{"paper":{"id":"site-2669","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":6,"source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-2669/zad/3182","paper_id":"site-2669","number":"3182","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Wykaż, że prawdziwa jest nierówność pierwiastek z {2^{50}+1} + pierwiastek z {2^{50}-1} < 2^{26}.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2669/zad-3182.svg","solution_image":"img/site/2669/rozw-3182.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 (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2669/zad/3183","paper_id":"site-2669","number":"3183","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Udowodnij, że jeśli a) x, y są liczbami rzeczywistymi, to x^2+y^2 >= 2xy, b) x, y, z są liczbami rzeczywistymi takimi, że x+y+z=1, to x^2+y^2+z^2 >= 1/3","answer":null,"answer_text":null,"solution":null,"image":"img/site/2669/zad-3183.svg","solution_image":"img/site/2669/rozw-3183.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 (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2669/zad/5328","paper_id":"site-2669","number":"5328","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Wykaż, że dla każdych dwóch różnych liczb rzeczywistych a i b prawdziwa jest nierówność a(a-2b) + 2b^2 > 0","answer":null,"answer_text":null,"solution":null,"image":"img/site/2669/zad-5328.svg","solution_image":"img/site/2669/rozw-5328.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 (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2669/zad/5497","paper_id":"site-2669","number":"5497","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Funkcja kwadratowa f(x) = x²+bx+c nie ma miejsc zerowych. Wykaż, że 1+c>b.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2669/zad-5497.svg","solution_image":"img/site/2669/rozw-5497.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 (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2669/zad/5787","paper_id":"site-2669","number":"5787","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Rozwiązanie zadania. Wykaż, że dla dowolnych różnych liczb rzeczywistych a i b prawdziwa jest nierówność a(a+b) + b² > 3ab.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2669/zad-5787.svg","solution_image":"img/site/2669/rozw-5787.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 (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2669/zad/6859","paper_id":"site-2669","number":"6859","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Dana jest liczba a = 1/1+√2 + 1/√2+√3 + 1/√3+√4. Wykaż, że a jest liczbą całkowitą. Wskazówka. Usuń niewymierności z mianowników.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2669/zad-6859.svg","solution_image":"img/site/2669/rozw-6859.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 (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"}]}