Mastering Olympiad Mathematics

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Showing posts with label rule out the possibility. Show all posts

Showing posts with label rule out the possibility. Show all posts

Saturday, July 18, 2015

Solve in positive integers the equation $ab(a+b-10)+21a-3a^2+16b-2b^2=60$: First Solution

Solve in positive integers the equation $ab(a+b-10)+21a-3a^2+16b-2b^2=60$.

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Labels: discriminant a perfect square, discriminant greater than or equal to zero, discriminant of quadratic, interval, positive small number, rule out the possibility, set up inequality, solve in positive integers

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((ax+by)^3-3axby(ax+by)) (\cos^4 x+\sin x\cos^3x+\sin^2 x\cos^2 x+\sin^3x\cos x)(\sin^4 x+\cos^4 x+2\sin x\cos^3 x+2\sin^2 x\cos^2 x+2\sin^3 x\cos x) (1+tan 1)(1+tan 2)(1+tan 3)(1+tan 4)...(1+tan 45)=2^(23) (1+tan n)(1+tan(45-n)) (a^3-b^3)(x^3-y^3) (b-1/2)² (b²-1)² (b²-2)² (b²+1)² (n^3-1)/(n^3+1) (x-y)(y-z)(z-x)/(x+y)(y+z)(z+x) (x+y+z)^2=x^2+y^2+z^2+2(xy+xz+yz) |sin x+cos x+tan x+cot x+ sec x+csc x| √1000- √998> √998- √996 √2 √2-1 √2+√5+√11 √2+1 √2cos(x/5-π/12)-√6sin(x/5-π/12)=2(sin(x/5-2π/3)-sin(3x/5+π/6)) 1+√2 1+tan n 2^x/y! 2015 Edexcel 2015 IMO contest problem 21st century learners 21th USA mathematical olympiad 1992 problem 22/7>pi 45^2>1996 6 of the sweets are orange 7 9/12+5/34+7/68=1 A a circle centered at with radius a is rational a is real parameter a quadratic a square a total of 26 possible cases a-c+b=2y a-c=y-z a-c>b-d a^3-b^3=(a-b)(a^2+ab+b^2) a^3x^3+b^3y^3 a^3y^3+b^3x^3 a^4+a^3 x+a^2 (x^2+1)+a x^3+x^4+x^2+a x>0 a+b=9 a+c>b+d a>b>0 ab+c+d=29 abilities abstractly ac>bd accessible ad+bc=39 add add pairs up add them up add up to adding adding both we obtain adding quantity adding up addition addition/subtraction additive number theory adopting different point of view advanced and critical problem solving skills Albert Einstein algebra algebraic manipulate algebraic manipulation algebraic manipulation skill algebraic manipulation skills algebraic method algebraically algebraically manipulate alternative altitude always greater than or equal to zero always learn more always less than zero AM-GM AM-GM inequality ambiguity AMPO mock problem analogous to the odd function rule analytic geometry analytical and critical thinking angle angle approaches zero angle sum and difference identities another expression of x^4 as a quadratic equation in x anti-derivative any one of the term to the right side APMO Olympiad application applications apply AM-GM inequality approach approach the problem arc length arc-length are roots to the equation area will be positive argument argument for logarithmic arithmetic arithmetic sequence arithmetic series Asian Pacific Mathematics Olympiad Mock Problem assessment assume is true attacking problem attempt attention to detail auxiliary equation average ax^2+by^2=49 ax^3+by^3=133 ax^4+by^4=406 ax^5+by^5 ax+by=7 axis of symmetry b B are acute angles b-a+c=2z b-a=z-x b^2=4ac is a real root of ax^2+bx+c=0 b>3 back substitute back substituting basic fundamental behavior of graph best observation best solutions binomial cube binomial probability formula binomial sum binomial theorem both a and b are positive integers both sides equal to zero bounded by brain power enrichment quiz brain power enrichment quiz II breaking problem into cases Breakthrough Prize in Mathematics brone bronze by observation c c-b+a=2x c-b=x-y Calculate calculus calculus method calculus optimization can't be a square cancel out all the terms leaving Cardano's cubic formula Cauchy Schwarz inequality cd=18 certainty challenge change of base change the inequality Chebyshev's theorem check comprehension check maximum and minimum points check validity by substituting back into original given equation checking it for each case choose wisely circle classic example classic proof cleaning up clear clear fraction clearing all of the square roots clearing fractions closed form co-function identities co-function identity coefficient coefficients coefficients of x^2 and x collect like terms collect terms with same power combinatorial formula n choose r combinatorics combining steps combining the results combining the signs common factor common ratio common term compare compare predictions compete effectively complete the square completes the proof completing the square complex complex roots complexity complicated compound angle compute the differential concave down concave down on internal (0 concave up conceptual understanding conclude conclude by now conditions confusion conjecture conjectures consecutive considering continue in such a manner continuous contradicting our assumption contradiction converse must be true convert cosecant function to sine function convert exponential to logarithmic form coordinate cos 1°+cos 2°+...+cos 44° cos 1°+cos2°+...+cos 44° cos 2A=(1-tan²A)/(1+tan²A) cos ex cos n+sin n cos^4x-sin^2x cosecant cosine Cosine rule cosine terms counter example counter-example creative creative intelligence creative teaching methodology creativity credible plan of attack critical thinker critical value critically and creatively cross multiply cube both sides cubic equation cubic polynomial cultivate cultivated curious curve cyclic symmetry d d are negative or positive real number data deep content knowledge definite integral degrees denominator Descartes' Rule of Signs Descartes's Rule descending powers Determine if n^2-21n+111 is or is not a perfect square determine if n²-21n+111 is or is not a perfect square determining develop power of observation developing ideas developing students critical-thinking skills diagram diameter difference of cubes difference of fifth exponents difference of logarithm terms difference of squares differentials differentiate the rational function differentiating differentiatio differentiation differentiation method difficult difficult to relate math to real world directly proves discern a pattern discover pattern discovery discovery and invention discriminant discriminant a perfect square discriminant greater than or equal to zero discriminant is less than zero discriminant must be greater than zero discriminant of quadratic distance distance between (2 distance formula distributing divide by sin x sin(x+1) divide into cases divide through divide/multiply dividing out common factors divisible divisor do not limit your thought and option domain double angle formulas double-angle identity dummy variable dynamism easy to check easy to deduce eats the sweet eats two orange sweets educational development Edward Witten effective effective instructional skills effective mathematics instruction elegant proof for inequality involving pi elegantly elegantly and neatly elementary elementary method elements eliminate elimination method elimination route enlighting and inspiring ensure country's long-term economic prosperity equality is attained when 2A+B is a right angle equality must hold equality occurs in AM-GM inequality equals zero equate equate the coefficients equating equating the coefficients equation equation has integer roots equation in terms of equilateral triangle equivalent establish validity estimate estimate the sum evaluate evaluate a+2b evaluate ab+bc+ca evaluate ax^5+by^5 evaluate x/(x+y)+y/(y+z)+z/(z+x) evaluate x/y evaluate xyz evaluating definite integral exact exact solution exact values examine excellent exists root expand expand sin 9x expand the first two factors of the RHS and then compare the coefficients expand the LHS expanding expanding and grouping like terms expanding and simplifying expanding equation exploration and experimentation. explore explore consequences Exponential exponentiate exponentiate both sides exposing students express in the form a+b√c expressions exquisite problem extreme cases f(x) is symmetric w.r.t. the vertical line x=pi/4 factor factor polynomial using the rational root test factor the quartic factored factoring factoring sum and difference of squares or cubes factoring we get factorization factorize factorized format familiarity comfort zones holding us back family of curves Field Medal Fields Medal Find all positive integers n where √(n+√1996) exceeds √(n-1) by an integer find all possible values for find exact real root find greatest possible value of tan B find maximum f(x)=(x^4-x^2)/(x^6+2x^3-1) Find minimum find minimum xy find nth term for the series Find sum find the product of all possible values of x find the range of values of x find the sum of all possible finding pattern finite interval first assume the given inequality holds first case first constraint becomes first derivative first derivative test first equation becomes first limit goes to zero first quadrant first term first two expressions floor function form a quartic equation as the product of two quadratic equations foster from the inequality formula FTOC function fundamentals gap between the approximation and the original function quickly vanishes general case general term of the series generate geometric geometric series get rid of c or d get rid of fourth root get rid of the square root given sum is zero given x is an integer gives three real roots gold medal good math challenge GPS tracking system gradient graph graph behaves on interval graphical graphical and integration method greater opportunities greater than greater than zero greatest integer function greatest possible value of tan B Green-Tao theorem group pair grouping grows faster than guidance Hannah takes at random Hard indefinite integral hard indefinite integral problem hard Olympiad mathematics problems hard to factorize harmonic analysis has integer roots has integer solutions only if the discriminant is a perfect square heuristic heuristic approach heuristic mathematical problem solving skills heuristic problem solving skills heuristic skills heuristic solution heuristic strategies heuristic strategy hypotenuse ICT ICT tool identity image IMO IMO contest IMO contest problem IMO difficult problem IMO integration problem IMO Mock Trigonometric IMO Mock Trigonometric Math Quiz IMO Optimization IMO problem IMO problems IMO solve equation IMO Trigonometric Problem implies improve improve thinking skill in descending order of x in the same way in turn suggest in-depth understanding indefinite integral independent independent learning independent problem solvers induction method industry saver inequality inequality method infinitely many prime numbers Infinitely many primes infinity infinity) inflexion points initiate learning insightful insights and limitations instill genuine interest in creative problem solving from an early age Institute for Advanced Study integer integers integral integral calculus integral from 0 to pi/2 integrand integrand of rational function integrating integrating factor integration method intelligent guessing interest intermediate value theorem International Mathematical Olympiad contest problems intersect one and only one intersection interval interval notation intervals intriguing intriguing math problem invaluable inverse tangent inverse tangent formula with pi inverses of each other irrational is equivalent to Isaac Newton isolate all terms to the left it must be true that it's obvious that Jensen's inequality Jensen's inequlaity k=1 key to success Lagrange multipliers method larger than the integral largest addend possible launch angle learning let k=1 LHS LHS and RHS of the inequality LHS can be simplified LHS of the identity LHS set to take the zero value limit limit property limit theorems linear linear combination linear homogeneous recursion linear power list out all possible cases ln x<x-1 for all x>1 ln(2)) and (4 ln(4)) locus logarithmic inequality logarithms logarithms of the bases of logic logical implication logical thinking long division long queue look from all angles Los Angeles lower and upper bound lower bound luck M-theory Maclaurin series make as the subject make c+d the subject make full use make x^3 the subject making a drawing mapping math challenge math educators math joke math professor Math quizzes math series math teacher guide mathematical tree fall over mathematics proficient students maxima maximal maximize maximum maximum attained mean meaningful problem-solving investigations measure growth in knowledge method midpoint minimize minimum minimum or inflexion minimum value is attained minimum xy mock IMO algebra Mock IMO contest problem modular arithmetic more accurate Mozart of Math Mozart of Maths multiply multiply both sides multiply together multiplying both sides n is integer n sweets in a bag n^2-1=(n-1)(n+1) n^2-2n+1=(n-1)^2 n^2+2n+1=(n+1)^2 narrow gap nasty denominator natural logarithm function natural number nature of function network new perspective Newton's method next square number after no real roots normal not straightforward note 4n+1 and 4n+3 are neither square at the same time notice that nth term number of cases possible numerator numerator and denominator are symmetric about the same value of x objective function observation observe observing that obtuse ODE Olympiad Algebra Olympiad Algebra Problem Olympiad Math trigonometry one of the solutions one or both factors must be 0 only squares differ by 7 opposite sign optimization hard problem or x + x^2 + ... + x^(10) = 8 - 10x^(11) ordered pairs organizing data original equations are satisfied orthogonal trajectory parabola parameter partial derivative partial differential equations partial fraction pattern peak period perfect squares periodic perpendicular perseverance pick plan of attack plug in the values point point-slope formula polynomial polynomial couldn't be factorized polynomial has other real roots polynomial long division polynomial long division to factor positive attitudes positive integer positive integers positive real numbers positive real root positive root positive roots positive small number possibility previously established result to prove prime prime factor probability problem solver problem solving problem solving skills problem-approach behaviors procedural literacy product product from n=2 to infinity product is the same product of the integers root must be another integer product of three positive integers product of two factors product to sum identity for tangent function product-to-sum formulas product-to-sum identities productive thinking proficiency in math proficient problem solver profound impact projectile Proof by contradiction properties of definite integrals property of symmetry for the integrand prove prove (a+b)/2>(a-b)/(ln a-ln b) prove ⌊√n+√(n+1)⌋=⌊√(4n+2)⌋ for all n is natural number prove \tan^2 20°+\tan^2 40°+\tan^2 80°=33 prove √2+√3>pi prove by contradiction prove has no root greater than 1 prove inequalities IMO problem prove that is the case Prove x^7-2x^5+10x^2-1 has no root greater than 1 proving by contradiction putnam contest putnam contest problem Putnam Definite Integral Problem putting all together Pythagorean identity pythagorean theorem quadratic quadratic equation quadratic expressions quadratic formula quadratic polynomial quality standard quantities quantity quantum gravity quartic Quiz quotient being summed quotient rule radians radius raise both sides of the equation by a quantity raise both sides to the third power raising it to the third power range ratio rational function rational number rational roots test rationalization of the numerator of the sine function rationalize the denominator rationalizing denominator reached to a contradiction real number line real numbers real roots have no exact values real scenarios real world application of math rearrange rearrange terms in descending powers of x rearranging reciprocal reciprocals rectangular recursion reduces to reflective teaching model reformation relate relation replace replace x by x+1 replaceing the pieces in the original interal restrict results in complex values retailers reversed order rewrite rewrite the equation rewrite the function rewriting the equation RHS RHS can be simplified rich reward right angled triangle right-angled triangle right-angled triangles role model answer roots roots of equations roots of the quadratic equation rule out the possibility same argument same length satisfactory solution satisfy satisfy the system saving lives second case second derivative test second factor second limit is bounded self reflective thinking separable series series telescopes set up inequality sextic polynomial shift thinking style shortcut shortcut solution show n^2-n-90=0 show that sum show that sum from n=0 to 88 of 1/(cos nk cos (n+1)k)=(cos k)/sin^2 k) show with proof shown as desired silver similarity simplicity's sake simplification simplified simplified down to simplify simplify (ay+bx)^3-(ax+by)^3+(a^3-b^3)(x^3-y^3) simplifying sin 1°+sin2°+...+sin 44° sin 1=sin((x+1)-x)=sin(x+1)cosx-cos(x+1)sinx sin 2A=2tanA/(1+tan²A) sin^4x-cos^2x sine Singapore skills slope slope-intercept small angle approximation solution solutions solve solve complex and unfamiliar problems solve difficult solve each case Solve for all 6 complex roots of the equation x^6+10x^5+70x^4+288x^3+880x^2+1600x+1792=0 solve for polynomial of degree seven solve for positive integers solve for real solutions solve for the values solve in positive integers solve in positive integers the equation ab(a+b-10)+21a-3a^2+16b-2b^2=60 solve n^2-n-90=0 solve system of equations solve system of equations in real a solve the equation x+a^3=\sqrt[3]{a-x} solve the system solve trigonometric equation solving a simple analogous problem solving equations solving for solving it for k solving problems solving this system solving trigonometric equation spot a pattern square square both sides square given equation square root squared term on LHS squaring both sides squaring both sides of the inequality standard deviation static thinking stationary point step 1 step 2 step 3 stimulate straightforward application of the identity strategic competence strictly increasing Strictly increasing/decreasing function subject to the constraints substititue into substitute substituting substitution substitution method substitution or elimination substitution skill substitution technique subtracted subtraction sufficiently small sum sum is 1 sum of 2 cubes sum of all possible a sum of first n terms sum of logarithm is less than zero sum of squares sum of the exponents sum of the given expression sum of the roots sum of the three roots sum of three square terms sum of two logarithm terms sum of two squares sum the series sum-to-product sum-to-product formula sum-to-product identity sum/difference summation of series super symmetric quantum field theories superstring theory symmetric system of equations system of two nonlinear equations tackle problem tackling problem take advantage take natural logarithm take square of both sides taking square root taking square root of both sides tan 45° tan 55°tan65°tan75°=tan85° tan(45-n) tan^2 x+tan^2(60-x)+tan^2(60+x)=9tan^2 3x+6 tangent tangent line target expression teach with example teaching teaching for mastery teaching methodology teaching pedagogy technique of substitution techniques of counting technology tedious way telescoping series telescoping sum Terence Tao the area bounded by the difference is less than zero the gradient the graph of y=log_3 x the number of customers the only perfect squares between the only way can be zero the product of the two integer roots the real root of + x^2 + ... + x^8 = 8 - 10x^9 the real root of x+ x^2 + ... + x^8 = 8 - 10x^9 or the real root of x+ x^2 + ... + x^(10) = 8 - 10x^(11) the rest are yellow the sum of first n terms the sum of the two integer roots theoretical physicist there is only a solution think logically think out of the box bring new perspective in thinking flexibly and creatively thinking outside the box thinking strategy thought-provoking three smaller triangles tracking down stolen cell phones traditional traditional practice Training for heuristic problem solving skills traits trapezoid rule trial and error trials trials and errors triangle trigonometric trigonometric identities trigonometric identity trigonometric substitution trigonometry problem triple angle formula triple angle formula for sin 3x triple angle formula for tan x triple angle formulas for cosine and sine function Twin Prime Conjecture two integer real roots UCLA ultimate goal undefined understand the issues unfamiliar experiences University of California unproductive upon expanding equation upper bound for inequality USA Mathematical Olympiad 1989 useful identity in telescoping useful math using recursive definition variable x variables varying assumptions versatile talents vertices visualization effect we are hence done we can conclude by now that is a square we have proved that we may compute we may state which is larger which is smaller which isn't an integer which one is greater which represents y= √x without the help of calculator Without using calculator wolfram alpha wolfram to solve polynomial functions wonder work backwards method working backwards wrong substitution x x_i x/((sin x+cos x)cos x) x^2+4y^2+9z^2=12 x^2+y^2+z^2=7 x^3+20x x^4+x^3-x^2-x+1>0 for x>1 x^5-20x x+2y+3z=6 x+a^3=\sqrt[3]{a-x} x=2y=3z x=a xy+xz+yz=4 y y are positive integers y=ln(x) y=x yields yields two results z are real z real numbers

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