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Question Number 17053 Answers: 1 Comments: 0

Question Number 19167 Answers: 1 Comments: 1

Two particles A and B move with constant velocities v_1 and v_2 along two mutually perpendicular straight lines towards the intersection point O. At moment t = 0, the particles were located at distances d_1 and d_2 from O respectively. Find the time, when they are nearest and also this shortest distance.

$$\mathrm{Two}\:\mathrm{particles}\:{A}\:\mathrm{and}\:{B}\:\mathrm{move}\:\mathrm{with} \\ $$$$\mathrm{constant}\:\mathrm{velocities}\:{v}_{\mathrm{1}} \:\mathrm{and}\:{v}_{\mathrm{2}} \:\mathrm{along}\:\mathrm{two} \\ $$$$\mathrm{mutually}\:\mathrm{perpendicular}\:\mathrm{straight}\:\mathrm{lines} \\ $$$$\mathrm{towards}\:\mathrm{the}\:\mathrm{intersection}\:\mathrm{point}\:{O}.\:\mathrm{At} \\ $$$$\mathrm{moment}\:{t}\:=\:\mathrm{0},\:\mathrm{the}\:\mathrm{particles}\:\mathrm{were} \\ $$$$\mathrm{located}\:\mathrm{at}\:\mathrm{distances}\:{d}_{\mathrm{1}} \:\mathrm{and}\:{d}_{\mathrm{2}} \:\mathrm{from}\:{O} \\ $$$$\mathrm{respectively}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{time},\:\mathrm{when}\:\mathrm{they} \\ $$$$\mathrm{are}\:\mathrm{nearest}\:\mathrm{and}\:\mathrm{also}\:\mathrm{this}\:\mathrm{shortest} \\ $$$$\mathrm{distance}. \\ $$

Question Number 17169 Answers: 2 Comments: 0

The base of a pyramid is an equilateral triangle of side length 6 cm. The other edges of the pyramid are each of length (√(15)) cm. Find the volume of the pyramid.

$$\mathrm{The}\:\mathrm{base}\:\mathrm{of}\:\mathrm{a}\:\mathrm{pyramid}\:\mathrm{is}\:\mathrm{an}\:\mathrm{equilateral} \\ $$$$\mathrm{triangle}\:\mathrm{of}\:\mathrm{side}\:\mathrm{length}\:\mathrm{6}\:\mathrm{cm}.\:\mathrm{The}\:\mathrm{other} \\ $$$$\mathrm{edges}\:\mathrm{of}\:\mathrm{the}\:\mathrm{pyramid}\:\mathrm{are}\:\mathrm{each}\:\mathrm{of}\:\mathrm{length} \\ $$$$\sqrt{\mathrm{15}}\:\mathrm{cm}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{pyramid}. \\ $$

Question Number 16983 Answers: 1 Comments: 0

If 15 men or 24 women or 36 boys do a piece of work in 12 days, working 8 hours a day, how many men must associated with 12 women and 6 boys to do another piece of work 2(1/4) times as great in 30 days working 6 hrs a day?

$$\mathrm{If}\:\mathrm{15}\:\mathrm{men}\:\mathrm{or}\:\mathrm{24}\:\mathrm{women}\:\mathrm{or}\:\mathrm{36}\:\mathrm{boys}\:\mathrm{do}\:\mathrm{a}\: \\ $$$$\mathrm{piece}\:\mathrm{of}\:\mathrm{work}\:\mathrm{in}\:\mathrm{12}\:\mathrm{days},\:\mathrm{working}\:\mathrm{8}\:\mathrm{hours} \\ $$$$\mathrm{a}\:\mathrm{day},\:\mathrm{how}\:\mathrm{many}\:\mathrm{men}\:\mathrm{must}\:\mathrm{associated} \\ $$$$\mathrm{with}\:\mathrm{12}\:\mathrm{women}\:\mathrm{and}\:\mathrm{6}\:\mathrm{boys}\:\mathrm{to}\:\mathrm{do}\:\mathrm{another} \\ $$$$\mathrm{piece}\:\mathrm{of}\:\mathrm{work}\:\mathrm{2}\frac{\mathrm{1}}{\mathrm{4}}\:\mathrm{times}\:\mathrm{as}\:\mathrm{great}\:\mathrm{in}\:\mathrm{30} \\ $$$$\mathrm{days}\:\mathrm{working}\:\mathrm{6}\:\mathrm{hrs}\:\mathrm{a}\:\mathrm{day}? \\ $$

Question Number 17096 Answers: 1 Comments: 0

The total number of solutions of the equation tan 3x − tan 2x − tan 3x tan 2x = 1 in [0, 2π] is

$$\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation} \\ $$$$\mathrm{tan}\:\mathrm{3}{x}\:−\:\mathrm{tan}\:\mathrm{2}{x}\:−\:\mathrm{tan}\:\mathrm{3}{x}\:\mathrm{tan}\:\mathrm{2}{x}\:=\:\mathrm{1}\:\mathrm{in} \\ $$$$\left[\mathrm{0},\:\mathrm{2}\pi\right]\:\mathrm{is} \\ $$

Question Number 16980 Answers: 0 Comments: 1

To Q16066: I have posted my solution there. Those who are intetested in this interesting question please have a critical view at it. Maybe there are alternative solutions which are easier and more direct and straight on.

$$\mathrm{To}\:\mathrm{Q16066}: \\ $$$$\mathrm{I}\:\mathrm{have}\:\mathrm{posted}\:\mathrm{my}\:\mathrm{solution}\:\mathrm{there}. \\ $$$$\mathrm{Those}\:\mathrm{who}\:\mathrm{are}\:\mathrm{intetested}\:\mathrm{in}\:\mathrm{this}\:\mathrm{interesting} \\ $$$$\mathrm{question}\:\mathrm{please}\:\mathrm{have}\:\mathrm{a}\:\mathrm{critical}\:\mathrm{view}\:\mathrm{at} \\ $$$$\mathrm{it}.\:\mathrm{Maybe}\:\mathrm{there}\:\mathrm{are}\:\mathrm{alternative}\:\mathrm{solutions} \\ $$$$\mathrm{which}\:\mathrm{are}\:\mathrm{easier}\:\mathrm{and}\:\mathrm{more}\:\mathrm{direct}\:\mathrm{and} \\ $$$$\mathrm{straight}\:\mathrm{on}. \\ $$

Question Number 16973 Answers: 1 Comments: 0

5^(log(x)) = x^(log(2)) , find x.

$$\mathrm{5}^{\mathrm{log}\left(\mathrm{x}\right)} \:=\:\mathrm{x}^{\mathrm{log}\left(\mathrm{2}\right)} ,\:\:\:\:\:\mathrm{find}\:\:\mathrm{x}. \\ $$

Question Number 16990 Answers: 1 Comments: 4

If for positive integers a and b, a + b = (a/b) + (b/a), find a^2 + b^2 .

$$\mathrm{If}\:\mathrm{for}\:\mathrm{positive}\:\mathrm{integers}\:{a}\:\mathrm{and}\:{b}, \\ $$$${a}\:+\:{b}\:=\:\frac{{a}}{{b}}\:+\:\frac{{b}}{{a}},\:\mathrm{find}\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} . \\ $$

Question Number 16958 Answers: 0 Comments: 0

Let ABCD be a quadrilateral with an inscribed circle. Prove that the circles inscribed in triangles ABC and ADC are tangent to each other.

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{quadrilateral}\:\mathrm{with}\:\mathrm{an} \\ $$$$\mathrm{inscribed}\:\mathrm{circle}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{circles} \\ $$$$\mathrm{inscribed}\:\mathrm{in}\:\mathrm{triangles}\:{ABC}\:\mathrm{and}\:{ADC} \\ $$$$\mathrm{are}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other}. \\ $$

Question Number 16951 Answers: 0 Comments: 0

Let M be a point in interior of ΔABC. Three lines are drawn through M, parallel to triangle′s sides, thereby producing three trapezoids. Suppose a diagonal is drawn in each trapezoid in such a way that the diagonals have no common endpoints. These three diagonals divide ABC into seven parts, four of them being triangles. Prove that the area of one of the four triangles equals the sum of the areas of the other three.

$$\mathrm{Let}\:{M}\:\mathrm{be}\:\mathrm{a}\:\mathrm{point}\:\mathrm{in}\:\mathrm{interior}\:\mathrm{of}\:\Delta{ABC}. \\ $$$$\mathrm{Three}\:\mathrm{lines}\:\mathrm{are}\:\mathrm{drawn}\:\mathrm{through}\:{M}, \\ $$$$\mathrm{parallel}\:\mathrm{to}\:\mathrm{triangle}'\mathrm{s}\:\mathrm{sides},\:\mathrm{thereby} \\ $$$$\mathrm{producing}\:\mathrm{three}\:\mathrm{trapezoids}.\:\mathrm{Suppose}\:\mathrm{a} \\ $$$$\mathrm{diagonal}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{in}\:\mathrm{each}\:\mathrm{trapezoid}\:\mathrm{in} \\ $$$$\mathrm{such}\:\mathrm{a}\:\mathrm{way}\:\mathrm{that}\:\mathrm{the}\:\mathrm{diagonals}\:\mathrm{have}\:\mathrm{no} \\ $$$$\mathrm{common}\:\mathrm{endpoints}.\:\mathrm{These}\:\mathrm{three} \\ $$$$\mathrm{diagonals}\:\mathrm{divide}\:{ABC}\:\mathrm{into}\:\mathrm{seven} \\ $$$$\mathrm{parts},\:\mathrm{four}\:\mathrm{of}\:\mathrm{them}\:\mathrm{being}\:\mathrm{triangles}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{four} \\ $$$$\mathrm{triangles}\:\mathrm{equals}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{areas} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{other}\:\mathrm{three}. \\ $$

Question Number 16947 Answers: 0 Comments: 0

Through the vertices of the smaller base AB of the trapezoid ABCD two parallel lines are drawn, intersecting the segment CD. These lines and the trapezoid′s diagonals divide it into seven triangles and a pentagon. Show that the area of the pentagon equals the sum of the areas of the three triangles that share a common side with the trapezoid.

Question Number 16946 Answers: 0 Comments: 0

Consider the quadrilateral ABCD. The points M, N, P and Q are the midpoints of the sides AB, BC, CD and DA. Let X = AP ∩ BQ, Y = BQ ∩ CM, Q = CM ∩ DN and T= DN ∩ AP. Prove that [XYZT] = [AQX] + [BMY] + [CNZ] + [DPT].

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{quadrilateral}\:{ABCD}. \\ $$$$\mathrm{The}\:\mathrm{points}\:{M},\:{N},\:{P}\:\mathrm{and}\:{Q}\:\mathrm{are}\:\mathrm{the} \\ $$$$\mathrm{midpoints}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sides}\:{AB},\:{BC},\:{CD} \\ $$$$\mathrm{and}\:{DA}. \\ $$$$\mathrm{Let}\:{X}\:=\:{AP}\:\cap\:{BQ},\:{Y}\:=\:{BQ}\:\cap\:{CM}, \\ $$$${Q}\:=\:{CM}\:\cap\:{DN}\:\mathrm{and}\:{T}=\:{DN}\:\cap\:{AP}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\left[{XYZT}\right]\:=\:\left[{AQX}\right]\:+\:\left[{BMY}\right] \\ $$$$+\:\left[{CNZ}\right]\:+\:\left[{DPT}\right]. \\ $$

Question Number 16944 Answers: 0 Comments: 2

Find the number of digits in the number 2^(2005) × 5^(2000) when written in full.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{digits}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{2}^{\mathrm{2005}} \:×\:\mathrm{5}^{\mathrm{2000}} \:\mathrm{when}\:\mathrm{written}\:\mathrm{in} \\ $$$$\mathrm{full}. \\ $$

Question Number 16969 Answers: 0 Comments: 0

Question Number 16942 Answers: 1 Comments: 0

A distance of 200 km is to be covered by car in less than 10 hours. Yash does it in two parts. He first drives for 150 km at an average speed of 36 km/hr, without stopping. After taking rest for 30 minutes, he starts again and covers the remaining distance non-stop. His average for the entire journey (including the period of rest) exceeds that for the second part by 5 km/hr. Find the speed at which he covers the second part.

$$\mathrm{A}\:\mathrm{distance}\:\mathrm{of}\:\mathrm{200}\:\mathrm{km}\:\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{covered}\:\mathrm{by} \\ $$$$\mathrm{car}\:\mathrm{in}\:\mathrm{less}\:\mathrm{than}\:\mathrm{10}\:\mathrm{hours}.\:\mathrm{Yash}\:\mathrm{does}\:\mathrm{it} \\ $$$$\mathrm{in}\:\mathrm{two}\:\mathrm{parts}.\:\mathrm{He}\:\mathrm{first}\:\mathrm{drives}\:\mathrm{for}\:\mathrm{150}\:\mathrm{km} \\ $$$$\mathrm{at}\:\mathrm{an}\:\mathrm{average}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{36}\:\mathrm{km}/\mathrm{hr}, \\ $$$$\mathrm{without}\:\mathrm{stopping}.\:\mathrm{After}\:\mathrm{taking}\:\mathrm{rest}\:\mathrm{for} \\ $$$$\mathrm{30}\:\mathrm{minutes},\:\mathrm{he}\:\mathrm{starts}\:\mathrm{again}\:\mathrm{and}\:\mathrm{covers} \\ $$$$\mathrm{the}\:\mathrm{remaining}\:\mathrm{distance}\:\mathrm{non}-\mathrm{stop}.\:\mathrm{His} \\ $$$$\mathrm{average}\:\mathrm{for}\:\mathrm{the}\:\mathrm{entire}\:\mathrm{journey} \\ $$$$\left(\mathrm{including}\:\mathrm{the}\:\mathrm{period}\:\mathrm{of}\:\mathrm{rest}\right)\:\mathrm{exceeds} \\ $$$$\mathrm{that}\:\mathrm{for}\:\mathrm{the}\:\mathrm{second}\:\mathrm{part}\:\mathrm{by}\:\mathrm{5}\:\mathrm{km}/\mathrm{hr}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{at}\:\mathrm{which}\:\mathrm{he}\:\mathrm{covers}\:\mathrm{the} \\ $$$$\mathrm{second}\:\mathrm{part}. \\ $$

Question Number 16940 Answers: 1 Comments: 1

Six points A, B, C, D, E, and F are placed on a square rigid, as shown. How many triangles that are not right-angled can be drawn by using 3 of these 6 points as vertices?

$$\mathrm{Six}\:\mathrm{points}\:\mathrm{A},\:\mathrm{B},\:\mathrm{C},\:\mathrm{D},\:\mathrm{E},\:\mathrm{and}\:\mathrm{F}\:\mathrm{are} \\ $$$$\mathrm{placed}\:\mathrm{on}\:\mathrm{a}\:\mathrm{square}\:\mathrm{rigid},\:\mathrm{as}\:\mathrm{shown}. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{triangles}\:\mathrm{that}\:\mathrm{are}\:\boldsymbol{\mathrm{not}} \\ $$$$\mathrm{right}-\mathrm{angled}\:\mathrm{can}\:\mathrm{be}\:\mathrm{drawn}\:\mathrm{by}\:\mathrm{using}\:\mathrm{3} \\ $$$$\mathrm{of}\:\mathrm{these}\:\mathrm{6}\:\mathrm{points}\:\mathrm{as}\:\mathrm{vertices}? \\ $$

Question Number 16938 Answers: 1 Comments: 2

The Object shown in the diagram is made by gluing together the adjacent faces of six wooden cubes, each having edges of length 2 cm. Find the total surface area of the object in square centimetres.

$$\mathrm{The}\:\mathrm{Object}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{the}\:\mathrm{diagram}\:\mathrm{is} \\ $$$$\mathrm{made}\:\mathrm{by}\:\mathrm{gluing}\:\mathrm{together}\:\mathrm{the}\:\mathrm{adjacent} \\ $$$$\mathrm{faces}\:\mathrm{of}\:\mathrm{six}\:\mathrm{wooden}\:\mathrm{cubes},\:\mathrm{each}\:\mathrm{having} \\ $$$$\mathrm{edges}\:\mathrm{of}\:\mathrm{length}\:\mathrm{2}\:\mathrm{cm}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{total} \\ $$$$\mathrm{surface}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{object}\:\mathrm{in}\:\mathrm{square} \\ $$$$\mathrm{centimetres}. \\ $$

Question Number 16936 Answers: 0 Comments: 5

In the diagram, it is possible to travel only along an edge in the direction indicated by the arrow. How many different routes from A to B are there in all?

$$\mathrm{In}\:\mathrm{the}\:\mathrm{diagram},\:\mathrm{it}\:\mathrm{is}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{travel} \\ $$$$\mathrm{only}\:\mathrm{along}\:\mathrm{an}\:\mathrm{edge}\:\mathrm{in}\:\mathrm{the}\:\mathrm{direction} \\ $$$$\mathrm{indicated}\:\mathrm{by}\:\mathrm{the}\:\mathrm{arrow}.\:\mathrm{How}\:\mathrm{many} \\ $$$$\mathrm{different}\:\mathrm{routes}\:\mathrm{from}\:\mathrm{A}\:\mathrm{to}\:\mathrm{B}\:\mathrm{are}\:\mathrm{there} \\ $$$$\mathrm{in}\:\mathrm{all}? \\ $$

Question Number 16935 Answers: 1 Comments: 0

A two-digit number has the property that the square of its tens digit plus ten times its units digit is equal to the square of its units digit plus ten times its tens digit. Find all two digit numbers which have this property, and are prime numbers.

$$\mathrm{A}\:\mathrm{two}-\mathrm{digit}\:\mathrm{number}\:\mathrm{has}\:\mathrm{the}\:\mathrm{property} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{square}\:\mathrm{of}\:\mathrm{its}\:\mathrm{tens}\:\mathrm{digit}\:\mathrm{plus} \\ $$$$\mathrm{ten}\:\mathrm{times}\:\mathrm{its}\:\mathrm{units}\:\mathrm{digit}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{square}\:\mathrm{of}\:\mathrm{its}\:\mathrm{units}\:\mathrm{digit}\:\mathrm{plus}\:\mathrm{ten}\:\mathrm{times} \\ $$$$\mathrm{its}\:\mathrm{tens}\:\mathrm{digit}.\:\mathrm{Find}\:\mathrm{all}\:\mathrm{two}\:\mathrm{digit} \\ $$$$\mathrm{numbers}\:\mathrm{which}\:\mathrm{have}\:\mathrm{this}\:\mathrm{property},\:\mathrm{and} \\ $$$$\mathrm{are}\:\mathrm{prime}\:\mathrm{numbers}. \\ $$

Question Number 16919 Answers: 1 Comments: 0

ax2+bx+c=0

$${ax}\mathrm{2}+{bx}+{c}=\mathrm{0} \\ $$

Question Number 16918 Answers: 1 Comments: 0

2x2+9x=10

$$\mathrm{2}{x}\mathrm{2}+\mathrm{9}{x}=\mathrm{10} \\ $$

Question Number 16917 Answers: 1 Comments: 0

If x + iy = (1/(a + ib)) prove that : (x^2 + y^2 )(a^2 + b^2 ) = 1

$$\mathrm{If}\:\:\:\mathrm{x}\:+\:\mathrm{iy}\:=\:\frac{\mathrm{1}}{\mathrm{a}\:+\:\mathrm{ib}} \\ $$$$\mathrm{prove}\:\mathrm{that}\::\:\:\:\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \right)\left(\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \right)\:=\:\mathrm{1} \\ $$

Question Number 16909 Answers: 1 Comments: 0

(√x^x^6 ) = 729 , Find x