# 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

## July 2012

### Monday, July 30, 2012

from **breeden**

Using only "calculus 1" methods, show that . (trig substitution and polar coordinates is not allowed, but we will allow "improper" integrals). Conclude that .

## June 2012

### Saturday, June 30, 2012

from **hochs**

Let *L* / *K* be a galois extension of degree 5. If there exists an element 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, 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 . Prove that every exact sequence is split.

### Tuesday, June 12, 2012

from **hochs**

Suppose is a complete noetherian local ring, and is a decreasing sequence of ideals of *A* such that . Then the linear topology defined by 's is finer than the -adic topology on *A*. That is, for any *n* > 0, there exists *i* > 0 such that .

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 without resorting to the Dominating Convergence theorem.

### Wednesday, May 23, 2012

from **lhrrwcc**

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

### 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 .

Solution by **hochs**

## April 2012

### Thursday, April 19, 2012

from **hochs**

For *n* > 1, let . 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 is continuous and satisfies for all . Then *f*(*x*) is identically 0 or for some .

### Sunday, April 15, 2012

from **hochs**

Suppose are nonzero complex numbers such that the set is a finite set. Prove that α_{i} is a root of unity for all *i*.

### Tuesday, April 10, 2012

from "spal"

Prove that 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 or , 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 is continuously differentiable and injective, then there exists a non-empty open subset of where | det(*f*'(*x*)) | > 0.

## March 2012

### Saturday, March 31, 2012

from **breeden**

(a) Show that there exists a (necessarily non-measurable) function with the property that for any function such that for all then *g* is non-measurable.

(b) Show that there exists a continuous function and a Lebesgue-measurable set such that 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 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 are nonnegative?

### Saturday, March 24, 2012

from **hochs**

For a function on a finite field with *q* elements, one can find a polynomial such that *f*(*c*) = *F*(*c*) for all elements (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 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 be any nonzero polynomial. Let , 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 .

### Saturday, March 17, 2012

from **breeden**

Let 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}, is exactly 2.

### Saturday, March 17, 2012

from **hochs**

Let , where denotes the 2-adic rationals (completion of w.r.t. the 2-adic valuation). Show that is Galois, find its galois group, and find a uniformizer for *L* (recall that is complete, hence the existence and uniqueness of discrete valuation extending that of .

### Thursday, March 15, 2012

from **hochs**

Denote by *S*_{n} the group of permutations of the sequence . Suppose that *G* is a subgroup of *S*_{n}, such that for every , there exists a unique for which π(*k*) = *k*. Show that *k* is the same for all .

### 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 10^{9}

### 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 are continuous functions where . Let and . Show that there exists such that *f*(*y*) = *M*.

### Tuesday, February 28, 2012

from **breeden**

Show that if is holomorphic where | *f*(*z*) | = 1 whenever | *z* | = 1 then *f*(*z*) = λ*z*^{n} where | λ | = 1 and .

### Friday, February 24, 2012

from **hochs**

Prove that if are coprime polynomials then there are infinitely many positive integers *n* such that *n**f* + *g* is irreducible in .

*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 is irreducible over . As usual, 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 such that *f*(*x* + *y*) = *f*(*x*) + *f*(*y*) for all ?

### Monday, February 20, 2012

from **breeden**

Show that there exists a constant *C* such that for all and .

*Hint:* Break the sum into two parts for and , respectively.

### Saturday, February 18, 2012

from **breeden**

Suppose that 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 and converges.

### Thursday, February 16, 2012

from **Zabrien**

Let be a set of positive integers with
and *x*_{n} = 2*n* − 1. Show that the complement of *X* in the naturals is closed under addition iff for all and for all we have .

### Wednesday, February 15, 2012

from **Zabrien**

Consider the vector space of real polynomials in *n* variables. Let

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 such that *f*(0) = 0 and , where denotes the open unit disc in .

Determine the value of:

## May 2009

### Saturday, May 30, 2009

For prime *p*, prove that , for all

**Variation:** For *odd* prime *p*, prove that , for all

## 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 . Suppose that . Show that .

## 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 in the plane so that for each two points , show that there are at most 3n pairs of points of distance exactly 1.

from **Crito**

The sequence is defined by . Prove that

a) for all ,

b) is not periodic.

## March 2008

### Friday, March 29, 2008

from **Polytope**

Solve the following system of equations:

### 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 and *p* does not divide *B*.

### Sunday, 13th of January

posted by **Crito**

1. A hyper-primitive root is a k-tuple and with the following property:

For each , that (*a*,*m*) = 1, has a unique representation in the following form:

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 . 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 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 such that for every reals x,y.

### Monday, 10th of December

Let *S* be a simple random walk on starting at the origin. Show that the probability that *S* returns to the origin before hitting (1,0) is .

### Tuesday, 4th of December

posted by **Crito**

Let *f*(*x*) = *x*^{2} + 2007*x* + 1. Prove that for every positive integer n, the equation 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

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 be a reduced residue system for *n*. Prove that:

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, . How many integers *k* exist, , such that but ?

(2) How many pairs of integers (*x*,*y*) exist such that and?

Bonus by **nerdy2**

Given a function , suppose exists everywhere. Show that g is continuous.

### Sunday, 11th of November

Find the number of integers c such that and there exists an integer *x* such that *x*^{2} + *c* is a multiple of 2^{2007}.

## October 2007

### Monday, 22nd of October

Posted by **HiLander**

Let and let *F* be a collection of subsets of with | *F* | = 2^{n − 1} + *n* + 1. Show that there are with , but .

### 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 , there exists a sequence in S such that for each n, is irreducible in if and only if .

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 be a sequence of natural number such that and be a sequence such that . Prove that the sequence:

is convergent and its limit is in (1,2].

Define to be this limit.

b) Prove that for each there exist sequences and and , such that and , and

### 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

### Wednesday, 3rd of October

posted by **Crito**

Sequence {*a*_{n}} is defined by for . Prove that for , where 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 and abcd=1. Find the maximum value of .

### Wednesday, 26th of September

posted by **Crito**

Let a,b,c,d be positive real numbers with a+b+c+d = 4. Prove that .

### 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 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 of positive real numbers such that for each natural m:

Solution by **int-e**.

### Wednesday, 19th of September

posted by **Crito**

Given an integer m, define the sequence as follows:
if
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 denote that *H* < *G* such that for every automorphism and , . Let denote that *H* is a normal subgroup of *G*.
If , show that .

### 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

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 be a prime.

(a) Show that exists a prime such that *q* | (*p* − 1)^{p} + 1

(b) Factoring in prime numbers show that:

### 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

For a gold star try it for with

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 such that is divisible by

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

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 such that and are all prime numbers.

and are prime numbers. has distinct rational roots. Find all and which work.

Solutions by **int-e**.

### Monday, 25th of June

Posted by **Crito**

Let be real numbers. Prove that: .

Find the maximal real constant α that can replace 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 , where gcd(*F*) = 1, show that the product divides
σ^{ | F | − 2}.

### Friday, 22nd of June

Posted by **Crito**

Does there exist a a sequence in , such that for each , and for each n, the polynomial is irreducible in ?

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: . 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 be a function such that for all . Prove that f is constant.

### Sunday, 3rd of June

Posted by **Crito**

i) Find all infinite arithmetic progressions of positive integers such that *d*_{p} is prime for all sufficiently large primes *p*.

ii) Find all polynomials such that 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

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: .

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 for each ?

Solution by **int-e**

**Bonus** posted by **Crito**

Prove that the function 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 there is a positive integer such that 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:

### Friday, 11th of May

from **Magnuss**

Determine whether there exists a function such that for all positive integers