Saturday, March 17, 2012
5.6 Saturday, March 17, 2012
5.7 Thursday, March 15, 2012
5.8 Wednesday, March 14, 2012
5.9 Saturday, March 3, 2012

6 February 2012

6.1 Wednesday, February 29, 2012
6.2 Tuesday, February 28, 2012
6.3 Friday, February 24, 2012
6.4 Thursday, February 23, 2012
6.5 Tuesday, February 21, 2012
6.6 Monday, February 20, 2012
6.7 Saturday, February 18, 2012
6.8 Thursday, February 16, 2012
6.9 Wednesday, February 15, 2012
6.10 Sunday, February 12, 2012
6.11 Friday, February 10, 2012

7 May 2009

7.1 Saturday, May 30, 2009

8 August 2008

8.1 Saturday, August 30, 2008

9 May 2008

9.1 Thursday, May 19, 2008

10 April 2008

10.1 Monday, April 7, 2008

11 March 2008

11.1 Friday, March 29, 2008
11.2 Sunday, March 23, 2008

12 February 2008

13 January 2008

13.1 Friday, 18th of January
13.2 Sunday, 13th of January

14 December 2007

14.1 Monday, 17th of December
14.2 Monday, 10th of December
14.3 Tuesday, 4th of December

15 November 2007

15.1 Thursday, 29th of November
15.2 Wednesday, 28th of November
15.3 Sunday, 18th of November
15.4 Monday, 12th of November
15.5 Sunday, 11th of November

16 October 2007

16.1 Monday, 22nd of October
16.2 Monday, 15th of October
16.3 Saturday, 13th of October
16.4 Friday, 12th of October
16.5 Friday, 5th of October
16.6 Wednesday, 3rd of October

17 September 2007

17.1 Friday, 28th of September
17.2 Thursday, 27th of September
17.3 Wednesday, 26th of September
17.4 Tuesday, 25th of Septemeber
17.5 Saturday, 22nd of September
17.6 Thursday, 20th of Setember
17.7 Wednesday, 19th of September
17.8 Friday, 14th of September
17.9 Tuesday, 11th of September
17.10 Saturday, 1st of September

18 August 2007

18.1 Wednesday, 29th of August
18.2 Saturday, 11th of August

19 July 2007

19.1 Thursday, 26th of July
19.2 Saturday, 21st of July
19.3 Tuesday, 3rd of July

20 June 2007

20.1 Friday, 29th of June
20.2 Thursday, 28th of June
20.3 Monday, 25th of June
20.4 Sunday, 24th of June
20.5 Friday, 22nd of June
20.6 Tuesday, 14th of June
20.7 Sunday, 10th of June
20.8 Tuesday, 5th of June
20.9 Sunday, 3rd of June
20.10 Saturday, 2nd of June

21 May 2007

21.1 Wednesday, 30th of May
21.2 Tuesday, 29th of May
21.3 Friday, 25th of May
21.4 Tuesday, 22nd of May
21.5 Friday, 18th of May
21.6 Friday, 11th of May

July 2012

Monday, July 30, 2012

from breeden

Using only "calculus 1" methods, show that $\int_{-1}^1 2\sqrt{1 - x^2} dx = \pi$ . (trig substitution and polar coordinates is not allowed, but we will allow "improper" integrals). Conclude that $\int_{-r}^r 2\sqrt{r^2 - x^2} dx = \pi r^2$ .

June 2012

Saturday, June 30, 2012

from hochs

Let $L / K$ be a galois extension of degree $5$ . If there exists an element $a \in L, a \neq 0$ such that $a$ and $5 a$ are conjugates over $K$ , then find all possible characteristics of $K$ . What if $5$ is replaced by an arbitrary prime?

Wednesday, June 27, 2012

from zeno

Prove that if $A$ is a non-commutative ring with $1$ , $x \in A$ has a right inverse but no left inverse, then $x$ has infinitely many right inverses.

Thursday, June 14, 2012

from hochs

Suppose a ring $R$ is a $k$ -algebra, where $k$ is a field. Suppose $A, B, C$ are left $R$ -modules that are finite dimensional over $k$ , and that there's a split exact sequence $0 \to A \to B \to C \to 0$ . Prove that every exact sequence $0 \to A \to B \to C \to 0$ is split.

Tuesday, June 12, 2012

from hochs

Suppose $(A, \mathfrak{m})$ is a complete noetherian local ring, and $\mathfrak{a}_1 \supset \mathfrak{a}_2 \supset \cdots$ is a decreasing sequence of ideals of $A$ such that $\cap_{i} \mathfrak{a}_i = (0)$ . Then the linear topology defined by $\mathfrak{a}_i$ 's is finer than the $\mathfrak{m}$ -adic topology on $A$ . That is, for any $n > 0$ , there exists $i > 0$ such that $\mathfrak{m}^n \supset \mathfrak{a}_i$ .

It's easy to cook up an example where this fails if $A$ is not noetherian. Find a counterexample when $A$ is not complete.

May 2012

Tuesday, May 29, 2012

from Y0UrShAD0

Evaluate $\lim_{n \to \infty} \int_0^{\infty} \frac{\sin(x/n)}{(1 + x/n)^n} dx$ without resorting to the Dominating Convergence theorem.

Wednesday, May 23, 2012

from lhrrwcc

Let $M = (a i j)$ be a $n \times n$ -matrix over a local ring $A$ such that for all $i, j$ : $a i i$ is a unit and $a_{ij} (i \neq j)$ is not a unit. Show that $M$ is a unit in $M a t n (A)$ .

Solution

Monday, May 21, 2012

from lhrrwcc

Let $M$ be a projective left module over a ring, then there exists a free left module $F$ such that $F \simeq M \oplus F$ .

Solution by hochs

April 2012

Thursday, April 19, 2012

from hochs

For $n > 1$ , let $P_n(x) = 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}$ . For $n$ odd, this polynomial has exactly one real root (prove this). Prove that this root is irrational for all odd $n > 1$ .

Tuesday, April 17, 2012

from joo & Karlo

Suppose that $f: \mathbb{R} \to \mathbb{R}$ is continuous and satisfies $f(\sqrt{x^2 + y^2}) = f(x)f(y)$ for all $x,y \in \mathbb{R}$ . Then $f (x)$ is identically 0 or $f(x) = e^{cx^2}$ for some $c \in \mathbb{R}$ .

Sunday, April 15, 2012

from hochs

Suppose $\alpha_1, \ldots, \alpha_n \in \mathbb{C}^{*}$ are nonzero complex numbers such that the set $\{ \alpha_1^{k} + \cdots + \alpha_n^{k} : k \geq 1 \}$ is a finite set. Prove that $α i$ is a root of unity for all $i$ .

Tuesday, April 10, 2012

from "spal"

Prove that $10^{10^{10^{n}}} + 10^{10^{n}} + 10^{n} - 1$ is not a prime for any nonnegative integer $n$ .

Friday, April 6, 2012

TGIF

For prime $p$ , show that $2 p + 3 p$ is not a perfect power (i.e. not of the form $m k$ for naturals $m > 1, k > 1$ ).

Wednesday, April 4, 2012

from breeden

You are given digits 1, 3, 4, 6 and allowed to add, multiply, subtract, divide and use brackets. For example, 4*(6/3+1) gives 12. Can you get 24? No uniting of digits or using powers is allowed, such as in 3*(14-6) or 6*(14+3). You also have to use each digit exactly once, so 4*6 does not work either.

Monday, April 2, 2012

from hochs

Let $(K, | | )$ be any complete normed field. Prove that any finite dimensional vector space over $K$ has unique norm (up to equivalence of norms), and therefore it is again complete. Remark. This is well-known when $K =\mathbb{R}$ or $\mathbb{C}$ , i.e. when $V$ is a Banach space. The point is that one does not need compactness to prove this slightly more general version.

Sunday, April 1, 2012

from brett1479

Show that if $f: \mathbb{R}^n \to \mathbb{R}^n$ is continuously differentiable and injective, then there exists a non-empty open subset of $\mathbb{R}^n$ where $| det(f'(x)) | > 0$ .

March 2012

Saturday, March 31, 2012

from breeden

(a) Show that there exists a (necessarily non-measurable) function $f: \mathbb{R} \to \mathbb{R}$ with the property that for any function $g: \mathbb{R} \to \mathbb{R}$ such that $|f(x) - g(x)| \leq 1$ for all $x \in \mathbb{R}$ then $g$ is non-measurable.

(b) Show that there exists a continuous function $f:[0,1] \to [0,1]$ and a Lebesgue-measurable set $B \subseteq [0,1]$ such that $f(B) \subseteq [0,1]$ is non-measurable. Can you take $B$ to be Borel-measurable? (Hint: One can find such an $f$ that takes a set of null-measure onto $[0,1]$ )

Monday, March 26, 2012

Happy Birthday Paul Erdős

How many sequences $a_1, a_2, \ldots, a_n$ of $1$ 's and $- 1$ 's exist such that the number of $i$ with $a i = 1$ is equal to the number of $j$ with $a j = - 1$ and all the partial sums $a_1, a_1 + a_2, \ldots, a_1 + a_2 + \cdots + a_n$ are nonnegative?

Saturday, March 24, 2012

from hochs

For a function $f: \mathbb{F} \rightarrow \mathbb{F}$ on a finite field $\mathbb{F}$ with $q$ elements, one can find a polynomial $F \in \mathbb{F}[x]$ such that $f (c) = F (c)$ for all elements $c \in \mathbb{F}$ (evaluation at $c$ ). If we insist that the degree of this polynomial be less than $q$ , then it is uniquely determined by the function $f$ . We can thus speak of degree of nonzero $f$ as the degree of this associated polynomial.

Prove that if $q$ is odd, then no permutation (i.e. bijective) map $f: \mathbb{F} \rightarrow \mathbb{F}$ has degree $q - 1$ . What if $q$ is even?

And an optional unsolved problem: What are the possible degrees of permutations? Classify all "permutation" polynomials.

Sunday, March 18, 2012

from hochs

Let $f \in \mathbb{R}[x]$ be any nonzero polynomial. Let $g_n = {n \choose 0} f + {n \choose 1} f' + {n \choose 2} f'' + \cdots + {n \choose n} f^{(n)}$ , where $f (k)$ denotes the $k$ -th derivative of $f$ . Prove that there exists a positive integer $N$ such that all roots of $g n$ are real for all $n \geq N$ .

Saturday, March 17, 2012

from breeden

Let $P_1, P_2, \ldots, P_n$ be vertices of a regular $n$ -gon inscibed in a unit circle. Let $O$ denote the center of the circle. Suppose $P$ is a point on the unit circle such that the line segment $O P$ bisects one of the sides $P i P i + 1$ . Show that the product of distances from $P$ to $P i$ , $i = 1, 2, \ldots, n$ is exactly $2$ .

Saturday, March 17, 2012

from hochs

Let $L = \mathbb{Q}_2 \left(\sqrt{-1}, \sqrt{2}, \sqrt{\sqrt{2}(4 + \sqrt{-1})} \right)$ , where $\mathbb{Q}_2$ denotes the $2$ -adic rationals (completion of $\mathbb{Q}$ w.r.t. the $2$ -adic valuation). Show that $L/ \mathbb{Q}_2$ is Galois, find its galois group, and find a uniformizer for $L$ (recall that $\mathbb{Q}_2$ is complete, hence the existence and uniqueness of discrete valuation extending that of $\mathbb{Q}_2$ .

Thursday, March 15, 2012

from hochs

Denote by $S n$ the group of permutations of the sequence $(1, 2, \ldots, n)$ . Suppose that $G$ is a subgroup of $S n$ , such that for every $\pi \in G \setminus\{e \}$ , there exists a unique $k \in \{ 1, 2, \ldots, n \}$ for which $π(k) = k$ . Show that $k$ is the same for all $\pi \in G \setminus\{e \}$ .

Wednesday, March 14, 2012

from hochs

Let $P (x)$ be a polynomial with real coefficients. Show that there exists a nonzero polynomial $Q (x)$ with real coefficients such that $P (x) Q (x)$ has terms that are all of a degree divisible by $109$

Saturday, March 3, 2012

from hochs

There are two kinds of coins, genuine and counterfeit. A genuine coin weighs $X$ grams and a counterfeit coin weighs $X + δ$ grams, where $X$ is positive integer and $δ$ is non-zero real number strictly between $5$ and $- 5$ . You are presented with $13$ piles of $4$ coins each. All of the coins are genuine, except for one pile, in which all $4$ coins are counterfeit. You are given a precise scale (say, a digital scale capable of displaying any real number - wow!). You are to determine three things: $X,δ$ and which pile contains the counterfeit coins. But you’re only allowed to use the scale twice!

February 2012

Wednesday, February 29, 2012

from breeden

Assume that $f_n: [0,1] \to \mathbb{R}^+$ are continuous functions where $f_1(x) \geq f_2(x) \geq f_3(x) \geq \cdots$ . Let $f(x) = \lim_{n \to \infty} f_n(x)$ and $M = \sup\{ f(x) : x \in [0,1] \}$ . Show that there exists $y \in [0,1]$ such that $f (y) = M$ .

Tuesday, February 28, 2012

from breeden

Show that if $f: \mathbb{C} \to \mathbb{C}$ is holomorphic where $| f (z) | = 1$ whenever $| z | = 1$ then $f (z) = λ z n$ where $| λ | = 1$ and $n \in \mathbb{N}$ .

Friday, February 24, 2012

from hochs

Prove that if $f,g \in \mathbb{Z}[x]$ are coprime polynomials then there are infinitely many positive integers $n$ such that $n f + g$ is irreducible in $\mathbb{Z}[x]$ .

Remark: Elementary solution exists. No complex analysis, no L-functions,... are needed.

Thursday, February 23, 2012

from hochs

Show that if $p$ is prime, then $\frac{x^{n_1} + x^{n_2} + \cdots + x^{n_p} - p}{x^{gcd(n_1, \ldots, n_p)} - 1}$ is irreducible over $\mathbb{Q}[x]$ . As usual, $n_1, n_2, \ldots, n_p$ denote positive integers.

Solution (http://www.artofproblemsolving.com/Forum/viewtopic.php?f=36&t=395023)

Tuesday, February 21, 2012

from breeden

Does there exist a nowhere continuous function $f: \mathbb{R} \to \mathbb{R}$ such that $f (x + y) = f (x) + f (y)$ for all $x,y \in \mathbb{R}$ ?

Monday, February 20, 2012

from breeden

Show that there exists a constant $C$ such that $\left| \sum_{k=1}^n \frac{\sin(kx)}{k} \right| < C$ for all $n = 1,2,3,\dots$ and $x \in [0,2\pi]$ .

Hint: Break the sum into two parts for $kx \leq 1$ and $kx \geq 1$ , respectively.

Saturday, February 18, 2012

from breeden

Suppose that $\sum_{k=1}^{\infty} a_k = \infty$ and ${b n}$ is a bounded sequence, where $a n$ and $b n$ are real. Show that there exists an increasing sequence of positive integers, $n k$ , such that $\sum_{k=1}^{\infty} a_{n_k} = \infty$ and $\lim_{k \to \infty} b_{n_k}$ converges.

Thursday, February 16, 2012

from Zabrien

Let $X = \{x_1,\dots,x_n\}$ be a set of positive integers with $x_1 < x_2 < \dots < x_n$ and $x n = 2 n - 1$ . Show that the complement of $X$ in the naturals is closed under addition iff for all $1 \leq i \leq n$ and for all $n - x_i + i \leq j \leq n$ we have $x_i + x_j - x_n \in X$ .

Wednesday, February 15, 2012

from Zabrien

Consider the vector space $\mathbb{R}[x_1, \cdots, x_n]$ of real polynomials in $n$ variables. Let

$P=\prod_{1 \leq i < j \leq n}(x_i - x_j).$

Let $X$ be the subspace spanned by all partial derivatives of $P$ . Show that $X$ has dimension at least $n!$ .

Sunday, February 12, 2012

from hochs

Suppose $X$ is a finite set such that $| X | = 2 k$ for some positive integer $k$ . Suppose there's a family $F$ of subsets of $X$ , where each element of $F$ has cardinality $k$ , and such that every subset of $X$ having cardinality $k - 1$ is uniquely contained in some element of $F$ . Prove that $k + 1$ is prime.

Solution by hochs

Friday, February 10, 2012

from breeden

Let $S$ be the set of holomorphic functions $f: \mathbb{D} \to \mathbb{D}$ such that $f (0) = 0$ and $f\Big(\frac{1}{2}\Big) = \frac{1}{3}$ , where $\mathbb{D}$ denotes the open unit disc in $\mathbb{C}$ .

Determine the value of: $\sup \Big\{ \Big|f\Big(\frac{i}{2}\Big)\Big| : f \in S \Big\}.$

May 2009

Saturday, May 30, 2009

For prime $p$ , prove that ${p^n \choose p} \equiv p^{n-1} \pmod {p^n}$ , for all $n \in \mathbb{N}$

Variation: For odd prime $p$ , prove that ${p^n \choose p} \equiv p^{n-1} \pmod {p^{2n}}$ , for all $n \in \mathbb{N}$

Solution

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.

Solution by int-e

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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