MathDB

1

Part of 2017 Iran Team Selection Test

Problems(3)

Inequality from Iranian TST 2017

Source: Iran TST 2017, Exam 1, Day 1, Problem 1

4/5/2017
Let a,b,c,da,b,c,d be positive real numbers with a+b+c+d=2a+b+c+d=2. Prove the following inequality: (a+c)2ad+bc+(b+d)2ac+bd+4≥4(a+b+1c+d+1+c+d+1a+b+1).\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).
Proposed by Mohammad Jafari
algebrainequalitiesIranIranian TST
2017 Iran TST2 P1

Source: 2017 Iran TST second exam day1 p1

4/23/2017
ABCDABCD is a trapezoid with AB∥CDAB \parallel CD. The diagonals intersect at PP. Let ω1\omega _1 be a circle passing through BB and tangent to ACAC at AA. Let ω2\omega _2 be a circle passing through CC and tangent to BDBD at DD. ω3\omega _3 is the circumcircle of triangle BPCBPC. Prove that the common chord of circles ω1,ω3\omega _1,\omega _3 and the common chord of circles ω2,ω3\omega _2, \omega _3 intersect each other on ADAD.
Proposed by Kasra Ahmadi
geometryIranIranian TSTtrapezoidcircumcircle
Number Theory from Iran TST 2017

Source: 2017 Iran TST third exam day1 p1

4/26/2017
Let n>1n>1 be an integer. Prove that there exists an integer n−1≥m≥⌊n2⌋n-1 \ge m \ge \left \lfloor \frac{n}{2} \right \rfloor such that the following equation has integer solutions with am>0:a_m>0: amm+1+am+1m+2+⋯+an−1n=1lcm(1,2,⋯ ,n)\frac{a_{m}}{m+1}+\frac{a_{m+1}}{m+2}+ \cdots + \frac{a_{n-1}}{n}=\frac{1}{\textrm{lcm}\left ( 1,2, \cdots , n \right )}
Proposed by Navid Safaei
number theoryIranIranian TST