\ Section {title} Chinese college entrance math finale
\ {the begin the enumerate}
\ Item 08 Effect of Jiangxi college entrance examination
\ Item 08 college entrance examination in Beijing in
\ item (2017 Tientsin) provided $ a \ in \ mathbb {Z } $, it is known in the definition of $ \ mathbb {R} function $ f (x) $ on = 2x ^ 4 + 3x ^ 3-3x ^ 2 -6x + a $ has a zero in the interval $ $ $ x_0 (2) the $, $ g (x) $ is $ f (x) $ derivative function.
\} the enumerate the begin {
\ $ G Item request ( X) $ monotonic interval;
\ Item provided $ m \ in [1, x_0) \ cup (x_0,2] $, the function $ h (x) = g (x) (m-x_0) -f (m) $, Prove: $ h (m ) h (x_0) <0 $;
\ item Prove: the presence of a constant greater than $ 0 $ $ A $, such that for any positive integer $ p, q $, and $ \ frac {p} {q } \ in [1, x_0) \ cup (x_0,2] $ meet $ \ left | \ FRAC {P} {Q} -x_0 \ right |. \ GEQ \ FRAC. 1 {{}}. 4 ^ Aq obtained $
\ End {} the enumerate
\ Item 1999 college entrance examination roll
\ Item 2003 college entrance examination three-dimensional geometry problems
\ Item 08 Guangdong college entrance examination
\ item (2003 entrance Jiangsu title finale volume) provided $ a> 0 $, as shown, a known line $ l: y = ax $ curve and $ C: 2 $, points on the $ C $ y = x ^ $ Q_1 the abscissa $ $ a_1 \, (0 <a_1 <a) $. Q_n from a point on the $ $ C $ \, (n \ geq 1) $ $ straight line parallel to the axis X $, $ L deposit $ linear at a point $ P_ {n + 1} $, and from the point $ P_ {n + 1} $ rectilinearly parallel to $ Y $ axis, cross curve $ C $ at a point $ Q_ {n + 1} $ . $ Q_n \ , (n = 1,2,3, \ cdots ) $ abscissa the number of columns constituting $ \ {A_N \} $.
\} the enumerate the begin {
\ $ A_ Item find the relationship {n + 1} $ a $ and $ A_N , and determining the $ \ {a_n \} $ of term formula;
\ item when $ a = 1, a_1 \ leq \ frac {1} {2} $ , the prover $ \ sum_ {k = 1} ^ {n} (a_k-a_ {k + 1 }) a_ {k + 2} <\ frac {1} {32} $;
\ item when $ a = 1 $, prove $ \ sum_ {k = 1} ^ {n} (a_k-a_ {k + 1}) a_ {k + 2} <\ frac {1} {3} $.
\ end {enumerate}
\ item (2010 National 2 derivative) provided function $ F (X) =. 1-E ^ {- X} $.
\ the begin {the enumerate}
\ Item show: when $ x> -1 $, $ f (x) \ geq \ frac {x} { x + 1} $;
\ item provided when $ x \ geq 0 $ time, $ f (x) \ leq \ frac {x} {ax + 1} $, $ A $ find the range.
\ the enumerate End {}
\ item (2014 National 2 derivative) known function F $ (X) = ^ E ^ {XE - X} $ -2X.
\ the enumerate the begin {}
\ Item discussed $ f (x) $ Monotonicity;
\ Item provided $ g (x) = f (2x) -4bf (x) $, when the maximum value of $ x> 0 $ time, $ g (x)> 0 $, $ b $ to find;
\ item known $ 1.4142 <\ sqrt {2} <1.4143 $, estimated $ \ ln 2 $ approximation (from 0.001 to the nearest $ $).
\ the enumerate End {}
\ item (2013 years Anhui Mathematics for Science) provided function $ f_n (x) = - 1 + x + \ frac {x ^ 2} {2 ^ 2} + \ frac {x ^ 3} {3 ^ 2} + \ cdots + \ frac {x ^ n} {n ^ 2} \, (x \ in \ mathbb {R}, n \ in \ mathbb {N} _ +) $, proved:
\} the enumerate the begin {
\ $ for each n-Item \ in \ mathbb {N} _ + $, there exists a unique $ x_n \ in \ left [\ frac {2} {3}, 1 \ right] $, satisfying $ f_n (x_n) = 0 $ ;
\ item for any $ p \ in \ mathbb {N } _ + $, by the (1) $ x_n $ constituting columns $ \ {x_n \} $ satisfies $ 0 <x_n-x_ {n + p} <\ frac {1 n-$ {}}.
\ the enumerate End {}
\ item (2014 years Anhui Mathematics for Science) provided real $ c> 0 $, integer $ P>. 1, n-\ in \ mathbb {N} ^ + $.
\ the begin {the enumerate}
\ Item proof: when $ x> 1, x \ neq when 0 $, $ (1 + x ) ^ p> 1 + px $;
\ item column number $ \ {a_n \} $ satisfy $ a_1> c ^ {\ frac {1} {p}}, a_ {n + 1} = \ frac {p-1} {p} a_n + \ frac {c} {p} a_n ^ {1- p} $, proved:. $ A_N> A_. 1} + {n-> {C ^ \ FRAC. 1 {{}}} $ P
\ the enumerate End {}
\ item (2013 Nian Anhui Science Mathematics) Department of Mathematics university plans to hold a different one for each of the main themes of psychological testing activities on Saturday and Sunday respectively responsible for Li and Zhang, known to the department a total of $ n $ student each activity required of the system $ k $ students participate ($ n $ and $ k $ are fixed positive integer), assuming that Li and Zhang were notified of information on their activities independently, randomly distributed to the Department of $ k $ students and the sent messages can be received, the department received a record number of student activity information Li or Zhang issued by the X-$ to $.
\ {the begin the enumerate}
\ Item ask the student armor Zhang Li received or issued by the probability of event notification information;
\ seeking to make item $ P (X = m) $ obtain the maximum value of the integer m $ $.
\ the enumerate End {}
\ Item (2010 Effect of Jiangxi)
\ item (2010 Jiangsu) known $ \ triangle ABC $ trilateral rational length.
prove:
\ the enumerate the begin {}
\ Item $ \ A $ COS is a rational number;
\ Item $ \ $ COS rational numbers nA.
\ End {enumerate}
\ item (2011 Zhejiang ') provided function $ F (X) = (XA) ^ 2 \ LN X, A \ in \ mathbb {R & lt} $.
\ the begin {the enumerate}
\ Item if $ x = e $ to $ y = f (x) $ extreme point, $ a $ realistic number;
\ item number realistic range of $ A $, such that for any $ x \ in (0,3a] $ , constancy $ f (x) \ leq 4e ^ 2 $ established Note:. $ e $ is the natural the base of the logarithm.
\ the enumerate End {}
\ Item (2018 Zhejiang ')
\ Item (2009 Effect of Jiangxi)
\ Item (Jiangsu 2004) known function $ f (x) \, ( x \ in \ mathbb {R}) $ satisfies the following condition: an arbitrary real number $ x_1, x_2 $ both
\ [\ lambda (x_1- x_2) ^ 2 \ leq (x_1 -x_2) [f (x_1) -f (x_2)] \]
and
\ [| f (x_1) -f (x_2) | x_1-x_2 | | \ leq, \]
where $ \ the lambda $ is greater than $ 0 $ constant. set of real numbers $ a_0, a, b $ satisfy $ f (a_0) = 0 $ and $ B = A- \ the lambda F (a) $.
\ the begin {the enumerate}
\ Item proof $ \ lambda \ leq 1 $, and not $ b_0 \ neq a_0 $ exists, such that $ f (b_0) = 0 $ ;
\item 证明$(b-a_0)^2\leq (1-\lambda^2)(a-a_0)^2$;
\ item proved: $ [F (B)] ^ 2 \ Leq (l- \ the lambda ^ 2) [F (A)] ^ 2 $.
\ the enumerate End {}
\ Item (2006 Jiangsu) provided the number of columns $ \ {a_n \}, \ {b_n \}, \ {c_n \} $ satisfy: $ b_n = a_n-a_ {n + 2}, c_n = a_n + 2a_ {n + 1} + 3a_ {n + 2} \, (n = 1,2,3, \ cdots) $.
Demonstrate: $ \ {a_n \} $ necessary and sufficient conditions for the arithmetic sequence is $ \ {c_n \} $ is the arithmetic sequence, and $ b_n \ leq b_ {n + 1} \, (n = 1,2, 3, \ cdots) $. (heard this question a full year, Jiangsu Province, only dozens of candidates get more than half of the scores, fewer than 10 people take out)
\ item (2011 Jiangsu) provided $ M $ is the set portion of the positive integers, the number of columns $ \ {a_n \} $ the first item $ a_1 = 1 $, before $ n and is $ S_n $ $ item is known for any integer $ k \ in M $, when the integer $ n> k $, $ S_ {+ k n} + S_ {nk} = 2 (S_n + S_k) $ are established.
\ the begin {the enumerate}
\ Item provided $ M = \ {1 \} , a_2 = 2 $, $ A_5 evaluation of $;
\ item set $ M = \ {3,4 \} $, find the number of columns $ \ {a_n \} $ of term formula.
\ the enumerate End {}
The subject of conclusions can be generalized to:
Sequence $ \ {a_n \} $ of the cross-$ k_1 prime, k_2 \ in \ mathbb {N } ^ \ ast, k_1> k_2> 0 $ satisfies:
\ [A_ {n-+ k_1} + A_ {n--k_1} = 2a_n \, (n> k_1 ), \ quad a_ {n + k_2} + a_ {n-k_2} = 2a_n \, (n> k_2), \]
at this time series $ \ {a_n \} $ is like the difference between the number of columns.
\ item (2015 Jiangsu) provided $ a_1, a_2, a_3, a_4 $ is a positive number, and the tolerance of $ d \, (d \ neq 0) $ of the arithmetic sequence.
\ the enumerate the begin {}
\ proof Item : $ 2 ^ {a_1}, 2 ^ {a_2}, 2 ^ {a_3}, 2 ^ {a_4} $ sequentially into geometric series;
\ Item if there $ a_1, d $, such that $ a_1, a_2 ^ 2, a_3 ^ 3, a_4 ^ 4 $ sequentially into geometric series, and the reasons;
\ item if there $ a_1, d $ and a positive integer $ n, k $, such that $ a_1 ^ n, a_2 ^ { n + k}, a_3 ^ {n + 2k}, a_4 ^ {n + 3k} $ sequentially into geometric sequence, and justified.
\ the enumerate End {}
\ Item (2012 Anhui) series $ \ {x_n \} $ satisfies $ x_1 = 0, x_ {n + 1} = - x_n + x_n + c \, (n \ in \ mathbb {N} ^ \ ast) $ .
\ the begin {the enumerate}
\ Item proof: $ \ {x_n \} $ is the necessary and sufficient conditions for decreasing the number of columns is $ C <0 $;
\ Item request $ c $ ranges, so $ \ {x_n \} $ is incremented column number.
\ the enumerate End {}
\ item (2010 Guangdong) provided $ A (x_1, y_1), B (x_2, y_2) $ $ plane rectangular coordinate system is two points on the $ xOy, now defined by the point to point $ A $ a $ B $ species fold line distance $ \ rho (a, B) $ is $ p (a, B) = | x_2-x_1 | + | y_2-y_1 |. $ $ xOy plane for a given $ $ different points a ( x_1, Y_1), B (x_2, Y_2) $.
\ the begin {the enumerate}
\ Item If the point $ C (x, y) $ is a point on a plane $ xOy $, prove $ \ rho (a, C) + \ rho (C, B) \ geq \ rho (A, B) $;
\ Item if there point $ C (x, y) in a plane xOy $ $ $ satisfies
\ding{172} $\rho (A,C)+\rho(C,B)\geq \rho(A,B)$; \qquad \ding{173} $\rho (A,C)=\rho(C,B)$.
If there is a request that all eligible point, to be proved.
\ {End} the enumerate
\ Item (2009 Hunan)
\ Item (2015 Guangdong) harmonic series, similar to the 2014 Shaanxi.
\ item (2014 Liaoning) known function $ f (x) = (\ cos xx) (\ pi + 2x) - \ frac {8} {3} (\ sin x + 1) $, $ g (x) . = 3 (x- \ pi) \ cos x-4 (1+ \ sin x) \ ln \ left (3- \ frac {2x} {\ pi} \ right) $
demonstrated:
\ the enumerate the begin {}
\ Item The only present $ x_0 \ in \ left (0 , \ frac {\ pi} {2} \ right) $, so $ f (x_0) = 0 $ ;
\ item exists a unique $ x_1 \ in \ left (\ frac {\ pi} {2}, \ pi \ right) $, so $ g (x_1) = 0 $ , and the pair (1) $ x_0 $ has $ x_1 + x_0 <\ $ PI.
\ the enumerate End {}
\item
\item
\item
\end{enumerate}