Có bao nhiêu số nguyên \(x\) sao cho tồn tại số thực \(y\) thỏa mãn \(\log _{3}(x+y)=\log _{4}\left(x^{2}+y^{2}\right)\)?
Hãy suy nghĩ và trả lời câu hỏi trước khi xem đáp án
Đặt \(t=\log _{3}(x+y)=\log _{4}\left(x^{2}+y^{2}\right) \Rightarrow\left\{\begin{array}{l}x+y=3^{t} \\ x^{2}+y^{2}=4^{t}\end{array}\right. (1)\)
Áp dụng bất đẳng thức Cauchy, ta có:
\(9^{t}=(x+y)^{2} \leq 2\left(x^{2}+y^{2}\right)=4^{t} \Rightarrow\left(\frac{9}{4}\right)^{t} \leq 2 \Rightarrow t \leq \log _{\frac{9}{4}} 2\).
Khi đó: \(x^{2}+y^{2}=4^{t} \Rightarrow x^{2} \leq 4^{t} \leq 4^{\log _{9} 2} \approx 1,89 \Rightarrow x \in\{-1; 0; 1\}\).
+ Trường hợp 1: \(x=0 \Rightarrow\left\{\begin{array}{l}y=3^{t} \\ y^{2}=4^{t}\end{array} \Rightarrow\left\{\begin{array}{l}t=0 \\ y=1\end{array}\right.\right.\).
+ Trường hợp 2: \(x=1 \Rightarrow\left\{\begin{array}{l}y=3^{t}-1 \\ y^{2}=4^{t}-1\end{array} \Rightarrow\left\{\begin{array}{l}t=0 \\ y=0\end{array}\right.\right.\).
+ Trường hợp 3: \(x=-1 \Rightarrow\left\{\begin{array}{l}y=3^{t}+1 \\ y^{2}+1=4^{t} \geq 1\end{array} \Rightarrow\left\{\begin{array}{l}t \geq 0 \\ y=3^{t}+1\end{array} \Rightarrow x^{2}+y^{2} \geq 5\right.\right.\) mâu thuẫn với \(x^{2}+y^{2} \leq 4^{\log _{\frac{3}{2}} \sqrt{2}}\).
Vậy có hai giá trị \(x \in\{0; 1\}\).
Đề Thi Tham Khảo Đánh Giá Năng Lực Năm 2025 – ĐHQG Hà Nội – Đề Số 05 là tài liệu luyện thi toàn diện, giúp học sinh làm quen với cấu trúc và độ khó của kỳ thi ĐGNL theo chuẩn mới. Bộ test gồm đầy đủ 3 phần: Toán học và Xử lý số liệu, Văn học – Ngôn ngữ, và Khoa học/Tiếng Anh, được thiết kế với thời lượng, số câu hỏi và thang điểm đúng theo đề thi chính thức. Các câu hỏi trong bộ test không chỉ kiểm tra kiến thức mà còn rèn luyện tư duy phản biện, lập luận logic và khả năng vận dụng thực tiễn. Đây là tài liệu quan trọng giúp thí sinh tự đánh giá năng lực, luyện tốc độ, nâng cao kỹ năng giải đề và chuẩn bị vững vàng cho kỳ thi Đánh Giá Năng Lực năm 2025 của ĐHQG Hà Nội.
Câu hỏi liên quan
Nguồn cung cấp nhiệt chủ yếu cho mặt đất là năng lượng của yếu tố nào?
Trong điều kiện lý tưởng, giống lúa A cho năng suất tối đa là 10,5 tấn/ha, trong khi đó giống lúa B chỉ cho năng suất tối đa là 6 tấn/ha. Điều này cho thấy vai trò của nhân tố nào đối với năng suất cây trồng?
Xác định ý đúng: Tài nguyên không bị hao kiệt cơ bản?
Read the following passage and mark the letter A, B, C or D on your answer sheet to indicate the best answer to each of the following questions.
Taking Home the World Cup
Standing just 36 centimeters tall, the FIFA World Cup Trophy symbolizes the pinnacle of football achievement. Every four years, teams worldwide compete for it, but no one ever takes the trophy home. Made of 18-carat gold, the original trophy remains under FIFA's protection, with winners receiving only a replica. This measure safeguards it from theft, as the trophy has been stolen twice in its history.
The original trophy was made by a French sculptor, Abel LaFleur, and was called the “Jules Rimet Cup,” in honor of the founder of the World Cup tournament. Sometime during the first three World Cup events (1930, 1934, and 1938), the name changed to simply the “World Cup.” Then during World War II, not much was seen or heard of the trophy. It was being kept hidden in a shoe box under the bed of Dr. Ottorino Barassi, the Italian vice-president of FIFA, to prevent it from falling into the hands of the Nazi army. In 1966, the Cup was stolen during a public showing of the trophy prior to the World Cup tournament in England. Luckily, it was found a short time later, none the worse for wear, in a trash container by a little dog named Pickles. Four years later, Brazil earned permanent possession of the original trophy by winning its third World Cup title. Unfortunately, the trophy was stolen a second time in 1983 and was never recovered. The Brazilian Football Association had to have a duplicate trophy made.
(I) After the first trophy became the possession of Brazil’s football association, a new World Cup Trophy for FIFA was designed by an Italian artist, Silvio Gazzaniga, in 1974. (II) This trophy cannot be won outright, but remains in the possession of FIFA, and rest assured they are watching it very closely. Today, World Cup winners are awarded a replica of the trophy that is gold-plated, rather than solid gold like the real one. (III) Gazzaniga’s World Cup trophy weighs almost five kilograms. (IV) Its base contains two layers of a semi-precious stone called malachite, and has room for seventeen small plaques bearing the names of the winning teams---enough space to honor all the World Cup champions up to the year 2038. After that, a new trophy will have to be made.
Which of the following best paraphrases the underlined sentence?
Read the following passage about The Elderly in Multicultural Societies: Challenges and Opportunities and mark the letter A, B, C or D on your answer sheet to indicate the best answer to each of the following questions from 23 to 30.
In multicultural societies, elderly individuals face unique challenges and important opportunities. As communities diversify rapidly, traditional support systems are constantly evolving. Seniors often navigate unfamiliar cultural environments while facing significant language barriers that lead to social isolation. These persistent obstacles frequently prevent full access to essential healthcare services or meaningful community participation. Fortunately, innovative local organizations address these critical issues by offering comprehensive translation services and culturally sensitive care options.
Technology presents both remarkable benefits and considerable difficulties for elderly individuals from various cultural backgrounds. Digital tools effectively connect seniors with distant family members globally, yet many older adults struggle substantially with these new communication methods. Recent research shows approximately 65% of elderly immigrants experience digital exclusion compared to native-born seniors. In response, numerous community centers have successfully established specialized technology workshops for older adults from diverse backgrounds, helping them maintain vital connections with overseas relatives and friends.
Cultural differences significantly impact elderly care approaches across communities. In collectivist cultures, elderly family members traditionally live with their children, while Western societies typically emphasize personal independence. This fundamental divergence creates tension when immigrant families attempt to adapt to new societal norms. However, vibrant multicultural environments provide valuable opportunities for exchanging diverse perspectives on aging. Many communities benefit tremendously from adopting hybrid approaches blending traditional cultural practices with modern care techniques, creating more comprehensive support systems.
Economic aspects of aging in multicultural contexts matter significantly for overall wellbeing. Elderly immigrants frequently face severe financial hardships due to limited retirement savings and persistent employment discrimination throughout their lives. Some progressive governments implement supportive policies addressing these vulnerable individuals' unique needs. Certain forward-thinking regions offer specialized employment programs for seniors with diverse cultural backgrounds, recognizing their valuable skills and extensive life experiences. Additionally, creative intergenerational initiatives connecting elderly immigrants with younger community members effectively foster mutual understanding and practical support, successfully transforming demographic challenges into meaningful community enrichment opportunities.
The word “divergence” in paragraph 3 could be best replaced by _________.
Read the following passage and mark the letter A, B, C, or D on your answer sheet to indicate the best answer to each of the following questions from 23 to 30.
Vietnamese Women’s Achievements
Women have always played an important role in Vietnam’s history. Some fight for their country, some inspire others, and some make positive changes to society. Their contributions have shaped the nation in various ways, leaving a lasting impact on culture, politics, and the arts.
Trưng Trắc and Trưng Nhị are famous examples of women who fought for their country’s freedom. Known in Vietnamese as “Hai Bà Trưng,” the Trưng sisters successfully led their people in a rebellion against foreign invaders in 40 A.D. After their victory, they ruled the country for three years, demonstrating remarkable leadership and bravery. As the first female rulers of Vietnam, the Trưng sisters have become symbols of patriotism and resilience. Their significance is even greater considering that Vietnamese women at the time had very limited access to education and politics. Their story continues to inspire generations.
Sương Nguyệt Anh is another example of an influential Vietnamese woman. As the daughter of the patriotic poet Nguyễn Đình Chiểu, she mastered literature from a young age. She worked as a teacher before becoming the editor-in-chief of Nữ Giới Chung, the first women’s newspaper in Sài Gòn, in 1918. As the first female editor in Vietnam, she made significant contributions to the press, feminism, and poetry. Through her work, she promoted women’s rights and encouraged female empowerment.
A third example is Điềm Phùng Thị. She graduated from Hà Nội Medical University in 1946 and earned a PhD in dentistry in France in 1954. However, she later pursued a career in sculpture, creating artworks that reflected her love for Vietnam. She exhibited her work throughout Europe and became the first female Vietnamese member of the European Academy of Arts and Sciences in 1992. Her name was later recorded in the Larousse Dictionary of Painters, recognizing her as a remarkable artist.
According to the passage, which of the following statements is TRUE?
Phương án nào sau đây thể hiện rõ nhất sự thay đổi về giao tiếp cộng đồng khi mạng máy tính được sử dụng rộng rãi?
Đại lượng nào dưới đây phải đo bằng phép đo gián tiếp?
Cư dân của nền văn hóa nào đã mở đầu cho nền nông nghiệp sơ khai ở Việt Nam?
Nhận định nào sau đây không phản ánh đúng ý nghĩa của các quy định pháp luật về quyền tố cáo của công dân?
CHỦ ĐỀ: HÀNH VI TÌM KIẾM SỰ HỖ TRỢ CỦA PHỤ NỮ KHI MẮC TRẦM CẢM SAU SINH
(Tham khảo bài văn của chủ đề trong đường link dưới đây ở trang 21)
https://tvsthpt.dlib.vn/app/doc-sach-ebook/d00b996754ec
Và trả lời câu hỏi sau:
Theo đoạn [4], phụ nữ luôn gặp khó khăn khi chia sẻ về vấn đề trầm cảm sau sinh, đúng hay sai?
Pháp luật có đặc điểm gì?
Chúng ta nên làm gì đề tự bảo vệ bản thân khi sử dụng mạng xã hội?
Enthalpy tạo thành chuẩn của một đơn chất bền là
Bước 3 của quy trình thiết kế kĩ thuật là:
Read the following leaflet and mark the letter A, B, C or D on your answer sheet to indicate the option that bestfits each of the numbered blanks from 7 to 12.
PROMOTE MENTAL WELL-BEING AT HOME!
Nurturing Your Mind and Spirit
Taking care of our mental health is just as important as our physical health, which helps us cope with stress, build resilience, and lead fulfilling lives.
Facts and Figures
• Nature’s Impact: Studies show that spend time outdoors can significantly reduce stress levels. However, (7)_____ people feel they lack access to green spaces.
• Screen Time Concerns: Excessive daily (8)_____ time is linked to increased anxiety and depression. (9)_____ spending too much time on the phone screen, why don’t we go outside and get some fresh air?
Positive Actions You Can Take
• Designate a cozy corner in your home for reading, meditation, or simply unwinding. Make it a daily habit with soft lighting, plants, and comfortable seating.
• Start a gratitude journal! (10)_____ three things you're thankful for each day to cultivate a positive mindset.
• Make time for meaningful (11)_____ with family and friends. Connect with (12)_____ regularly help strengthens your support network and combats feelings of isolation.
Cho phương trình lượng giác \(2\text{sin}x=\sqrt{2}\).
Thành phần tham gia đông đảo nhất trong phong trào Hội kín ở Nam Kì là
Biết \({C_\mathbb{R}}A = [ - 3;11)\) và \({C_\mathbb{R}}B = ( - 8;1]\). Khi đó \({C_\mathbb{R}}\left( {A \cap B} \right)\) bằng?
Để khắc phục tình trạng lạm phát do lượng tiền mặt trong lưu thông đã vượt quá mức cần thiết, nhà nước cần làm gì?