{"paper":{"id":"site-4915","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":5,"source_label":"Matematyka · Zbiory zadań (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-4915/zad/5344","paper_id":"site-4915","number":"5344","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"rozszerzona","text":"Liczby dodatnie a i b spełniają równość a^2+2a=4b^2+4b. Wykaż, że a=2b.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4915/zad-5344.svg","solution_image":"img/site/4915/rozw-5344.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4915/zad/5505","paper_id":"site-4915","number":"5505","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"rozszerzona","text":"Rozwiązanie zadania. Wykaż, że dla każdej liczby rzeczywistej x większej od 2 i dla każdej liczby rzeczywistej y prawdziwa jest nierówność 5x^2 - 6xy + 3y^2 - 2x - 4 > 0.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4915/zad-5505.svg","solution_image":"img/site/4915/rozw-5505.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4915/zad/6708","paper_id":"site-4915","number":"6708","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"rozszerzona","text":"Rozważmy dwie kolejne liczby naturalne a i b takie, że obie są niepodzielne przez 3. Udowodnij, że liczba a^3+b³ jest podzielna przez 9.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4915/zad-6708.svg","solution_image":"img/site/4915/rozw-6708.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4915/zad/6829","paper_id":"site-4915","number":"6829","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"rozszerzona","text":"Wykaż, że dla każdej liczby rzeczywistej x oraz dla każdej liczby rzeczywistej y, spełniających warunek x+y ≥ 1, prawdziwa jest nierówność x³ + 2xy + y³ ≥ x² + xy(x+y) + y^2.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4915/zad-6829.svg","solution_image":"img/site/4915/rozw-6829.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4915/zad/7226","paper_id":"site-4915","number":"7226","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"rozszerzona","text":"Udowodnij, że dla każdej liczby rzeczywistej x i każdej liczby rzeczywistej m prawdziwa jest nierówność 20x²-24mx+18m² ≥ 4x+12m-5.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4915/zad-7226.svg","solution_image":"img/site/4915/rozw-7226.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"}]}