Algebra (Olympiad)32 questions

Algebra (Olympiad) — JEE Maths Practice Questions & Solutions

32 questions on Algebra (Olympiad) with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

hard
Let {an}\{a_n\} be a sequence of positive integers such that a1=a2=2a_1=a_2=2. For n≥1n\ge1,
an+2an−an+12=an+1an.a_{n+2}a_n-a_{n+1}^2=a_{n+1}a_n.
 Determine the value of a1a2a1032 a7\dfrac{a_1a_2a_{10}}{32\,a_7}.
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hard
Consider the polynomial
(x2+1)(x2+4)(x2−2x+2)(x2+2x+2).(x^2+1)(x^2+4)(x^2-2x+2)(x^2+2x+2).
 Let f(x)=(x4+2x2+2x)2+(P(x))2f(x)=\big(x^4+2x^2+2x\big)^2+\big(P(x)\big)^2, where P(x)P(x) is a cubic polynomial. Find the value of P(1) P(2)P(1)\,P(2)
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hard
Let
f(n)=12n3−5n2−251n+3896n2−37n+45.f(n) = \frac{12n^3 - 5n^2 - 251n + 389}{6n^2 - 37n + 45}.
 There is a unique positive integer nn for which f(n)f(n) is an integer. Denote this unique value of nn by pp. Find the value of p+f(p)p + f(p).
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hard
Let
x=1+112+122+1+122+132+1+132+142+⋯+1+1992+11002.x=\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+\cdots+\sqrt{1+\frac{1}{99^2}+\frac{1}{100^2}}.
 If [x][x] denotes the greatest integer not exceeding xx, then find the value of 100 (x−[x])100\,(x-[x])
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hard
Solve the system in real numbers:
(x−1)(y−1)(z−1)=xyz−1and(x−2)(y−2)(z−2)=xyz−2.(x - 1)(y - 1)(z - 1) = xyz - 1 \quad \text{and} \quad (x - 2)(y - 2)(z - 2) = xyz - 2.
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hard
The system of equations
log⁡10(2000xy)−(log⁡10x)(log⁡10y)=4,\log_{10}(2000xy)-(\log_{10}x)(\log_{10}y)=4,
log⁡10(2yz)−(log⁡10y)(log⁡10z)=1,\log_{10}(2yz)-(\log_{10}y)(\log_{10}z)=1,
log⁡10(zx)−(log⁡10z)(log⁡10x)=0\log_{10}(zx)-(\log_{10}z)(\log_{10}x)=0
 has two solutions in real numbers (x1,y1,z1)(x_1,y_1,z_1) and (x2,y2,z2)(x_2,y_2,z_2). Evaluate y1+y2y_1+y_2.
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hard
Find all positive integers xx such that 2x+12x + 1 is a perfect square but none of the integers 2x+2,2x+3,…,3x+32x + 2, 2x + 3, \ldots, 3x + 3 is a perfect square.
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hard
Find the number of positive integral solutions of the equation
x1+2(x1+x2)+3(x1+x2+x3)+4(x1+x2+x3+x4)+⋯+15(x1+x2+⋯+x15)1255.x_1 + 2(x_1 + x_2) + 3(x_1 + x_2 + x_3) + 4(x_1 + x_2 + x_3 + x_4) + \cdots + 15(x_1 + x_2 + \cdots + x_{15}) 1255.
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hard
If xx is a real number such that
x+(x+1)(x+2)+(x+2)(x+3)+(x+3)(x+1)=4,x + \sqrt{(x+1)(x+2)} + \sqrt{(x+2)(x+3)} + \sqrt{(x+3)(x+1)} = 4,
 find the value of PP, where x+37=P840x + \dfrac{3}{7} = \dfrac{P}{840}.
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hard
Evaluate
(20144+4×2013420132+40272)−(20124+4×2013420132+40252).\left(\frac{2014^4 + 4 \times 2013^4}{2013^2 + 4027^2}\right) - \left(\frac{2012^4 + 4 \times 2013^4}{2013^2 + 4025^2}\right).
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hard
There exist unique positive integers xx and yy such that x2+84x+2008=y2x^2 + 84x + 2008 = y^2. Find x+yx + y.
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hard
Determine all real numbers x>1x > 1, y>1y > 1, z>1z > 1 satisfying
x+y+z+3x−1+3y−1+3z−1=2(x+2+y+2+z+2).x + y + z + \frac{3}{x - 1} + \frac{3}{y - 1} + \frac{3}{z - 1} = 2\big(\sqrt{x + 2} + \sqrt{y + 2} + \sqrt{z + 2}\big).
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hard
The total number of all ordered triples (x,y,z)(x, y, z) of real numbers such that
x2−yz+xy+zx=82,y2−zx+xy+yz=−18,z2−xy+yz+zx=18.x^2 - yz + xy + zx = 82, \qquad y^2 - zx + xy + yz = -18, \qquad z^2 - xy + yz + zx = 18.
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hard
The number of pairwise distinct rational numbers x,y,zx, y, z such that
1(x−y)2+1(y−z)2+1(z−x)2=2014 \frac{1}{(x - y)^2} + \frac{1}{(y - z)^2} + \frac{1}{(z - x)^2} = 2014\,
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hard
If
Sn=∑k=1nk(2n−2k+1)(2n−k+1)andTn=∑k=1n1k,S_n=\sum_{k=1}^{n}\frac{k}{(2n-2k+1)(2n-k+1)}\qquad\text{and}\qquad T_n=\sum_{k=1}^{n}\frac{1}{k},
 then find the value of TnSn\dfrac{T_n}{S_n}.
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hard
The number of real values of x,y,zx, y, z for satisfying the equations
x2−y=z−1,y2−z=x−1,z2−x=y−1.\sqrt{x^2 - y} = z - 1, \qquad \sqrt{y^2 - z} = x - 1, \qquad \sqrt{z^2 - x} = y - 1.
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medium
The all pairs of real numbers (x,y)(x, y) satisfying (2x+1)2+y2+(y−2x)2=13(2x + 1)^2 + y^2 + (y - 2x)^2 = \dfrac{1}{3}.
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medium
The number of all real pairs (a,b)(a, b) satisfying a2+b2=25a^2 + b^2 = 25 and 3(a+b)−ab=153(a + b) - ab = 15.
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medium
If n+20n + 20 and n−21n - 21 are both perfect squares, where nn is a natural number, find nn.
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medium
The nonzero real numbers a,b,ca, b, c satisfy the system
a2+a=b2,b2+b=c2,c2+c=a2.a^2 + a = b^2, \qquad b^2 + b = c^2, \qquad c^2 + c = a^2.
 find the value of the expression (a−b)(b−c)(c−a)(a-b)(b-c)(c-a).
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medium
Let MM be the maximum value of the expression
A=x4y+x3y+x2y+xy+xy2+xy3+xy4,A=x^4y+x^3y+x^2y+xy+xy^2+xy^3+xy^4,
 subject to x+y=3x+y=3, where xx and yy are real numbers. Find the greatest integer not exceeding MM.
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medium
Calculate the value of 114+1004+11142\sqrt{\dfrac{11^4 + 100^4 + 111^4}{2}}.
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medium
Find the greatest value of xx for which
(13x−x2x+1)(x+13−xx+1)=42.\left(\frac{13x-x^2}{x+1}\right)\left(x+\frac{13-x}{x+1}\right)=42.
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medium
Find all real numbers xx such that
9x+4x+1=6x+3x+2x.9^x + 4^x + 1 = 6^x + 3^x + 2^x.
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medium
The product of the expression
(5+6+7)(−5+6+7)(5−6+7)(5+6−7)(\sqrt5+\sqrt6+\sqrt7)(-\sqrt5+\sqrt6+\sqrt7)(\sqrt5-\sqrt6+\sqrt7)(\sqrt5+\sqrt6-\sqrt7)
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medium
The number of all positive integers aa and bb such that a2−b2=18a^2 - b^2 = 18.
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medium
The number of all real numbers xx satisfying 2x+3x−4x+6x−9x=12^x + 3^x - 4^x + 6^x - 9^x = 1.
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medium
The number of pairs of positive integers pp and qq that satisfy p2=q2+p+q+2018p^2 = q^2 + p + q + 2018.
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medium
Let aa and bb be real numbers such that a4+a2b2+b4=900a^4 + a^2 b^2 + b^4 = 900 and a2+ab+b2=45a^2 + ab + b^2 = 45. Find the value of 2ab2ab.
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medium
If a2x3+b2y3+c2z3=p na^2x^3+b^2y^3+c^2z^3=p^{\,n}, where ax2=by2=cz2ax^2=by^2=cz^2 and 1x+1y+1z=1p\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{p}, and it is given that a+b+c=p\sqrt{a}+\sqrt{b}+\sqrt{c}=\sqrt{p} (all quantities positive), then find the value of nn.
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medium
Determine the unique pair of real numbers (x,y)(x, y) that satisfy (4x2+6x+4)(4y2−12y+25)=28(4x^2 + 6x + 4)(4y^2 - 12y + 25) = 28.
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medium
If a,b,c,da, b, c, d are real numbers such that a+b+c+d=20a + b + c + d = 20 and ab+ac+ad+bc+bd+cd=150ab + ac + ad + bc + bd + cd = 150, then find the value of
a3+b3+c3+d3a2+b2+c2+d2.\frac{a^3 + b^3 + c^3 + d^3}{a^2 + b^2 + c^2 + d^2}.
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