Show Digest for last: 2 Days | 3 Days | 5 Days | 1 Week | 2 Weeks

Find all possible values of 1/x+1/y, if x,y are real numbers not equal to 0 that satisfy

x³+y³+3x²y²=x³y³

Ms. Langston and four other executives are experiencing the downturn in the economy. While traveling on Wednesday of last week, they had to fly on commercial aircraft instead of enjoying the comfort of private jets. Each executive is an expert in a particular area of business. Use the following clues to determine the first and last name of each executive along with her position, company and her special expertise. No two executives share the same name, position, company or forte.

First Names: Ann, Arlice, Meg, Patricia, Rene
Last Names: Barnes, Langston, Mulcahy, Russo, Whitman
Positions: Chief Executive Officer (CEO), Chief Financial Officer (CFO), Chairperson, Commissioner, President
Companies: ieTrade, Lenamar Tech, Licent Co., Sabancci Ltd., The Surat Group
Fortes: Invention, Leadership, Political
Connections, Profit Margin, Stock Options

1. Neither the CEO (who is not the leadership guru) nor the one with the excellent political connections (who is not a CFO or with Licent Co.) is surnamed Russo (who is neither Patricia nor Rene.)

2. Both Ann and Ms. Langston boarded in San Francisco and deplaned in Chicago; the leadership specialist and the chairperson flew from Chicago to Washington, D.C.; the flight of the Sabancci Ltd. executive (who is not surnamed Mulcahy) was from D.C. to New York.

3. The Lenamar Tech. executive (who has no special political connections) and the inventor (who is neither CEO nor CFO) are either the commissioner and chairperson in some order, or are surnamed Russo and Barnes, in some order; only one of these alternatives is true, the other is false.

4. Arlice is neither the ieTrade CFO nor Ms. Barnes; the profit margin specialist is not amongst the three executives.

5. Neither Meg (who is neither the stock options specialist nor the inventor) nor Ms. Mulcahy (who is not Arlice, who is not the chairperson) is affiliated with either Licent Co. or The Surat Group (who is not Rene).

6. Of the CEO (who is not Ms. Russo) and the profit margin specialist (who is not Rene, who is not the chairperson), one is Ms. Whitman and the other is associated with Lenamar Tech (who is not Ann).

7. The stock option specialist is either Ms. Barnes or the executive that works with Sabancci Ltd. (who is neither Patricia, who is not the commissioner, nor Rene.)

8. The president (who is not with Sabancci Ltd.) is not the inventor (who is neither surnamed Whitman nor Barnes.) Ms. Barnes is not the leadership guru.

9. The Surat Group executive is neither surnamed Langston nor Whitman.

MAI is the square of ST.
And:

                               
            . . . . . A 
                  M A I 
          --------------
          . . . . . . .
        . . . . . . .
      . . E N . . . 
     -------------------
      P R I N T E M P S 

How many pairs of integers (m,n) satisfy the equation (3m+4n)(4m+3n)=3⁶³?

What is the largest possible rational root of the equation ax² + bx + c = 0 where a, b and c are positive integers that do not exceed 100?

Given that:

        3√3
x= --------------
    1+ 3√3 + 3√9

Find the value of:

4x3 + 9x+1
------------
8x3 + 18x-1

Determine all possible values of a nonnegative integer k, such that k is NOT expressible in the form:

N+floor(√N+1/2)

for some integer N.

*** Adapted from a problem appearing in the IMO longlist.

For a real number x, how many solutions does the equation 4^x + 7^x + 8^x + 10^x + 14^x + 15^x = 17^x + 19^x have?

Let ABCD be a trapezoid with AD || BC. A point M is chosen inside the trapezoid, and a point N is chosen inside the triangle BMC such that AM || CN, BM || DN. Prove that triangles ABN and CDM have equal areas.

For what values of n is

(n²-1)!

not divisible by

(n!)ⁿ ?

How many prime numbers p make the number 3^p2+p+1 + 7^p2+p+1 divisible by p?

Determine all possible triplets (a, b, c) of positive integers that satisfy this equation:

  (c-a-b)2 = 24*a*b*c

A rectangle, HOMF, has sides HO=11 and OM=5. A triangle ABC has H as orthocentre, O as circumcentre, M be the midpoint of BC, F is the foot of altitude from A. What is the length of BC?

Solve in real numbers:

     
               1
x - ⌊x⌋ = -----------
          1/x + 1/⌊x⌋

Note:⌊x⌋ is the floor of x, which is the greatest integer less than or equal to x.

Let us consider this quartic equation:

x⁴- 2x² - 400x = 9999

Find all possible roots of the above equation.

Let p(x) be a fifth degree polynomial with real coefficients such that p(x)+1 is divisible by (x-1)³ and p(x)-1 is divisible by (x+1)³. Find the value of ∫p(x)dx from -10 to 10.

How many multiples of 2013 have 2013 factors?

Factorize the given polynomial

P(x)=(3x+2)(4x-3)(x-1)(12x+11)-14

I is the incenter of △ ABC.

Will the circumcenter of △ BIC always lie on the circumcenter of △ ABC?

• If so, prove it.
• If not, provide an example.

Find the minimum value of

cos²(x)-2sin(x)+tan²(x)

for x∈R.

Given that:

27^x + 64^x   193
----------- = ----
48^x + 36^x   144

Find all possible real values of x, that satisfy the given conditions.


        

Consider two positive integers x and y, and determine all possible pairs (x,y) that satisfy this equation:
x³ = 3^y - 1

Provide valid reasoning for your answer.

Let f(x)=(x+√3)/(1-x√3). Compute the following integral

∫f(f(f(x)))dx

It is known that:

x3 + 4x = 8

Determine the value of:

x7 + 16x2

Show that f(z)=z⁷-5z⁴+z²-2 has exactly 4 zeros inside the unit circle.

A is an nxn matrix such that:

A² = p*A²+q*A+r*I

Find |A-I|^-1 assuming that it exists.

*** Adapted from CMJ 278 by M.J. Klamkin.