SMO(senior)-2014 Problem 2 Number Theory

[et_pb_section fb_built="1" _builder_version="3.22.4" fb_built="1" _i="0" _address="0"][et_pb_row _builder_version="3.25" _i="0" _address="0.0"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.0.0"][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_padding="20px|20px|20px|20px" _i="0" _address="0.0.0.0"]

Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" _i="1" _address="0.0.0.1"]Find, with justification, all positive real numbers   $a,b,c$   satisfying the system of equations:    \[a\sqrt{b}=a+c,b\sqrt{c}=b+a,c\sqrt{a}=c+b.\][/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.27.4" hover_enabled="0" _i="1" _address="0.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.1.0"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.27.4" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" hover_enabled="0" _i="0" _address="0.1.0.0"][et_pb_accordion_item title="Source of the problem" open="off" _builder_version="3.27" hover_enabled="0" _i="0" _address="0.1.0.0.0"]SMO (senior)-2014 stage 2 problem 2

[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.0.1" open="on"]Number Theory[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.27.4" hover_enabled="0" _i="2" _address="0.1.0.0.2" open="off"]Easy [/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.0.3" open="off"]Excursion in Mathematics

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Start with hints

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[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27.4" hover_enabled="0" _i="1" _address="0.1.0.2.1"]Given all three relations are cyclic and symmetric . So without loss of generality it can be assumed that \( a \geq b \geq c >0 \) .     Then proceed .               [ Note \( (0, 0, 0) \) can't be a solution since \( a , b , c \) are positive reals .] [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27.4" hover_enabled="0" _i="2" _address="0.1.0.2.2"]So \( a \sqrt b = a + c \Rightarrow a(\sqrt b - 1) = c \ [and \ we \ have \ a \geq c]   \Rightarrow ( \sqrt b - 1 ) \leq 1 \Rightarrow b \leq 4 \)[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27.4" hover_enabled="0" _i="3" _address="0.1.0.2.3"]Similarly \( b \sqrt c = b + a \Rightarrow b(\sqrt c - 1) = a \  [and \ we \  have \ a \geq b ] \Rightarrow \sqrt c - 1 \geq 1 \Rightarrow c \geq 4 \)[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.27.4" hover_enabled="0" _i="4" _address="0.1.0.2.4"]

Till now we have \( b \leq 4 \ and  \  c \geq 4 \) , but we assumed that \( b \geq c \) . So it is clear that \( b = c =4 \)  \( \Rightarrow a = 4 \) also. So the only triplet \( (a , b , c)\) is \( (4,4 ,4) \) .  [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" min_height="12px" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" _i="3" _address="0.1.0.3"]

Connected Program at Cheenta

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Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.[/et_pb_blurb][et_pb_button button_url="https://cheenta.com/matholympiad/" url_new_window="on" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3" background_layout="dark" _i="5" _address="0.1.0.5"][/et_pb_button][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" _i="6" _address="0.1.0.6"]

Similar Problems

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Real Surds - Problem 2 Pre RMO 2017

Problem

Suppose (a, b) are positive real numbers such that (a \sqrt{a}+b \sqrt{b}=183 . a \sqrt{b}+b \sqrt{a}=182). Find (\frac{9}{5}(a+b)).

Hint 1

This problem will use the following elementary algebraic identity:

$(x+y)^3=x^3+y^3+3 x^2 y+3 x y^2$

Can you identify what is x and what is y?

Hint 2

background_video_pause_outside_viewport="on" tab_text_shadow_style="none" body_text_shadow_style="none"] Set $x=\sqrt{a}, y=\sqrt{b}$. Then the given information translates to

$$
x^3+y^3=183, x^2 y+x y^2=182
$$

This implies $(x+y)^3=(\sqrt{a}+\sqrt{b})^3=x^3+y^3+3\left(x^2 y+x y^2\right)=183+3 \times 182=729$ Finally taking cube root on both sides, we have $\sqrt{a}+\sqrt{b}=9$

Hint 3

Note that $\sqrt{a} b+a \sqrt{b}=182 \Rightarrow \sqrt{a} \sqrt{b}(\sqrt{a}+\sqrt{b})=182 \Rightarrow \sqrt{a b} \times 9=182$ So at this point we know $(\sqrt{a}+\sqrt{b})=9, \sqrt{a b}=\frac{182}{9}$. It should be easy to find the value of $\left.\frac{9}{( } 5\right)(a+b)$ from these relations.

Final Answer

$a+b=(\sqrt{a}+\sqrt{b})^2-2 \sqrt{a b}=9^2-2 \times \frac{182}{9}=\frac{365}{9}$

Hence $\frac{9}{5}(a+b)=\frac{9}{5} \times \frac{365}{9}=73$

Pre RMO 2017 Problem 13 Solution - Kite in a Circle

Cyclic Quadrilaterals are often important objects in a Geometry problem. Recognizing them can lead to a path to the solution. Pre RMO 2017 Problem 13 Solution is a part of our Pre-RMO problem solving series.

Also visit: Math Olympiad Program of Cheenta

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Pre RMO 2017

  1. How many positive integers less than \(1000\) have the property that the sum of the digits of each such number is divisible by \(7\) and the number itself is divisible by \(3\) ?
  2. Suppose \(a,b\) are positive real numbers such that \(a\sqrt{a}+b\sqrt{b}=183\). \(a\sqrt{b}+b\sqrt{a}=182\). Find \(\frac{9}{5}(a+b)\).
  3. A contractor has two teams of workers: team A and team B. Team A can complete a job in \(12\) days and team B can do the same in \(36\) days.Team A starts working on the job and team B joins team A after four days.The team A withdraws after two more days. For how many more days should team B work to complete the job?
  4. Let \(a,b\) be integers such that all the roots of the equation \((x^2+ax+20)(x^2+17x+b)=0\) are negative integers.What is the smallest possible value of \(a+b\)?
  5. Let \(u,v,w\) be real numbers in geometric progression such that \(u>v>w\). Suppose \(u^{40}=v^n=w^{60}\).Find the value of \(n\).
  6. Let the sum \(\sum_{n=1}^{9}\frac{1}{n(n+1)(n+2)}\) written in its lowest terms be \(\frac{p}{q}\).Find the value of \(q-p\).
  7. Find the number of positive integers \(n\), such that \(\sqrt{n}+\sqrt{n+1}<11\).
  8. A pen costs ₹ \(11\) and a notebook costs ₹ \(13\).find the number of ways in which a person can spend exactly ₹ \(1000\) to buy pens and notebooks.
  9. There are five cities \(A,B,C,D,E\) on a certain island.Each city is connected to every other city by road.In how many ways can a person starting from city \(A\) come back to \(A\) after visiting some cities without visiting a city more than once and without taking the same road more than once.? (The order in which he visits the cities also matters; e.g., the routes \({A}\to{B}\to{C}\to{A}\) and \({A}\to{C}\to{B}\to{A}\) are different. )
  10. There are eight rooms on the first floor of a hotel,with four rooms on each side of the corridor,symmetrically situated (that is each room is exactly oposite to one other room).Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in oposite rooms.In how many ways can the guests be accommodated?
  11. Let \(f(x)=sin\frac{x}{3}+cos\frac{3x}{10}\) for all real \(x\). Find the least natural number \(n\) such that \(f(n\pi+x)=f(x)\) for all real \(x\).
  12. In a class, the total number of boys and girls are in the ratio \(4:3\). On one day it was found that \(8\) boys and \(14\) girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class?
  13.  In a rectangle \(ABCD\). \(E\) is the midpoint of \(AB\): \(F\) is a point on \(AC\) such that \(BF\) is perpendicular to \(AC\): and \(FE\) perpendicular to \(BD\). Suppose \(BC=8\sqrt{3}\). Find \(AB\).
  14. Suppose \(x\) is a positive real number such that \(\{x\}.[x]\) and \(x\) are in a geometric progression. Find the least positive integer \(n\) such that \(x^n>100\). (Here \([x]\) denotes the integer part of \(x\) and \(\{x\}=x-[x]\).)
  15. Integers \(1,2,3,...,n\), where \(n>2\), are written on a board. Two numbers \(m,k\) such that \(1<{m}<{n}\), \(1<{k}<{n}\) are removed and the averege of the remaining numbers is found to be \(17\). What is the maximum sum of the two removed numbers?
  16. Five distinct \(2-digit\) numbers are in a geometric progression. Find the middle term.
  17. Suppose the altitudes of a triangle are \(10\), \(12\) and \(15\). What is its semi-perimeter?
  18. If the real numbers \(x,y,z\) are such that \(x^2+4y^2+16z^2=48\) and \(xy+4yz+2zx=24\), what is the value of \(x^2+y^2+z^2\)?
  19. Suppose \(1,2,3\) are the roots of the equation \(x^4+ax^2+bx=c\). Find the value of \(c\).
  20.  What is the number of triples \((a,b,c)\) of positive integers such that (i) \(a<b<c<10\) and (ii) \(a,b,c,10\) form the sides of a quadrilateral?
  21. Find the number of ordered triples \((a,b,c)\) of positive integers such that \(abc=108\).
  22. Suppose in the plane \(10\) pairwise nonparallel lines intersect one another. What is the maximum possible number of polygons (with finite areas) that can be formed?
  23.  Suppose an integer \(x\), a natural number \(n\) and a prime number \(p\) satisfy the equation \(7x^2-44x+12=p^n\). Find the largest value of \(p\).
  24. Let \(P\) be an interior point of a triangle \(ABC\) whose side lengths are \(26,65,78\). The line through \(P\) parallel to \(BC\) meets \(AB\) in \(k\) and \(AC\) in \(L\). The line through \(P\) parallel to \(CA\) meets \(BC\) in \(M\) and \(BA\) in \(N\). The line through \(P\) parallel to \(AB\) meets \(CA\) in \(S\) and \(CB\) in \(T\). If \(KL,MN,ST\) are of equal lengths, find this common length.
  25. Let \(ABCD\) be a rectangle and let \(E\) and \(F\) be points on \(CD\) and \(BC\) respectively such that area \((ADE)=16\) area \((CEF)=9\) and area \((ABF)=25\). What is the area of triangle \(AEF\)?
  26. Let \(AB\) and \(CD\) be two parallel chords in a circle with radius \(5\) such that the centre \(O\) lies between these chords. Suppose \(AB=6\), \(CD=8\). Suppose further that the area of the part of the circle lying between the chords \(AB\) and \(CD\) is \((m\pi+n)/k\), where \(m,n,k\) are positive integers with \(gcd(m,n,k)=1\). What is the value of \(m+n+k\)?
  27. Let \({\Omega}_{1}\) be a circle with centre \(O\) and let \(AB\) be a diameter of \({\Omega}_{1}\). Let \(P\) be a point on the segment \(OB\) different from \(O\). Suppose another circle \({\Omega}_{2}\) with centre \(P\) lies in the interior of \({\Omega}_{1}\). Tangents are drawn from \(A\) and \(B\) to the circle \({\Omega}_{2}\) intersecting \({\Omega}_{1}\) again at \(A_1\) and \(B_1\) respectively such that \(A_1\) and \(B_1\) are on the opposite sides of \(AB\). Given that \({A_1}B=5\), \(A{B_1}=15\) and \(OP=10\), find the radius of \({\Omega}_{1}\).
  28. Let \(p,q\) be prime numbers such that \(n^{3pq}-n\) is a multiple of \(3pq\) for all positive integers \(n\). Find the least possible value of \(p+q\).
  29. For each positive integer \(n\), consider the highest common factor \(h_n\) of the two numbers \(n!+1\) and \((n+1)!\). For \(n<100\), find the largest value of \(h_n\).
  30.  Consider the areas of the four triangles obtained by drawing the diagonals \(AC\) and \(BD\) of a trapezium \(ABCD\). The product of this areas, taken two at time, are computed. If among the six products so obtained two products are \(1296\) and \(576\), determine the square root of the maximum possible area of the trapezium to the nearest integer.