148. Solve the equation

x3+3ax2+3 (a2 − bc) x + a3 +b3 +c3 — 3abc=0.

[INDIA CIVIL SERVICE.]

149. If n is a prime number which will divide neither a, b, nor a+b, prove that an-2b-an-3b2+ an¬4b3 — ...+abr-2 exceeds by 1 a multiple of n. [ST JOHN'S COLL. ČAMB.]

150. Find the nth term and the sum to n terms of the series whose sum to infinity is (1 − abx2)(1 − ax) −2(1 − bx)−2.

[OXFORD MODS.]

151. If a, b, c are the roots of the equation b2+c2 c2+a2 a2+b2 equation whose roots are

3+px+q=0, find the

a

b

с

[TRIN. COLL. CAMB.]

152. Prove that

(y+z-2x)+(z+x-2y)2+(x+y-22)+18(x2+ y2+z2-yz-zx-xy)2.

153. Solve the equations:

[CLARE COLL. CAMB.]

(1) x3-30x+133=0, by Cardan's method.

(2) x5 – 4x1 – 10x3 +40x2 + 9x – 36=0, having roots of the form

±a, ±b, c.

154. It is found that the quantity of work done by a man in an hour varies directly as his pay per hour and inversely as the square root of the number of hours he works per day. He can finish a piece of work in six days when working 9 hours a day at 1s. per hour. How many days will he take to finish the same piece of work when working 16 hours a day at 1s. 6d. per hour?

155. If s, denote the sum to n terms of the series

157. A cottage at the beginning of a year was worth £250, but it was found that by dilapidations at the end of each year it lost ten per cent. of the value it had at the beginning of each year after what number of years would the value of the cottage be reduced below £25? Given log103=4771213. [R. M. A. WOOLWICH.]

160. If n is a positive integer greater than 1, shew that

n5-5n3+60n2 - 56n

is a multiple of 120.

[WADHAM COLL. Ox.]

161. A number of persons were engaged to do a piece of work which would have occupied them 24 hours if they had commenced at the same time; but instead of doing so, they commenced at equal intervals and then continued to work till the whole was finished, the payment being proportional to the work done by each: the first comer received eleven times as much as the last; find the time occupied.

163. Solve the equation

a3 (b-c)(x-b)(x−c)+b3 (c− a) (x−c) (x − a) + c3 (a - b)(x-a) (x-b)=0;

also shew that if the two roots are equal

165. Shew that, if a, b, c, d be four positive unequal quantities and s=a+b+c+d, then

( s − a) ( s − b) (s — c) (s − d) >
> 81abcd.

166. Solve the equations:

[PETERHOUSE, CAMB.]

5
2

(1) √x+a-√y−a= √a, √x−a−√ÿ+a=23↓a.

167. Eliminate l, m, n from the equations:
lx+my+nz=mx+ny+lz=nx+ly +mz=k2 (12+m2+n2) = 1.

168. Simplify

a(b+c-a)2+...+...+ (b + c - a) (c+a-b) (a+b-c)
(b+c− (c+a−b)(a+b−c)
a2(b+c-a)+...+... - (b+c-a) (c+a−b) (a+b−c) *

169. Shew that the expression

[MATH. TRIPOS.]

(x2 − yz)3 + (y2 — zx)3 + (z2 − xy)3 − 3 (x2 — yz) (y2 — zx) (z2 — xy)

is a perfect square, and find its square root.

[LONDON UNIVERSITY.]

170. There are three towns A, B, and C'; a person by walking from A to B, driving from B to C, and riding from C to A makes the journey in 15 hours; by driving from A to B, riding from B to C, and walking from C to A he could make the journey in 12 hours. On foot he could make the journey in 22 hours, on horseback in 8 hours, and driving in 11 hours. To walk a mile, ride a mile, and drive a mile he takes altogether half an hour: find the rates at which he travels, and the distances between the towns.

171. Shew that n2 – 7n3+14n3 – 8n is divisible by 840, if n is an integer not less than 3.

172. Solve the equations:

(1) √x2+12y+√/y2+12x=33, x+y=23.

173.

If s be the sum of n positive unequal quantities a, b, c...,

then

174. A merchant bought a quantity of cotton; this he exchanged for oil which he sold. He observed that the number of cwt. of cotton, the number of gallons of oil obtained for each cwt., and the number of shillings for which he sold each gallon formed a descending geometrical progression. He calculated that if he had obtained one cwt. more of cotton, one gallon more of oil for each cwt., and 1s. more for each gallon, he would have obtained £508. 9s. more; whereas if he had obtained one cwt. less of cotton, one gallon less of oil for each cwt., and 18. less for each gallon, he would have obtained £483. 13s. less: how much did he actually receive?

175. Prove that

(b+c-a-x)(b−c) (a-x)=16(b−c) (c− a) (a - b) (x − a) (x —b) (x−c). [JESUS COLL. CAMB.]

176. If a, ß, y are the roots of the equation a3 − px2+r=0, find the B+y yta a+ß equation whose roots are

a В

γ

[R. M. A. WOOLWICH.]

177. If any number of factors of the form a2+b2 are multiplied together, shew that the product can be expressed as the sum of two squares.

Given that (a2+b2) (c2+d2) (e2+ƒ2) (y2+h2)=p2+q2, find p and q in terms of a, b, c, d, e, f, g, h. LONDON UNIVERSITY.]

178. Solve the equations

x2+y2=61, x3—y3=91.

[R. M. A. WOOLWICH.]

179. A man goes in for an Examination in which there are four papers with a maximum of m marks for each paper; shew that the number of ways of getting 2m marks on the whole is

(m+1) (2m2+4m+3).

[MATH. TRIPOS.]

180. If a, ẞ are the roots of x2+px+1=0, and y, & are the roots of x2+qx+1=0; shew that (a-y) (B− y) (a+8) (B+8)=q2-p2.

[R. M. A. WOOLWICH.]

181. Shew that if am be the coefficient of xm in the expansion of (1+x)”, then whatever n be,

182. A certain number is the product of three prime factors, the sum of whose squares is 2331. There are 7560 numbers (including unity) which are less than the number and prime to it. The sum of its divisors (including unity and the number itself) is 10560. Find the number. [CORPUS COLL. CAMB.]

183. Form an equation whose roots shall be the products of every two of the roots of the equation x3- ax2+ bx+c=0.

185. If (6/6+14)2n+1=N, and if F be the fractional part of N, prove that NF=202n+1.

186. Solve the equations:

[EMM. COLL. CAMB.]

(1) x+y+2=2, x2+y2+z2=0, 23+y3+23=-1.

(2) x2-(y-z)2=a2, y2 — (z−x)2= b2, z2 — (x − y)2=c2.

[CHRIST'S COLL. CAMB.]

187. At a general election the whole number of Liberals returned was 15 more than the number of English Conservatives, the whole number of Conservatives was 5 more than twice the number of English Liberals. The number of Scotch Conservatives was the same as the number of Welsh Liberals, and the Scotch Liberal majority was equal to twice the number of Welsh Conservatives, and was to the Irish Liberal majority as 2: 3. The English Conservative majority was 10 more than the whole number of Irish members. The whole number of members was 652, of whom 60 were returned by Scotch constituencies. Find the numbers of each party returned by England, Scotland, Ireland, and Wales, respectively. [ST JOHN'S COLL. CAMB.]

188. Shew that a(c-b)+b5 (a−c)+c (b-a)

=(b−c)(c− a)(a−b) (Σa3+Σa2b+abc).