MathDB

Problem 3

Part of 2019 LIMIT Category A

Problems(2)

side inequality in triangle

Source: LIMIT 2019 CAS1 P3

4/28/2021
In △ABC\triangle ABC, ∣AB‾∣=∣AC‾∣\left|\overline{AB}\right|=\left|\overline{AC}\right|, DD is the foot of the perpendicular from CC to ABAB and EE the foot of the perpendicular from BB to ACAC, then <spanclass=′latex−bold′>(A)</span> ∣BC‾∣3>∣BD‾∣3+∣BE‾∣3<span class='latex-bold'>(A)</span>~\left|\overline{BC}\right|^3>\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3 <spanclass=′latex−bold′>(B)</span> ∣BC‾∣3<∣BD‾∣3+∣BE‾∣3<span class='latex-bold'>(B)</span>~\left|\overline{BC}\right|^3<\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3 <spanclass=′latex−bold′>(C)</span> ∣BC‾∣3=∣BD‾∣3+∣BE‾∣3<span class='latex-bold'>(C)</span>~\left|\overline{BC}\right|^3=\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3 <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
inequalitiesGeometric Inequalitiesgeometry
counting finite sequences

Source: LIMIT 2019 CAS2 P3

4/28/2021
How many finite sequences x1,x2,…,xmx_1,x_2,\ldots,x_m are there such that xi=1x_i=1 or 22 and ∑i=1mxi=10\sum_{i=1}^mx_i=10? <spanclass=′latex−bold′>(A)</span> 89<span class='latex-bold'>(A)</span>~89 <spanclass=′latex−bold′>(B)</span> 73<span class='latex-bold'>(B)</span>~73 <spanclass=′latex−bold′>(C)</span> 107<span class='latex-bold'>(C)</span>~107 <spanclass=′latex−bold′>(D)</span> 119<span class='latex-bold'>(D)</span>~119
combinatorics