The 74th William Lowell Putnam Mathematical Competition
Saturday, 7 December 2013A1Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all
![$20$ $20$](https://dxdy-03.korotkov.co.uk/f/e/e/0/ee070bffef288cab28aad0517a35741b82.png)
integers is
![$39.$ $39.$](https://dxdy-01.korotkov.co.uk/f/8/b/f/8bf7cd3af993cbbcf73e5c73db2af38382.png)
Show that there are two faces that share a vertex and have the same integer written on them.
A2Let
![$S$ $S$](https://dxdy-03.korotkov.co.uk/f/e/2/5/e257acd1ccbe7fcb654708f1a866bfe982.png)
be the set of all positive integers that are
not perfect squares. For
![$n$ $n$](https://dxdy-02.korotkov.co.uk/f/5/5/a/55a049b8f161ae7cfeb0197d75aff96782.png)
in
![$S,$ $S,$](https://dxdy-02.korotkov.co.uk/f/1/6/1/161a3847e7eda2430d4ebb273364b3e982.png)
consider choices of integers
![$a_1,a_2,\dots, a_r$ $a_1,a_2,\dots, a_r$](https://dxdy-02.korotkov.co.uk/f/5/7/f/57f636ba65e24156840b84bef1e7c97c82.png)
such that
![$n<a_1<a_2<\cdots<a_r$ $n<a_1<a_2<\cdots<a_r$](https://dxdy-03.korotkov.co.uk/f/2/3/3/23328344e662140f92bc4dcc9d49acbf82.png)
and
![$n\cdot a_1\cdot a_2\cdots a_r$ $n\cdot a_1\cdot a_2\cdots a_r$](https://dxdy-01.korotkov.co.uk/f/c/5/4/c5452723b7b8347e4eb41c500a5c4eec82.png)
is a perfect square, and let
![$f(n)$ $f(n)$](https://dxdy-04.korotkov.co.uk/f/3/d/4/3d425a215e8eeb2a056f553633aaae4a82.png)
be the minimum of
![$a_r$ $a_r$](https://dxdy-02.korotkov.co.uk/f/d/8/3/d83219126893d166356367471d2327fe82.png)
over all such choices. For example,
![$2\cdot 3\cdot 6$ $2\cdot 3\cdot 6$](https://dxdy-01.korotkov.co.uk/f/4/4/c/44cf4aa5a356c1b1bab9fad9358be3ce82.png)
is a perfect square, while
![$2\cdot 3\cdot 5, 2\cdot 4\cdot 5,$ $2\cdot 3\cdot 5, 2\cdot 4\cdot 5,$](https://dxdy-03.korotkov.co.uk/f/e/2/c/e2c9095ba3a7c5f2da5cfd59ea8fb73c82.png)
and
![$2\cdot 3\cdot 4\cdot 5$ $2\cdot 3\cdot 4\cdot 5$](https://dxdy-01.korotkov.co.uk/f/8/e/9/8e999e2f137bf2a66c71b0949586bf5882.png)
are not, and so
![$f(2)=6.$ $f(2)=6.$](https://dxdy-03.korotkov.co.uk/f/6/a/1/6a17f30c67108da217afbd0bd4a20abb82.png)
Show that the function
![$f$ $f$](https://dxdy-02.korotkov.co.uk/f/1/9/0/190083ef7a1625fbc75f243cffb9c96d82.png)
from
![$S$ $S$](https://dxdy-03.korotkov.co.uk/f/e/2/5/e257acd1ccbe7fcb654708f1a866bfe982.png)
to the integers is one-to-one.
A3Suppose that the real numbers
![$a_0,a_1,\dots,a_n$ $a_0,a_1,\dots,a_n$](https://dxdy-02.korotkov.co.uk/f/d/d/b/ddb709244b56c01eeff875e0445d208d82.png)
and
![$x,$ $x,$](https://dxdy-04.korotkov.co.uk/f/3/8/0/380aab7befb490c9e8b8027e557ed54582.png)
with
![$0<x<1,$ $0<x<1,$](https://dxdy-04.korotkov.co.uk/f/f/4/9/f491553079916b775a0995127090768382.png)
satisfy
![\[\frac{a_0}{1-x}+\frac{a_1}{1-x^2}+\cdots+\frac{a_n}{1-x^{n+1}}=0.\] \[\frac{a_0}{1-x}+\frac{a_1}{1-x^2}+\cdots+\frac{a_n}{1-x^{n+1}}=0.\]](https://dxdy-01.korotkov.co.uk/f/8/7/2/872201219ea79501dff5b794ab3ed26182.png)
Prove that there exists a real number
![$y$ $y$](https://dxdy-02.korotkov.co.uk/f/d/e/c/deceeaf6940a8c7a5a02373728002b0f82.png)
with
![$0<y<1$ $0<y<1$](https://dxdy-01.korotkov.co.uk/f/c/2/5/c25934d47e7c7b309473aceaf16a824582.png)
such that
A4A finite collection of digits
![$0$ $0$](https://dxdy-03.korotkov.co.uk/f/2/9/6/29632a9bf827ce0200454dd32fc3be8282.png)
and
![$1$ $1$](https://dxdy-01.korotkov.co.uk/f/0/3/4/034d0a6be0424bffe9a6e7ac9236c0f582.png)
is written around a circle. An
arc of length
![$L\ge 0$ $L\ge 0$](https://dxdy-01.korotkov.co.uk/f/8/4/4/8446bd354507f2cc68c918447b0117d782.png)
consists of
![$L$ $L$](https://dxdy-02.korotkov.co.uk/f/d/d/c/ddcb483302ed36a59286424aa5e0be1782.png)
consecutive digits around the circle. For each arc
![$w,$ $w,$](https://dxdy-03.korotkov.co.uk/f/a/6/b/a6b9ce566687f1727adfb5d93e3f724382.png)
let
![$Z(w)$ $Z(w)$](https://dxdy-03.korotkov.co.uk/f/e/d/a/eda892749db9f8bbcb33d3f8fc83d0f482.png)
and
![$N(w)$ $N(w)$](https://dxdy-03.korotkov.co.uk/f/6/7/5/675a5201f3ea76f4a058c3ec1911574b82.png)
denote the number of
![$0$ $0$](https://dxdy-03.korotkov.co.uk/f/2/9/6/29632a9bf827ce0200454dd32fc3be8282.png)
's in
![$w$ $w$](https://dxdy-04.korotkov.co.uk/f/3/1/f/31fae8b8b78ebe01cbfbe2fe5383262482.png)
and the number of
![$1$ $1$](https://dxdy-01.korotkov.co.uk/f/0/3/4/034d0a6be0424bffe9a6e7ac9236c0f582.png)
's in
![$w,$ $w,$](https://dxdy-03.korotkov.co.uk/f/a/6/b/a6b9ce566687f1727adfb5d93e3f724382.png)
respectively. Assume that
![$|Z(w)-Z(w')|\le 1$ $|Z(w)-Z(w')|\le 1$](https://dxdy-01.korotkov.co.uk/f/8/7/8/878fbfa88bc00e8a6b57953a6f847fea82.png)
for any two arcs
![$w,w'$ $w,w'$](https://dxdy-03.korotkov.co.uk/f/a/c/9/ac92a6224f0eec17e5e38ae2bae88d4c82.png)
of the same length. Suppose that some arcs
![$w_1,\dots,w_k$ $w_1,\dots,w_k$](https://dxdy-03.korotkov.co.uk/f/6/f/8/6f8b3070b288f4f3a3ab1b17230854ec82.png)
have the property that
![\[Z=\frac1k\sum_{j=1}^kZ(w_j)\text{ and }N=\frac1k\sum_{j=1}^k N(w_j)\] \[Z=\frac1k\sum_{j=1}^kZ(w_j)\text{ and }N=\frac1k\sum_{j=1}^k N(w_j)\]](https://dxdy-04.korotkov.co.uk/f/f/8/c/f8c34e794a71eb71db1321466536651582.png)
are both integers. Prove that there exists an arc
![$w$ $w$](https://dxdy-04.korotkov.co.uk/f/3/1/f/31fae8b8b78ebe01cbfbe2fe5383262482.png)
with
![$Z(w)=Z$ $Z(w)=Z$](https://dxdy-03.korotkov.co.uk/f/a/6/9/a6948dcf404c3f8b191206b5b14716ae82.png)
and
A5For
![$m\ge 3,$ $m\ge 3,$](https://dxdy-04.korotkov.co.uk/f/f/d/0/fd0cd7c44bea124001889f3a0bf437a482.png)
a list of
![$\binom m3$ $\binom m3$](https://dxdy-03.korotkov.co.uk/f/a/6/a/a6a7ad4e4c86c61e0a515973b01ddeca82.png)
real numbers
![$(1\le i<j<k\le m)$ $(1\le i<j<k\le m)$](https://dxdy-02.korotkov.co.uk/f/d/1/f/d1fcb1c03f36e21fdb51902d85f197a282.png)
is said to be
area definite for
![$\mathbb{R}^n$ $\mathbb{R}^n$](https://dxdy-01.korotkov.co.uk/f/8/a/8/8a86f4a11e2fbfc03de61d587ba826de82.png)
if the inequality
![\[\sum_{1\le i<j<k\le m}a_{ijk}\text{Area}(\triangle A_iA_jA_k)\ge0\] \[\sum_{1\le i<j<k\le m}a_{ijk}\text{Area}(\triangle A_iA_jA_k)\ge0\]](https://dxdy-03.korotkov.co.uk/f/2/1/6/216b08ae5cff2bae0689790c8240a77f82.png)
holds for every choice of
![$m$ $m$](https://dxdy-01.korotkov.co.uk/f/0/e/5/0e51a2dede42189d77627c4d742822c382.png)
points
![$A_1,\dots,A_m$ $A_1,\dots,A_m$](https://dxdy-02.korotkov.co.uk/f/d/8/9/d8908ddfe9fae4b5020eb1982ff853ae82.png)
in
![$\mathbb{R}^n.$ $\mathbb{R}^n.$](https://dxdy-02.korotkov.co.uk/f/5/c/a/5ca4008c054183e5c89dea5de9bc10cb82.png)
For example, the list of four number
![$a_{123}=a_{124}=a_{134}=1, a_{234}=-1$ $a_{123}=a_{124}=a_{134}=1, a_{234}=-1$](https://dxdy-02.korotkov.co.uk/f/5/3/5/5353204028ba5d84bc479e4a5fd1fa1c82.png)
is area definite for
![$\mathbb{R}^2.$ $\mathbb{R}^2.$](https://dxdy-01.korotkov.co.uk/f/c/4/4/c44251231a908328644c708ebe6665e382.png)
Prove that if a list of
![$\binom m3$ $\binom m3$](https://dxdy-03.korotkov.co.uk/f/a/6/a/a6a7ad4e4c86c61e0a515973b01ddeca82.png)
numbers is area definite for
![$\mathbb{R}^2,$ $\mathbb{R}^2,$](https://dxdy-02.korotkov.co.uk/f/5/b/b/5bb37cd0f726d42f8de3f84cdae4c03782.png)
then it is area definite for
A6Define a function
![$w:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ $w:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$](https://dxdy-04.korotkov.co.uk/f/3/c/2/3c2a57716d7752a6ff8a76e57f8d794982.png)
as follows. For
![$|a|,|b|\le 2,$ $|a|,|b|\le 2,$](https://dxdy-02.korotkov.co.uk/f/d/7/8/d7845431c02d09c4adcbea0e99c7617482.png)
let
![$w(a,b)$ $w(a,b)$](https://dxdy-04.korotkov.co.uk/f/3/1/6/3168b2fad56bc9e8b0c823312499d9f682.png)
be as in the table shown; otherwise, let
![$w(a,b)=0.$ $w(a,b)=0.$](https://dxdy-03.korotkov.co.uk/f/e/8/0/e808480a5a78b501fb6d8890ccb0def982.png)
![\[\begin{array}{|lr|rrrrr|}\hline &&&&b&&\\
&w(a,b)&-2&-1&0&1&2\\ \hline
&-2&-1&-2&2&-2&-1\\
&-1&-2&4&-4&4&-2\\
a&0&2&-4&12&-4&2\\
&1&-2&4&-4&4&-2\\
&2&-1&-2&2&-2&-1\\ \hline\end{array}\] \[\begin{array}{|lr|rrrrr|}\hline &&&&b&&\\
&w(a,b)&-2&-1&0&1&2\\ \hline
&-2&-1&-2&2&-2&-1\\
&-1&-2&4&-4&4&-2\\
a&0&2&-4&12&-4&2\\
&1&-2&4&-4&4&-2\\
&2&-1&-2&2&-2&-1\\ \hline\end{array}\]](https://dxdy-03.korotkov.co.uk/f/e/f/a/efaa523f3d6efa1d9f8ae65b52fedc9382.png)
For every finite subset
![$S$ $S$](https://dxdy-03.korotkov.co.uk/f/e/2/5/e257acd1ccbe7fcb654708f1a866bfe982.png)
of
![$\mathbb{Z}\times\mathbb{Z},$ $\mathbb{Z}\times\mathbb{Z},$](https://dxdy-02.korotkov.co.uk/f/5/b/5/5b5cb76465f09770e3fe002e998bd21f82.png)
define
![\[A(S)=\sum_{(\mathbf{s},\mathbf{s'})\in S\times S} w(\mathbf{s}-\mathbf{s'}).\] \[A(S)=\sum_{(\mathbf{s},\mathbf{s'})\in S\times S} w(\mathbf{s}-\mathbf{s'}).\]](https://dxdy-03.korotkov.co.uk/f/e/2/1/e21d22f9de37aa667ff8853119e1309f82.png)
Prove that if
![$S$ $S$](https://dxdy-03.korotkov.co.uk/f/e/2/5/e257acd1ccbe7fcb654708f1a866bfe982.png)
is any finite nonempty subset of
![$\mathbb{Z}\times\mathbb{Z},$ $\mathbb{Z}\times\mathbb{Z},$](https://dxdy-02.korotkov.co.uk/f/5/b/5/5b5cb76465f09770e3fe002e998bd21f82.png)
then
![$A(S)>0.$ $A(S)>0.$](https://dxdy-02.korotkov.co.uk/f/d/b/5/db5b02d310267a1c0780693e03a1d25e82.png)
(For example, if
![$S=\{(0,1),(0,2),(2,0),(3,1)\},$ $S=\{(0,1),(0,2),(2,0),(3,1)\},$](https://dxdy-02.korotkov.co.uk/f/d/5/c/d5c0c47688e01c817cffc55cbcb8084f82.png)
then the terms in
![$A(S)$ $A(S)$](https://dxdy-01.korotkov.co.uk/f/8/e/c/8ec314dc3db66d32ad78cda06122b2a982.png)
are
![$12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4.$ $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4.$](https://dxdy-04.korotkov.co.uk/f/f/4/9/f49605bd5cc057c66af86312f0decabc82.png)
)
B1For positive integers
![$n,$ $n,$](https://dxdy-01.korotkov.co.uk/f/8/5/b/85bc1f723bdc744666d4f2241b1031f782.png)
let the numbers
![$c(n)$ $c(n)$](https://dxdy-02.korotkov.co.uk/f/d/d/5/dd5b89823b96df7204768b86a4133a7182.png)
be determined by the rules
![$c(1)=1,c(2n)=c(n),$ $c(1)=1,c(2n)=c(n),$](https://dxdy-03.korotkov.co.uk/f/6/4/9/6493bf73703f69994d9dba83bda9a8da82.png)
and
![$c(2n+1)=(-1)^nc(n).$ $c(2n+1)=(-1)^nc(n).$](https://dxdy-04.korotkov.co.uk/f/b/5/0/b503320e46434da021bbd8f5789d71fa82.png)
Find the value of
B2Let
![$C=\bigcup_{N=1}^{\infty}C_N,$ $C=\bigcup_{N=1}^{\infty}C_N,$](https://dxdy-03.korotkov.co.uk/f/2/4/0/240dbccceed66464771f214497fbbd9682.png)
where
![$C_N$ $C_N$](https://dxdy-03.korotkov.co.uk/f/e/1/c/e1ce25b9f4788aeee0e7414ae72f77b982.png)
denotes the set of `cosine polynomials' of the form
![\[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\] \[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\]](https://dxdy-03.korotkov.co.uk/f/2/b/3/2b3220cc4a0cbfbb119dafb36d75ecd382.png)
for which:
(i)
![$f(x)\ge 0$ $f(x)\ge 0$](https://dxdy-01.korotkov.co.uk/f/c/f/f/cff0438e21bf8471d89d528cf381c6be82.png)
for all real
![$x,$ $x,$](https://dxdy-04.korotkov.co.uk/f/3/8/0/380aab7befb490c9e8b8027e557ed54582.png)
and
(ii)
![$a_n=0$ $a_n=0$](https://dxdy-04.korotkov.co.uk/f/f/f/c/ffc0153792a656bf90cfc32ce7117ac782.png)
whenever
![$n$ $n$](https://dxdy-02.korotkov.co.uk/f/5/5/a/55a049b8f161ae7cfeb0197d75aff96782.png)
is a multiple of
![$3.$ $3.$](https://dxdy-04.korotkov.co.uk/f/f/6/3/f635d39fb9a992e9e9dd82c94a5bca1c82.png)
Determine the maximum value of
![$f(0)$ $f(0)$](https://dxdy-02.korotkov.co.uk/f/d/0/8/d0803fa2e33abc6c4023640ac5a40f3c82.png)
as
![$f$ $f$](https://dxdy-02.korotkov.co.uk/f/1/9/0/190083ef7a1625fbc75f243cffb9c96d82.png)
ranges through
![$C,$ $C,$](https://dxdy-02.korotkov.co.uk/f/1/0/e/10e618f4e1d9c6045e850f882c07fdfd82.png)
and prove that this maximum is attained.
B3Let
![$P$ $P$](https://dxdy-02.korotkov.co.uk/f/d/f/5/df5a289587a2f0247a5b97c1e8ac58ca82.png)
be a nonempty collections of subsets of
![$\{1,\dots,n\}$ $\{1,\dots,n\}$](https://dxdy-04.korotkov.co.uk/f/f/8/4/f84dbbcc67018dffbd9a2e01dc012eb582.png)
such that:
(i) if
![$S,S'\in P,$ $S,S'\in P,$](https://dxdy-03.korotkov.co.uk/f/a/0/1/a01428d28a17e3461d402aa2fbca977982.png)
then
![$S\cup S'\in P$ $S\cup S'\in P$](https://dxdy-03.korotkov.co.uk/f/6/d/7/6d7aeef63d609b254f27d402d513d3c682.png)
and
![$S\cap S'\in P,$ $S\cap S'\in P,$](https://dxdy-01.korotkov.co.uk/f/c/a/8/ca84d6d240c02170f9cb017df1cb982382.png)
and
(ii) if
![$S\in P$ $S\in P$](https://dxdy-02.korotkov.co.uk/f/d/f/c/dfc9c4c620b528a2f83bd1876968ae8a82.png)
and
![$S\ne\emptyset,$ $S\ne\emptyset,$](https://dxdy-02.korotkov.co.uk/f/1/2/3/12336de1c3f4f62223f5fd43279c2e4a82.png)
then there is a subset
![$T\subset S$ $T\subset S$](https://dxdy-02.korotkov.co.uk/f/5/a/8/5a8b1c5d9a869052bd9bcdf40a422aa082.png)
such that
![$T\in P$ $T\in P$](https://dxdy-03.korotkov.co.uk/f/2/7/b/27bbd1aed83ad0c7b17c97b2e665a88982.png)
and
![$T$ $T$](https://dxdy-03.korotkov.co.uk/f/2/f/1/2f118ee06d05f3c2d98361d9c30e38ce82.png)
contains exactly one fewer element than
![$S.$ $S.$](https://dxdy-01.korotkov.co.uk/f/4/c/a/4ca775844092d020a10ba185bf3ce4fd82.png)
Suppose that
![$f:P\to\mathbb{R}$ $f:P\to\mathbb{R}$](https://dxdy-02.korotkov.co.uk/f/1/a/0/1a0a1cab3893dbebe4922a85f68d9bd682.png)
is a function such that
![$f(\emptyset)=0$ $f(\emptyset)=0$](https://dxdy-02.korotkov.co.uk/f/d/8/5/d850e94e4c0f8e8680ba8b05d6a90e6b82.png)
and
![\[f(S\cup S')= f(S)+f(S')-f(S\cap S')\text{ for all }S,S'\in P.\] \[f(S\cup S')= f(S)+f(S')-f(S\cap S')\text{ for all }S,S'\in P.\]](https://dxdy-02.korotkov.co.uk/f/1/f/a/1fa1ebccae0dfa067a421aec6fead51882.png)
Must there exist real numbers
![$f_1,\dots,f_n$ $f_1,\dots,f_n$](https://dxdy-02.korotkov.co.uk/f/1/6/9/169ee93c75052fad8e9a4fd78d21baae82.png)
such that
![\[f(S)=\sum_{i\in S}f_i\] \[f(S)=\sum_{i\in S}f_i\]](https://dxdy-02.korotkov.co.uk/f/d/1/f/d1f64bdabeb8d3d583a1ef7b4309e2f082.png)
for every
B4For any continuous real-valued function
![$f$ $f$](https://dxdy-02.korotkov.co.uk/f/1/9/0/190083ef7a1625fbc75f243cffb9c96d82.png)
defined on the interval
![$[0,1],$ $[0,1],$](https://dxdy-01.korotkov.co.uk/f/0/a/e/0ae2faac7f0b6f04f2d00a80b1a93c9582.png)
let
![\[\mu(f)=\int_0^1f(x)\,dx,\text{Var}(f)=\int_0^1(f(x)-\mu(f))^2\,dx, M(f)=\max_{0\le x\le 1}|f(x)|.\] \[\mu(f)=\int_0^1f(x)\,dx,\text{Var}(f)=\int_0^1(f(x)-\mu(f))^2\,dx, M(f)=\max_{0\le x\le 1}|f(x)|.\]](https://dxdy-02.korotkov.co.uk/f/5/f/a/5fa57038ca2ad58651a973a22921c4f782.png)
Show that if
![$f$ $f$](https://dxdy-02.korotkov.co.uk/f/1/9/0/190083ef7a1625fbc75f243cffb9c96d82.png)
and
![$g$ $g$](https://dxdy-04.korotkov.co.uk/f/3/c/f/3cf4fbd05970446973fc3d9fa3fe3c4182.png)
are continuous real-valued functions defined on the interval
![$[0,1],$ $[0,1],$](https://dxdy-01.korotkov.co.uk/f/0/a/e/0ae2faac7f0b6f04f2d00a80b1a93c9582.png)
then
B5Let
![$X=\{1,2,\dots,n\},$ $X=\{1,2,\dots,n\},$](https://dxdy-04.korotkov.co.uk/f/3/b/4/3b470bf6b2acefd4e8be4acea073014982.png)
and let
![$k\in X.$ $k\in X.$](https://dxdy-03.korotkov.co.uk/f/2/0/f/20fd50bf7f096e6ba3bd373db1411a5682.png)
Show that there are exactly
![$k\cdot n^{n-1}$ $k\cdot n^{n-1}$](https://dxdy-04.korotkov.co.uk/f/7/2/c/72cd1788e447e2600443563e300b1bea82.png)
functions
![$f:X\to X$ $f:X\to X$](https://dxdy-02.korotkov.co.uk/f/9/6/5/965611404e0fc4c2dcefbedf599fa9dc82.png)
such that for every
![$x\in X$ $x\in X$](https://dxdy-04.korotkov.co.uk/f/f/6/c/f6c4299940a91895633082ceb4df9c1982.png)
there is a
![$j\ge 0$ $j\ge 0$](https://dxdy-03.korotkov.co.uk/f/a/d/f/adf2e834e109ba3dbd0dde3a54205bc482.png)
such that
![$f^{(j)}(x)\le k.$ $f^{(j)}(x)\le k.$](https://dxdy-03.korotkov.co.uk/f/e/a/4/ea4fbe5c3fdb41a7b543e471eff7bf0982.png)
[Here
![$f^{(j)}$ $f^{(j)}$](https://dxdy-01.korotkov.co.uk/f/c/c/e/ccee8b011171022d4ea5ba36793d998d82.png)
denotes the
![$j$ $j$](https://dxdy-04.korotkov.co.uk/f/3/6/b/36b5afebdba34564d884d347484ac0c782.png)
th iterate of
![$f,$ $f,$](https://dxdy-03.korotkov.co.uk/f/6/f/6/6f60e7714a307b2095b24945797a63b482.png)
so that
![$f^{(0)}(x)=x$ $f^{(0)}(x)=x$](https://dxdy-02.korotkov.co.uk/f/5/c/1/5c14c658c88a1a46a88d9aad0ab0c16882.png)
and
![$f^{(j+1)}(x)=f\left(f^{(j)}(x)\right).$ $f^{(j+1)}(x)=f\left(f^{(j)}(x)\right).$](https://dxdy-01.korotkov.co.uk/f/4/8/b/48bcf58b1fbe9d2d6f9eeb865365d6c482.png)
]
B6Let
![$n\ge 1$ $n\ge 1$](https://dxdy-01.korotkov.co.uk/f/c/8/f/c8f1b9fc9504b4c36930135e7ff3161382.png)
be an odd integer. Alice and Bob play the following game, taking alternating turns, with Alice playing first. The playing area consists of
![$n$ $n$](https://dxdy-02.korotkov.co.uk/f/5/5/a/55a049b8f161ae7cfeb0197d75aff96782.png)
spaces, arranged in a line. At each turn, a player either
![\textbullet \textbullet](https://dxdy-04.korotkov.co.uk/f/3/9/5/395bacfb8ee603f116de32b46228219182.png)
places a stone in an empty space, or
![\textbullet \textbullet](https://dxdy-04.korotkov.co.uk/f/3/9/5/395bacfb8ee603f116de32b46228219182.png)
removes a stone from a nonempty space
![$s,$ $s,$](https://dxdy-04.korotkov.co.uk/f/b/2/e/b2e4b727b722462631879a1b313d8d2682.png)
places a stone in the nearest empty space to the left of
![$s$ $s$](https://dxdy-03.korotkov.co.uk/f/6/f/9/6f9bad7347b91ceebebd3ad7e6f6f2d182.png)
(if such a space exists), and places a stone in the nearest empty space to the right of
![$s$ $s$](https://dxdy-03.korotkov.co.uk/f/6/f/9/6f9bad7347b91ceebebd3ad7e6f6f2d182.png)
(if such a space exists).
Furthermore, a move is permitted only if the resulting position has not occurred previously in the game. A player loses if he or she is unable to move. Assuming that both players play optimally throughout the game, what moves may Alice make on her first turn?
-- Пн дек 09, 2013 20:25:41 --Список городов, где проходило зеркало:
http://putnam.ho.ua/schedule.htmlНекоторые фото:
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