2023年卡内基梅隆大学信息与数学竞赛（代数、数论组）

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\title{2023年卡内基梅隆大学信息与数学竞赛——中文翻译}
\author{程国根}
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\section*{代数、数论组}
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\noindent\href{https://artofproblemsolving.com/community/c204707h3041228p27382791}{\bf 1.}非负整数$a,b,c,d$满足
$$(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})^{2}=2023.$$
求$a^{3}+b^{3}+c^{3}+d^{3}$．
\noindent\href{https://artofproblemsolving.com/community/c204707h3041233p27382807}{\bf 2.}求$a$的最大值，使得存在实数$b,c>1$，满足
$$a^{\log_{b}c}\cdot b^{\log_{c}a}=2023.$$
\noindent\href{https://artofproblemsolving.com/community/c204707h3041234p27382811}{\bf 3.}求$$
2022^{\left(2022^{\cdot ^ {\cdot ^{\cdot ^{\left(2022^{2022}\right)}}}}\right)} \pmod{111}
$$
（一共有$2022$个$2022$，结果为$0$到$110$之间的整数．）
\noindent\href{https://artofproblemsolving.com/community/c204707h3041236p27382819}{\bf 4.}$10$个两两互素的不同正整数构成等差数列，求它们和的最小可能值．
\noindent\href{https://artofproblemsolving.com/community/c204707h3041240p27382826}{\bf 5.}抛物线$\mathcal{P}$过点$(0,0)$和$(12,5)$，且准线为$y=b$．求$|b|$的最小值．
\noindent\href{https://artofproblemsolving.com/community/c204707h3041242p27382830}{\bf 6.}三元非负整数组$(a,b,c)$满足$2a+3b+5c=200$，设$a+b+c=N$，求$N$的所有可能值之和．
\noindent\href{https://artofproblemsolving.com/community/c204707h3041246p27382838}{\bf 7.}计算：$$\sum\limits_{i=1}^{\varphi(2023)}\dfrac{\gcd(i,\varphi(2023))}{\varphi(2023)},$$其中$\varphi(n)$为欧拉函数．
\noindent\href{https://artofproblemsolving.com/community/c204707h3041249p27382846}{\bf 8.}已知$A,B,C,D$为$0$到$9$的自然数，且$A\neq 0$，$\overline{ABCD}=\overline{CD}^{2}-\overline{AB}^{2}$．求$A+B+C+D$的所有不同值之和．
\noindent\href{https://artofproblemsolving.com/community/c204707h3041255p27382856}{\bf 9.}自然数$n<2023$，且$2n^{2}+3n$为完全平方数．求$n$的所有可能值之和．
\noindent\href{https://artofproblemsolving.com/community/c204707h3041258p27382864}{\bf 10.}对给定的$n$，考虑点$(x,y)\in \mathbb{N}^2$，其中$x\le y\le n$．一只蚂蚁从$(0,1)$出发，每次它从$(a,b)$移动到$(c,d)$，满足$bc-ad=1$，且使得$d$最大．设$g_{n}$为蚂蚁能够移动的最大次数，求$g_{2023}-g_{2022}$．
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