1888　第二高等中学校予科第一学年MathJax

【１】　若干の分数を与へられて此等の分数の中の孰れを割りても商として完全数を得るようななる巳約分数の中に於て最も大なるものを求むる法及其理由如何．

【２】　或分数を小数に直すとき其結果が限りなき数字にて成りたる小数なるときは此小数は必ず循環小数なり其理如何．

【３】　十二人にて四日七分の二に四十五坪の地を耕せり今八人にて五十坪の地面を七日と二時間に耕せんと云ふ問ふ毎日勤むる時間如何．

【４】　米商あり金二千百円を以て米若干を買ひ一円に付三升騰げて之を売りしに七百円を利せりと云ふ問ふ買入し石高幾何．

【５】　体積百九十五万三千百二十五立方寸のものあり其外面を暑さ一寸の板を以て覆ふときは其板の外面積幾方寸なるや．

【１】 Prove that,if $a+b+c=0$,then $a^3+b^3+c^3-3abc=0$.

【２】 The least common multiple of two quantities is $x^4-5a^2{x}^2+4a^4$,and their greatest common measure is $x^2-a^2$.If one of the quantities be $x^3-2a{x}^2-a^{2}x+2a^3$,find the other quantity.

【３】 Factorise the two following expressions:

(a) $x^2+6x-16$

(b) $yz(y+z)+zx(z+x)+xy(x+y)+2xyz$

【４】 A is faster than B by $P$ miles an hour.He overtakes B who has a start of $L$ miles,after a run of $Q$ miles.Required the Speeds of A and B.

【５】 Solve

$\{\begin{array}{l}2x-7y+4z=0\\ 3x-3y+z=0\\ 9x+5y+3z=28\end{array}$

【１】 Determine the least path from one point to another subject to the condition that it shall meet a given straight line.

【２】 What is meant by the locus of the point under the given condition? Determine the locus of a point at a given distance from a given straight line.

【３】 Divide a given straight line into two parts,such that the square of one of the parts may be half the square on the whole line.

【４】 Prove that one circle and only one can be drawn through any three points,not in the same straight line.

【５】 With three given radii,$m$,$n$, and $p$ describe three circles which shall matually touch one another externally.

（編注）文中describeは原稿ではmiscribe．