- dan's math@home - problem
of the week - archives
-
-
- Problem Archives
page 24
- Problems Only.
For answers & winners click here.
-
- 1-10
. 11-20
. 21-30
. 31-40
. 41-50
. 51-60
. 61-70
. 71-80
. 81-90
. 91-100
- 101-110 . 111-120 . 121-130
. 131-140 . 141-150 . 151-160
. 161-170 . 171-180
- 181-190 . 191-200
. 201-210 . 211-220
. 221-230 . 231-240 .
241-250 . 251+ index
-
- 231 Obsessed w Squares
- 232 -- oNly oNe loNe N.
- 233- Lucas Square Sums
- 234 - Saturated Numbers
- 235 FromZeroToInfinity
- 236 - Gimme an Integer!
- 237- - - Nearly Integers!
- 238 - - Cube It Together
- 239 -- The 3-4-5 Square
- 240 - - The Pies Have It.
-
- problem #231 - posted
friday, oct 7, 2005
- first problem of this year's
contest!
- (this is my ninth season;
I can't believe it!)
- Obsessed with Squares (back to top)
- Find all integers n for
which 2^1994 + 2^1998 + 2^1999 + 2^2000
+ 2^2002 + 2^n
- is a perfect square. Show your reasoning.
-
-
-
- problem #232 - posted
friday, oct 21, 2005
- second problem of this year's
contest!
- oNly oNe loNe N (back to top)
- Consider the equation below,
where x and y are relatively prime positive integers:
- Show that there is only one
possible value for N. Find it. Show
your reasoning.
-
-
-
- problem #233 - posted
monday, nov 7, 2005
- Lucas Square Sums (back to top)
- The Lucas numbers are defined as L0 = 2, L1 = 1, Ln+1 = Ln + Ln-1 for
n > 1
- Find a closed form for the sum of the squares of
Lk , from
k = 0 to n ,
- in terms of the Ln's.
Verify your result numerically up to n = 10.
- "Closed form" is an algebraic
formula without "sum of" or ". . . " - Show your reasoning.
-
-
-
- problem #234 - posted
saturday, nov 19, 2005
- Saturated Numbers? (back to top)
- Ok, bear with me on this
. . . Every integer n > 1 has a prime factorization.
- If no primes are skipped; 2^a 3^b 5^c
. . . the n is called saturated (sat).
- If also exponents in order;
a >= b >= c . . . n is saturated ordered (s.o.).
- If n has more divisors than
any smaller number, it's supercomposite (s.c.).
- a) For n <= 20, 40, 60,
80, 100, 120: how many composite, sat, s.o.,
- . . and s.c. nos are there?
(ex: for n <= 10 there are 5,
4, 4, and 3 of each.)
- b) List all the n <= 200
that are sat but not s.o.
- c) List all the n <= 200
that are s.o. but not s.c.
- d) And finally, list all
the n <= 200 that are s.c.
- Show your reasoning.
-
-
-
- problem #235 - posted
sunday, dec
4, 2005
- From Zero to Infinity, Rationally!
(back to top)
- Prove there are no rational numbers x,
y such that
but there are infinitely
many rational
x, y such that 
- Show your reasoning.
-
-
-
- problem #236 - posted
wednesday, dec
14, 2005
- Gimme an Integer! (back to top)
- Find all positive real numbers
c such that the following is an integer.
- Show your reasoning.
-
-
-
- problem #237 - posted
tuesday, dec
27, 2005
- Nearly Integers! (back to top)
- Show that for any positive
integer n, the number
(1 + \/2)^n is
- less than 1/ 2^n away
from the nearest integer. \/2 means square root of 2.
- Show your reasoning.
-
-
-
- problem #238 - posted
monday, jan
9, 2006
- Cube It Together! (back to top)
- a) (warmup) Start with a 2" by 6"
rectangle. Say how to cut it up in
the fewest number
- of pieces and rearrange into
a 3" by 4" rect. b)
Start with a block 8cm by 8cm by 27cm.
- Find the fewest number of
pieces to cut it into to rearrange
into a 12 by 12 by 12 cube.
- Explain precisely how to
do it. Show your reasoning.
(Dan's note: I actually did this, with
cheese!)
-
-
-
- problem #239 - posted
thursday, jan
26, 2006
- The 3, 4, 5 . . . Square ? (back
to top)
- I recently got this e-mail
question sent to me: You are sitting inside a big square
- painted on the floor.
One corner is three
meters (3m) from you, another
corner
- is 4m away from you, and another is 5m away from you. How
big is the square?
- a) Is the problem solvable? . b)
Is the solution unique? .
- c) How many solutions are
there, and what are they? . . Show
your reasoning.
-
-
-
- problem #240 - posted
sunday, feb
5, 2006
: The Pies Have It :
(back
to top)
- Using only the number pi
(), the symbols for addition, multiplication,
square root (
),
- parentheses ( ), and greatest
integer brackets [ ],
but no other symbols or operations,
- construct each of the integers 1 through 10, using
as few total 's as you can.
- [n] is the largest integer less than
or equal to n. Show your reasoning.
-
-
-
- THANKS to all of you who
have entered, or even just clicked and looked.
- My website is now in its
ninth season - over 76,000 hits
so far! (Not factorial.)
- Help it grow by telling
your friends, teachers, and family about it.
- YOU CAN ALWAYS FIND ME AT dansmath.com - Dan the Man Bach
- 2006 A.D.
-
-
-
- Problem Archives
Index
-
- Probs &
answers . 1-10
. 11-20
. 21-30
. 31-40
. 41-50
. 51-60
. 61-70
. 71-80
. 81-90
. 91-100
- Problems only . .
. 1-10
. 11-20
. 21-30
. 31-40
. 41-50
. 51-60
. 61-70
. 71-80
. 81-90
. 91-100
- Probs &
answers . 101-110
. 111-120
. 121-130
. 131-140
. 141-150
. 151-160
. 161-170
. 171-180
- Problems only . .
. 101-110 . 111-120
. 121-130 .
131-140 . 141-150
. 151-160 . 161-170
. 171-180
- Probs &
answers . 181-190 . 191-200
. 201-210
. 211-220
. 221-230
. 231-240
. 241-250
. 251+
- Problems only . . . . 181-190 . 191-200 . 201-210
. 211-220 . 221-230
. 231-240 . 241-250 . 251+
-
- Browse the complete
problem list, check out the weekly
leader
board,
- or go back and work on
this week's problem!
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