Math topic中文

Maximum and minimum problems

Use principles like fixed sum -> closer factors give bigger product, plus enumeration and construction, to find extremes.

Practice questions

1Ricardo has $2020$ coins, some of which are pennies ( $1$ -cent coins) and the rest of which are nickels ( $5$ -cent coins).…
2已知首项为 $1$ 的数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}$ ，且 $a_{n\text{+}1}\text{=}\dfrac{1}{3}S_{n}$ ，数列 $\{b_{n}\}$ 满足 $b_{n}\text{=}\log_{3}a_{n}$ …
3In a tournament there are six teams that play each other twice. A team earns $3$ points for a win, $1$ point for a draw,…
4已知 $n$ 行 $n$ 列 $\left(n\geqslant 2\right)$ 的数表 $A=\left(\begin{matrix}{{a}_{11}}&{{a}_{12}}&\cdot\cdot\cdot&{{a}_{1n}}\\ {{a}_{21}}&{{a}_{22}}&\cdot\cdot\cdot&{{a}_{2n}}\\ \vdots &\vdots&\ddots&\vdots\\ {{a}_{n1}}&{{a}_{n2}}&\cdot\cdot\cdot&{{a}_{nn}}\\\end{matrix}\right)$ …
5The integers from 1 through 25 are arbitrarily separated into five groups of 5 numbers each. The median of each group is…
6已知圆 $C$ ： $x^{2}+y^{2}-4x-2y+3=0$ ，过原点的直线 $l$ 与圆 $C$ 相交于 $A$ ， $B$ 两点，则当 $\triangle ABC$ 的面积最大时，直线 $l$ 的方程为 $(\quad)$ A…
7Minh enters the numbers $1$ through $81$ into the cells of a $9 \times 9$ grid in some order. She calculates the product…
8A palindrome is a number that has the same value when read from left to right or from right to left. (For example 12321…
9(1)三角形 $ABC$ 中， $AB=2\sqrt{3},BC=2,\angle C={{60}^{0}}$ ，则三角形 $ABC$ 的面积为___ (2)设等比数列 $\left\{ {{a}_{n}} \right\}$ 满足 $a{\,\!}_{1}+a{\,\!}_{3}=10$ …
10One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square…
11Shauna takes five tests, each worth a maximum of 100 points. Her scores on the first three tests are 76, 94, and 87. In…
12Kei draws a 6-by-6 grid. He colors 13 of the unit squares silver and the remaining squares gold. Kei then folds the grid…
13The digits $2$ , $0$ , $2$ , and $3$ are placed in the expression below, one digit per box. What is the maximum possible va…
14Two integers are inserted into the list $3, 3, 8, 11, 28$ to double its range. The mode and median remain unchanged. Wha…