MathDB

4

Part of 2023 China Second Round

Problems(2)

A problem of Vectors

Source: 2023 China Second Round Test 1 P4

6/2/2024
if three non-zero vectors on a plane a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} satisfy: (a) a⃗⊥b⃗\vec{a}\bot\vec{b} (b) b⃗⋅c⃗=2∣a⃗∣\vec{b}\cdot\vec{c}=2|\vec{a}| (c) c⃗⋅a⃗=3∣b⃗∣\vec{c}\cdot\vec{a}=3|\vec{b}| find out the minimum of ∣c⃗∣|\vec{c}|
vectoralgebra
On the Ratio of Products of Sum of Each Row and Each Column

Source: Problem 4, China National High School Mathematics League 2023

9/10/2023
Let a=1+10−4a=1+10^{-4}. Consider some 2023×20232023\times 2023 matrix with each entry a real in [1,a][1,a]. Let xix_i be the sum of the elements of the ii-th row and yiy_i be the sum of the elements of the ii-th column for each integer i∈[1,n]i\in [1,n]. Find the maximum possible value of y1y2⋯y2023x1x2⋯x2023\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}} (the answer may be expressed in terms of aa).
algebrainequalities