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Part of 2012 Junior Balkan Team Selection Tests - Romania

Problems(4)

1/3 <= 1/p (p +1)+ 1/q (q +1) < 1/2, 1/p (p-1) + 1/q (q-1)<=1 if 1/p+1/q=1

Source: 2012 Romania JBMO TST 1.1, KoMaL Competition, 1999

6/2/2020
Prove that if the positive real numbers pp and qq satisfy 1p+1q=1\frac{1}{p}+\frac{1}{q}= 1, then a) 13≤1p(p+1)+1q(q+1)<12\frac{1}{3} \le \frac{1}{p (p + 1)} +\frac{1}{q (q + 1)} <\frac{1}{2} b) 1p(p−1)+1q(q−1)≥1\frac{1}{p (p - 1)} + \frac{1}{q (q - 1)} \ge 1
inequalitiesalgebra
a_{k+1} &lt;= (a_k + a_{k+2} )/2

Source: 2012 Romania JBMO TST 2.1

6/2/2020
Let a1,a2,...,ana_1, a_2, ..., a_n be real numbers such that a1=an=aa_1 = a_n = a and ak+1≤ak+ak+22a_{k+1} \le \frac{a_k + a_{k+2}}{2} , for all k=1,2,...,n−2k = 1, 2, ..., n - 2. Prove that ak≤a,a_k \le a, for all k=1,2,...,n.k = 1, 2, ..., n.
inequalitiesSequencealgebra
max of a/c+c/a+b/d+d/b. if a/b + b/c + c/d + d/a=4, ac = bd,

Source: 2012 Romania JBMO TST 3.1

6/2/2020
Let a,b,c,da, b, c, d be distinct non-zero real numbers satisfying the following two conditions: ac=bdac = bd and ab+bc+cd+da=4\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 4. Determine the largest possible value of the expression ac+ca+bd+db\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}.
algebramaxinequalitiessystem of equations
1/(1 + a^2 + (b + 1)^2) +1/(1 + b^2 + (c + 1)^2)+1/(1 + c^2 +(a + 1)^2) &lt;=1/2

Source: 2012 Romania JBMO TST 4.1

6/2/2020
Show that, for all positive real numbers a,b,ca, b, c such that abc=1abc = 1, the inequality 11+a2+(b+1)2+11+b2+(c+1)2+11+c2+(a+1)2≤12\frac{1}{1 + a^2 + (b + 1)^2} +\frac{1}{1 + b^2 + (c + 1)^2} +\frac{1}{1 + c^2 + (a + 1)^2} \le \frac{1}{2}
inequalitiesalgebra