{"status": "success", "data": {"description_md": "A real-valued function $f$ has the property that for all real numbers $a$ and $b,$$$f(a + b)  + f(a - b) = 2f(a) f(b).$$Which one of the following cannot be the value of $f(1)?$\n\n$\\textbf{(A) } 0 \\qquad \\textbf{(B) } 1 \\qquad \\textbf{(C) } -1 \\qquad \\textbf{(D) } 2 \\qquad \\textbf{(E) } -2$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>A real-valued function  <span class=\"katex--inline\">f</span>  has the property that for all real numbers  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b,</span>  <span class=\"katex--display\">f(a + b)  + f(a - b) = 2f(a) f(b).</span> Which one of the following cannot be the value of  <span class=\"katex--inline\">f(1)?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } 0 \\qquad \\textbf{(B) } 1 \\qquad \\textbf{(C) } -1 \\qquad \\textbf{(D) } 2 \\qquad \\textbf{(E) } -2</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2023 AMC 12B Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12B_p23", "prev": "/problem/23_amc12B_p21"}}