수악중독

Suppose \((a_1, \; a_2 , \; a_3 , \; a_4)\) is a 4-term sequence of real numbers satisfying the following two conditions:\(a_3=a_2+a_1\) and \(a_4=a_3+a_2\);there exist rean numbers \(a, \; b, \;c\) such that \[an^2+bn+c=\cos(a_n)\]for all \( n \in \{1, \;2, \;3, \;4\}\).Compute the maximum possible value of \[\cos(a_1)-\cos(a_4)\] over all such sequences \((a_1, \; a_2, \; a_3,\; a_4)\). 정답 \(-..

Let \(a, \; b,\; c, \; d, \; e\) be nonnegative integers such that \(625a+250b+100c+40d+16c=15^3\). What is the maximum possible value of \(a+b+c+d+e\)? 정답 \(153\)

Let \(a, \;b, \;c\) be positive real numbers such that \(a+b+c=10, \; ab+bc+ca=25\). Let \(m={\rm min} \{ab, \; bc, \; ca\}\). Find the largest possible value of \(m\). 정답 \( \dfrac{25}{9}\)

Compute the number of sequences of integers \((a_1, \; \cdots ,\; a_{200})\) such taht th following conditions hold.\(0 \le a_1 < a_2 < \cdots < a_{200} \le 202\). There exists a positive integer \(N\) with the following property: for every index \(i \in \{ 1, \; \cdots ,\; 200 \}\) there exists an index \( j \in \{1, \; \cdots ,\; 200\}\) such that \(a_i +a_j -N\) is divisible by \(203\). 정답 \(..

Let P be a real number and \(c \ne 0\) an integer such that\[ c-0.1 < x^p \left ( \dfrac{1-(1+x)^10}{1+(1+x)^10} \right ) < c+0.1 \] for all (positive) real numbers \(x\) with \(0

The fraction \(\dfrac{1}{2015}\) has a unique "(restricted) partial fraction decomposition" of the form \[\dfrac{1}{2015}=\dfrac{a}{5}+\dfrac{b}{13}+\dfrac{c}{31},\] whrer \(a, \; b, \; c\) are integers with \(0 \le a

Let \(Q\) be a polynomial \[Q(x)=a_0 +a_1 x + \cdots + a_n x^n,\] where \( a_0, \; \cdots,\; a_n\) are nonnegative integers. Given that \( Q(1)=4\) and \(Q(5)=152\), find \(Q(6)\). 정답 \(254\)