[{"content":"题目描述 已知： $a \u0026gt; 0, b \u0026gt; 0$ 且 $a + b = 2$。\n求证： $\\sqrt{a+1} + \\sqrt{b+1} \\le 2\\sqrt{2}$。\n证明过程 方法一：利用最基础的代数运算性质 要证：$\\sqrt{a+1}+\\sqrt{b+1} \\le 2\\sqrt{2}$\n即证：$(\\sqrt{a+1}+\\sqrt{b+1})^2 \\le (2\\sqrt{2})^2$\n化简可得：$a+1+2\\sqrt{(a+1)(b+1)}+b+1 \\le 8$\n整理可得：$a+b+2+2\\sqrt{ab+a+b+1} \\le 8$\n代入$a+b=2$可得：$4+2\\sqrt{ab+3} \\le 8$\n即$\\sqrt{ab+3} \\le 2$\n两边平方可得：$ab \\le 1$\n代入$a+b=2$可得：$a(2-a) \\le 1$\n由于$a\u0026gt;0$且$b=2-a\u0026gt;0$，故$0\u0026lt;a\u0026lt;2$\n由二次函数单调性可知当$a=1$时$a(2-a)$最大值为1\n即$a(2-a) \\le 1$成立，原不等式得证\n方法二：利用柯西不等式 (Cauchy-Schwarz Inequality) 根据柯西不等式：$(x_1y_1 + x_2y_2)^2 \\le (x_1^2 + x_2^2)(y_1^2 + y_2^2)$\n令 $x_1 = \\sqrt{a+1}, x_2 = \\sqrt{b+1}$, $y_1 = 1, y_2 = 1$。\n代入公式可得：$(\\sqrt{a+1} \\cdot 1 + \\sqrt{b+1} \\cdot 1)^2 \\le [(\\sqrt{a+1})^2 + (\\sqrt{b+1})^2] \\cdot (1^2 + 1^2)$\n化简不等式可得：$(\\sqrt{a+1} + \\sqrt{b+1})^2 \\le (a + 1 + b + 1) \\cdot 2$\n代入 $a + b = 2$，则：$(\\sqrt{a+1} + \\sqrt{b+1})^2 \\le (2 + 2) \\cdot 2 = 8$\n由于左侧大于0，两边同时开平方根可得：$\\sqrt{a+1} + \\sqrt{b+1} \\le \\sqrt{8} = 2\\sqrt{2}$\n当且仅当 $\\frac{\\sqrt{a+1}}{1} = \\frac{\\sqrt{b+1}}{1}$，即 $a = b = 1$ 时等号成立\n证毕\n方法三：利用算术-平方平均不等式 (QM-AM) 对于任意正数 $x, y$ 有 $\\frac{x+y}{2} \\le \\sqrt{\\frac{x^2+y^2}{2}}$。\n令 $x = \\sqrt{a+1}, y = \\sqrt{b+1}$。\n根据不等式可得：$\\frac{\\sqrt{a+1} + \\sqrt{b+1}}{2} \\le \\sqrt{\\frac{(\\sqrt{a+1})^2 + (\\sqrt{b+1})^2}{2}}$\n简化不等式可得：$\\frac{\\sqrt{a+1} + \\sqrt{b+1}}{2} \\le \\sqrt{\\frac{a + 1 + b + 1}{2}} = \\sqrt{\\frac{a + b + 2}{2}}$\n代入 $a + b = 2$ 可得：$\\frac{\\sqrt{a+1} + \\sqrt{b+1}}{2} \\le \\sqrt{\\frac{2 + 2}{2}} = \\sqrt{2}$\n两边同乘以2：$\\sqrt{a+1} + \\sqrt{b+1} \\le 2\\sqrt{2}$\n当且仅当 $a+1 = b+1$，即 $a = b = 1$ 时等号成立\n证毕\n","permalink":"https://theudas.github.io/posts/mathematical-proof-problem-1/","summary":"\u003ch2 id=\"题目描述\"\u003e题目描述\u003c/h2\u003e\n\u003cp\u003e\u003cstrong\u003e已知：\u003c/strong\u003e $a \u0026gt; 0, b \u0026gt; 0$ 且 $a + b = 2$。\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e求证：\u003c/strong\u003e $\\sqrt{a+1} + \\sqrt{b+1} \\le 2\\sqrt{2}$。\u003c/p\u003e\n\u003ch2 id=\"证明过程\"\u003e证明过程\u003c/h2\u003e\n\u003ch3 id=\"方法一利用最基础的代数运算性质\"\u003e方法一：利用最基础的代数运算性质\u003c/h3\u003e\n\u003cp\u003e要证：$\\sqrt{a+1}+\\sqrt{b+1} \\le 2\\sqrt{2}$\u003c/p\u003e\n\u003cp\u003e即证：$(\\sqrt{a+1}+\\sqrt{b+1})^2 \\le (2\\sqrt{2})^2$\u003c/p\u003e\n\u003cp\u003e化简可得：$a+1+2\\sqrt{(a+1)(b+1)}+b+1 \\le 8$\u003c/p\u003e\n\u003cp\u003e整理可得：$a+b+2+2\\sqrt{ab+a+b+1} \\le 8$\u003c/p\u003e\n\u003cp\u003e代入$a+b=2$可得：$4+2\\sqrt{ab+3} \\le 8$\u003c/p\u003e\n\u003cp\u003e即$\\sqrt{ab+3} \\le 2$\u003c/p\u003e\n\u003cp\u003e两边平方可得：$ab \\le 1$\u003c/p\u003e\n\u003cp\u003e代入$a+b=2$可得：$a(2-a) \\le 1$\u003c/p\u003e\n\u003cp\u003e由于$a\u0026gt;0$且$b=2-a\u0026gt;0$，故$0\u0026lt;a\u0026lt;2$\u003c/p\u003e\n\u003cp\u003e由二次函数单调性可知当$a=1$时$a(2-a)$最大值为1\u003c/p\u003e\n\u003cp\u003e即$a(2-a) \\le 1$成立，原不等式得证\u003c/p\u003e\n\u003ch3 id=\"方法二利用柯西不等式-cauchy-schwarz-inequality\"\u003e方法二：利用柯西不等式 (Cauchy-Schwarz Inequality)\u003c/h3\u003e\n\u003cp\u003e根据柯西不等式：$(x_1y_1 + x_2y_2)^2 \\le (x_1^2 + x_2^2)(y_1^2 + y_2^2)$\u003c/p\u003e\n\u003cp\u003e令 $x_1 = \\sqrt{a+1}, x_2 = \\sqrt{b+1}$, $y_1 = 1, y_2 = 1$。\u003c/p\u003e","title":"不等式证明"},{"content":"应用文 - Should We Establish a Graffiti Wall? 题目 你校英文报 Campus Culture 栏目正在开展关于是否设立涂鸦墙（graffiti wall）讨论。请你写一篇短文投稿，内容包括：\n1. 你的见解； 2. 你的建议。\n范文 Recently, our school newspaper Campus Culture has initiated a discussion regarding（发起了一场关于\u0026hellip;的讨论） the establishment of a graffiti wall on campus. Personally, I heartily endorse（由衷地支持） this proposal.\nTo begin with, a graffiti wall serves as an ideal vehicle（是\u0026hellip;的理想载体） for students to unleash their creativity（释放他们的创造力） and vent their emotions（宣泄他们的情绪）. Given the grueling academic schedule（鉴于繁重的学业安排）, a designated space for artistic expression acts as a stress-reliever（起到了减压作用）, fostering individuality and imagination. Moreover, a well-curated graffiti wall can breathe new life into（为\u0026hellip;注入新的活力） our campus culture, transforming dull corners into vibrant galleries. Rather than leaving random scribbles on desks, students are encouraged to showcase their talents in a more constructive and organized manner.\nHowever, to ensure the wall functions as intended（为确保这面墙能按预期发挥作用）, certain measures are indispensable. Primarily, a clear code of conduct（行为准则） should be implemented to filter out inappropriate content. Additionally, the school could collaborate with student unions to organize periodic themed competitions（定期的主题竞赛）, which would not only stimulate artistic enthusiasm but also guide students toward positive self-expression.\nIn conclusion, establishing a graffiti wall is a visionary initiative（有远见的举措）. With proper guidance and systematic management, it will undoubtedly become a landmark of our campus culture, contributing to a more dynamic and harmonious school life.\n写作积累 表示支持某个方案可以说heartily endorse，可以用于替代support 描述某个东西是实现目标、传播文化或表达情感的工具时可以说sth. serve as an ideal vehicle，类似于be a good way to do sth. 描述艺术、科研或课外活动对潜力的激发可以说unleash one\u0026rsquo;s creativity 讨论心理健康、压力调节或情绪管理可以说vent one\u0026rsquo;s emotions 某事情可以减缓压力可以说sth. act as a stress-reliever 说某新活动/新产品/新爱好/新朋友的好处时可以说sth. breathe new life into our lives 提出建议之前可以先说to ensure sth. functions as intended 当需要说rules/standards时可以用code of conduct替代 student unions 学生会 grueling表示极度难熬的、累死人的，可以替代very hard / tiring designated表示指定的、专门的，可以替代special，比如说designated smoking area（指定吸烟区） well-curated表示精心策划/挑选的，可以用于形容展览、博物馆等 dull表示枯燥的、暗淡的，可以替代boring vibrant表示充满活力的、鲜艳的，可以替代lively / colorful dull和vibrant这对反义词常用于对比\nconstructive表示建设性的、有益的，可以用于形容建议、活动等，可以替代helpful / useful organized表示有组织的、井井有条的 indispensable表示不可或缺的，可以替代necessary / very important等 visionary表示有远见的，可以用于形容提议、建议、计划等，跟far-seeing类似 proper表示恰当的、正规的，可以替代right systematic表示系统化的、有条理的，可以用来形容手段、方法等 dynamic表示动态的、不断发展的，类似于changing / energetic harmonious表示和谐的、融洽的，用于形容社会、关系、气氛等，类似于peaceful / friendly 书信 - 邀请参加春节活动 题目 假定你是李华，你的美国交换生好友Tom对中国传统节日很感兴趣。你所在的学校将要举办一场春节庆祝活动，你和同学想邀请Tom参加，并一起表演一个中西合璧的节目。请你写一封信给Tom，内容包括：\n1. 活动时间和地点； 2. 庆祝活动的内容； 3. 邀请Tom一起表演。\n注意：1. 词数100左右；2. 可以适当增加细节，以使行文连贯。\n范文 Dear Tom,\nKnowing that you have a keen interest in traditional Chinese festivals, I’m writing to invite you to our school’s Lunar New Year （农历新年）celebration.\nThe event will be held in the school auditorium at 7:00 p.m. next Friday. It promises to be a vibrant night filled with traditional activities, such as paper-cutting（剪纸）, riddle-solving（猜灯谜）, and a variety of stage performances（舞台表演）.More importantly, my classmates and I are planning to perform a \u0026ldquo;East meets West\u0026rdquo;（中西合璧） musical piece, and we would be thrilled if you could join us! We’re thinking of blending a traditional Chinese instrument with your amazing guitar skills. It would be a fantastic way to showcase cultural fusion（文化融合）.\nPlease let me know if you’re interested. Looking forward to your early reply!\nYours sincerely,\nLi Hua\n写作积累 have a keen interest in 对\u0026hellip;有浓厚兴趣 It promises to be \u0026hellip; 向别人介绍某活动时可以用这个句型 we would be thrilled if you could join us 虚拟语气表示委婉邀请 ","permalink":"https://theudas.github.io/posts/english-composition-1/","summary":"\u003ch2 id=\"应用文---should-we-establish-a-graffiti-wall\"\u003e应用文 - Should We Establish a Graffiti Wall?\u003c/h2\u003e\n\u003ch3 id=\"题目\"\u003e题目\u003c/h3\u003e\n\u003cp\u003e你校英文报 Campus Culture 栏目正在开展关于是否设立涂鸦墙（graffiti wall）讨论。请你写一篇短文投稿，内容包括：\u003c/p\u003e\n\u003cp\u003e1. 你的见解；\n2. 你的建议。\u003c/p\u003e\n\u003ch3 id=\"范文\"\u003e范文\u003c/h3\u003e\n\u003cp\u003eRecently, our school newspaper \u003cem\u003eCampus Culture\u003c/em\u003e has \u003cstrong\u003einitiated a discussion regarding（发起了一场关于\u0026hellip;的讨论）\u003c/strong\u003e the establishment of a graffiti wall on campus. Personally, I \u003cstrong\u003eheartily endorse（由衷地支持）\u003c/strong\u003e this proposal.\u003c/p\u003e\n\u003cp\u003eTo begin with, a graffiti wall \u003cstrong\u003eserves as an ideal vehicle（是\u0026hellip;的理想载体）\u003c/strong\u003e for students to \u003cstrong\u003eunleash their creativity（释放他们的创造力）\u003c/strong\u003e and \u003cstrong\u003event their emotions（宣泄他们的情绪）\u003c/strong\u003e. \u003cstrong\u003eGiven the grueling academic schedule（鉴于繁重的学业安排）\u003c/strong\u003e, a \u003cstrong\u003edesignated\u003c/strong\u003e space for artistic expression \u003cstrong\u003eacts as a stress-reliever（起到了减压作用）\u003c/strong\u003e, fostering individuality and imagination. Moreover, a \u003cstrong\u003ewell-curated\u003c/strong\u003e graffiti wall can \u003cstrong\u003ebreathe new life into（为\u0026hellip;注入新的活力）\u003c/strong\u003e our campus culture, transforming \u003cstrong\u003edull\u003c/strong\u003e corners into \u003cstrong\u003evibrant\u003c/strong\u003e galleries. Rather than leaving random scribbles on desks, students are encouraged to showcase their talents in a more \u003cstrong\u003econstructive and organized\u003c/strong\u003e manner.\u003c/p\u003e","title":"2025年寒假英语作文"}]