(2019 AMC 12B) Define a sequence recursively by $x_0=5$ and[x_{n+1}=frac{x_n^2+5x_n+4}{x_n+6}]for all nonnegative integers $n.$ Let $m$ be the least positive integer such that[x_mleq 4+frac{1}{2^{20}}.]In which of the following intervals does $m$ lie?
$ extbf{(A) } [9,26] qquad extbf{(B) } [27,80] qquad extbf{(C) } [81,242]qquad extbf{(D) } [243,728] qquad extbf{(E) } [729,infty)$
(2019 AMC 12A) A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = frac{3}{7}$, and[a_n=frac{a_{n-2} cdot a_{n-1}}{2a_{n-2} - a_{n-1}}]for all $n geq 3$ Then $a_{2019}$ can be written as $frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q ?$
$ extbf{(A) } 2020 qquad extbf{(B) } 4039 qquad extbf{(C) } 6057 qquad extbf{(D) } 6061 qquad extbf{(E) } 8078$
section{清华大学数学系2020年“大中衔接”研讨与教学活动}
egin{center}
extbf{2020年清华大学“大中衔接”试题}
end{center}
1.设指数$alpha_1,alpha_2,cdots,alpha_n$两两不同,系数$a_1,a_2,cdots,a_n$不全为零.证明:
$$f(x)=a_1x^{alpha_1}+a_2x^{alpha_2}+cdots+a_nx^{alpha_n}$$
在$(0,+infty)$上至多有$n-1$个零点.
2.设$f:[-1,1] o mathbb{R}$是连续函数,证明:
$$lim_{lambda o infty}
left[int_{-1}^{1}frac{|lambda|}{lambda^2+x^2}f(x)mathrm{d}x
ight]=pi f(0).$$
3.设整数$n>1$,证明:至多只有有限多个正整数$a$,使得方程
$$x_1^2+x_2^2+cdots+x_n^2=ax_1x_2cdots x_n$$
有非零整数解.
%letoldwideringwidering
%letwideringundefined
%usepackage{yhmath} %弧AB
%letwideringoldwidering
4.设$A$、$B$、$C$是单位球面$S$上三个不同点且都位于第一象限中,对于任何两个不同点$P,Qin S$,只要$PQ$不是$S$的直径,则平面$OPQ$与$S$的交集是$S$上的一个圆周, $P$、$Q$将此圆分成两段弧,将其中较短的那段弧记为$wideparen{PQ}$.设$wideparen{BC},wideparen{CA},wideparen{AB}$的中点分别为$D$、$E$、$F$.证明: $wideparen{AD},wideparen{BE},wideparen{CF}$经过同一个点.
5.设共有$kgeqslant 4$个不同的字母,可用它们构成单词.给定一族(记为$T$)禁用单词,其中任何两个禁用单词长度不等.称一个单词是“可用的”,如果它不含连续一段字母恰为某禁用单词.证明:至少有$left( frac{k+sqrt{k^2-4k}}{2} ight) ^n$个长为$n$的可用单词.
6.设$n,p$是正整数,集合$A_1,A_2,cdots,A_k$是${1,2,cdots,n}$的子集,满足对任何$i
eq j$都有$|A_iackslash A_j|geqslant p$.证明:
$$
kleqslant frac{left( p-1
ight) !cdot n!}{left( lfloor frac{n+p-1}{2}
floor
ight) !cdot left( lceil frac{n+p-1}{2}
ceil
ight) !}.
$$
section{2020年北京大学金秋营试题}
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extbf{2020年北京大学金秋营}
extbf{第一天}
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1.对于非负实数$a_1,a_2,cdots,a_n$,考虑如下$2^n$个实数
$$i_1a_1+i_2a_2+cdots+i_na_n,$$其中$i_k=pm 1,1leqslant kleqslant n$,记$S$为这$2^n$个数中所有正数之和,在$sum_{i=1}^{n}a_n=1$的条件下,求$S$的最小值.
2.在$ riangle ABC$中, $D$为$BC$, $E,F$分别为$wideparen{BC},wideparen{BAC}$, 取$ riangle ABC$外接圆$omega$, $ riangle ADE$外接圆与射线$AB,AC$交于点$J,K$, $ riangle ADF$外接圆与射线$AB,AC$交于点$L,M$,证明:若$AD$、$JK$、$LM$共点,则$JM$、$LK$交点在$omega$上.
3.数列${a_n}$满足: $a_0=2,a_1=5,a_{n+1}=5a_n-a_{n-1},ngeqslant 1$,已知$a_mmid a_{2n-2}+ a_{2n-1}$,求证: $3mid m,3mid n$.
4.求$k$的最小值,使得将$7 imes 7$方格挖去$k$个格后,剩余图形不存在$T$字形. ($T$字形指一个方格与其相邻的三个方格有公共边构成的图形)
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extbf{第二天}
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5. $ riangle ABC$内部取一点$O$,直线$AO,BO,CO$分别交对边于$D,E,F$,若四边形$AEFO,BDFO,CDEO$都有内切圆,求证: $O$在$ riangle ABC$内心和垂心所在直线上
6.若自然数$n$可以写成若干个自己的不同的因数的和,其中有个为$1$,就称$n$为好数,证明:对任意$m$大于1,存在无穷个$m$的正倍数为好数,且最小的倍数不大于$pm$,其中$p$是$m$最大的奇素因数(若$m$为二的幂,则$p$为$3$)
7.求证: $sum_{r=0}^{p}sum_{t=1}^{p}(-1)^{t-1}C_t^rC_p^tC_{p(p-t)}^{p(k-r)}equiv C_{p^2}^{kp}\,(mod p^p)$ ,其中$p$为素奇数, $kin [1,p]$.
8.求所有的$n$,使得平面上有$n$个完全相同的凸多边形,且满足对任意$k$个凸多边形,所有在它们之中且不在其余多边形中的点的集合为凸多边形(非退化).
(Austrian Regional Competition For Advanced Students 2019) Let $x,y$ be real numbers such that $(x+1)(y+2)=8.$ Prove that$$(xy-10)^2ge 64.$$
$$16=(2x+2)(y+2)le frac{(2x+2+y+2)^2}{4}=frac{(10-xy)^2}{4}$$
(Austrian Regional Competition For Advanced Students 2018) If $a, b$ are positive reals such that $a+b<2$. Prove that$$frac{1}{1+a^2}+frac{1}{1+b^2} le frac{2}{1+ab}$$and determine all $a, b$ yielding equality.
Proposed by Gottfried Perz
Because$$frac{2}{1+ab}-frac{1}{1+a^2}-frac{1}{1+b^2}=frac{(a-b)^2(1-ab)}{(1+ab)(1+a^2)(1+b^2)}geq0.$$
(Austria-Poland 2004 system of equations) Solve the following system of equations in $mathbb{R}$ where all square roots are non-negative:
$$
egin{matrix}
a - sqrt{1-b^2} + sqrt{1-c^2} = d \
b - sqrt{1-c^2} + sqrt{1-d^2} = a \
c - sqrt{1-d^2} + sqrt{1-a^2} = b \
d - sqrt{1-a^2} + sqrt{1-b^2} = c \
end{matrix}
$$
Apply the substitution $a=sinalpha$, ..., where $alpha,...in[-pi/2,pi/2]$.
Summing up (1) and (2) we obtain $sineta-coseta=sindelta-cosdelta$. It follows that $eta=delta$ OR $eta+delta=-pi/2$. The same for $alpha$ and $gamma$. Now it is routine to analyze 3 variants.
(2019 AMC 12B) Define a sequence recursively by $x_0=5$ and[x_{n+1}=frac{x_n^2+5x_n+4}{x_n+6}]for all nonnegative integers $n.$ Let $m$ be the least positive integer such that[x_mleq 4+frac{1}{2^{20}}.]In which of the following intervals does $m$ lie?
$ extbf{(A) } [9,26] qquad extbf{(B) } [27,80] qquad extbf{(C) } [81,242]qquad extbf{(D) } [243,728] qquad extbf{(E) } [729,infty)$
(2019 AMC 12A) A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = frac{3}{7}$, and[a_n=frac{a_{n-2} cdot a_{n-1}}{2a_{n-2} - a_{n-1}}]for all $n geq 3$ Then $a_{2019}$ can be written as $frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q ?$
$ extbf{(A) } 2020 qquad extbf{(B) } 4039 qquad extbf{(C) } 6057 qquad extbf{(D) } 6061 qquad extbf{(E) } 8078$
(Austria-Poland 1997 Problem) Numbers $frac{49}{1}, frac{49}{2}, ... , frac{49}{97}$ are writen on a blackboard. Each time, we can replace two numbers (like $a, b$) with $2ab-a-b+1$. After $96$ times doing that prenominate action, one number will be left on the board. Find all the possible values fot that number.
Cute; nice to find the invariant $Pi = Pi_n = prod_{i=1}^n (2a_i -1)$, where $n$ is the number of the numbers written on the blackboard! Since $Pi_{97} = 1$, it follows the last number will also be $Pi_1 = 1$.
(This is obviously so, because $2(2ab - a - b + 1) - 1 = (2a-1)(2b-1)$).