Olympiad problems

2016 District Olympiad, 3

[b]a)[/b] Prove that, for any integer $ k, $ the equation $ x^3-24x+k=0 $ has at most an integer solution. [b]b)[/b] Show that the equation $ x^3+24x-2016=0 $ has exactly one integer solution.

2020-IMOC, N1

Tags: equation , integer , number theory

$\textbf{N1.}$ Find all nonnegative integers $a,b,c$ such that \begin{align*} a^2+b^2+c^2-ab-bc-ca = a+b+c \end{align*} [i]Proposed by usjl[/i]

1949-56 Chisinau City MO, 54

Tags: algebra , equation

Solve the equation: $$\frac{x^2}{3}+\frac{48}{x^3}=10 \left(\frac{x}{3}-\frac{4 }{x} \right)$$

2011 Croatia Team Selection Test, 4

Tags: number theory , relatively prime , diophantine equation , equation

Find all pairs of integers $x,y$ for which \[x^3+x^2+x=y^2+y.\]

1984 IMO Shortlist, 3

Tags: number theory , equation , divisor

Find all positive integers $n$ such that \[n=d_6^2+d_7^2-1,\] where $1 = d_1 < d_2 < \cdots < d_k = n$ are all positive divisors of the number $n.$

2008 Hanoi Open Mathematics Competitions, 4

Tags: algebra , number theory , equation , relatively prime

Prove that there exists an infinite number of relatively prime pairs $(m, n)$ of positive integers such that the equation \[x^3-nx+mn=0\] has three distint integer roots.

1969 IMO Longlists, 17

Tags: arithmetic sequence , algebra , number theory , equation

$(CZS 6)$ Let $d$ and $p$ be two real numbers. Find the first term of an arithmetic progression $a_1, a_2, a_3, \cdots$ with difference $d$ such that $a_1a_2a_3a_4 = p.$ Find the number of solutions in terms of $d$ and $p.$

2005 Croatia National Olympiad, 1

Tags: algebra , equation , polynomial

Let $a \not = 0, b, c$ be real numbers. If $x_{1}$ is a root of the equation $ax^{2}+bx+c = 0$ and $x_{2}$ a root of $-ax^{2}+bx+c = 0$, show that there is a root $x_{3}$ of $\frac{a}{2}\cdot x^{2}+bx+c = 0$ between $x_{1}$ and $x_{2}$.

1991 Austrian-Polish Competition, 1

Tags: combination , algebra , equation , austrian polish

Show that there are infinitely many integers $m \ge 2$ such that $m \choose 2$ $= 3$ $n \choose 4$ holds for some integer $n \ge 4$. Give the general form of all such $m$.

1977 Chisinau City MO, 135

Tags: equation , algebra

Solve the equation: $$x=1978 - \dfrac{1977}{1978 - \dfrac{1977}{\frac{...}{...\dfrac{1977}{1978 -\dfrac{1977}{x}}}}}{}$$

2020 Tuymaada Olympiad, 1

Tags: equation , number theory , algebra , system of equations

Does the system of equation \begin{align*} \begin{cases} x_1 + x_2 &= y_1 + y_2 + y_3 + y_4 \\ x_1^2 + x_2^2 &= y_1^2 + y_2^2 + y_3^2 + y_4^2 \\ x_1^3 + x_2^3 &= y_1^3 + y_2^3 + y_3^3 + y_4^3 \end{cases} \end{align*} admit a solution in integers such that the absolute value of each of these integers is greater than $2020$?

2011 N.N. Mihăileanu Individual, 2

Tags: equation , inequalities , algebra

Determine the real numbers $ x,y,z $ from the interval $ (0,1) $ that satisfies $ x+y+z=1, $ and $$ \sqrt{\frac{x(1-y^2)}{2}} +\sqrt{\frac{y(1-z^2)}{2}} +\sqrt{\frac{z(1-x^2)}{2}} =\sqrt{1+xy+yz+zx} . $$ [i]Gabriela Constantinescu[/i]

1989 IMO Shortlist, 3

Tags: calculus , algebra , rectangle , equation , area

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?

2015 Ukraine Team Selection Test, 10

Tags: number theory , equation

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

1968 IMO, 6

Tags: floor function , number theory , equation , summation

Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \] [hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]

2006 Estonia Math Open Junior Contests, 1

Tags: algebra , equation

The paper is written on consecutive integers $1$ through $n$. Then are deleted all numbers ending in $4$ and $9$ and the rest alternating between $-$ and $+$. Finally, an opening parenthesis is added after each character and at the end of the expression the corresponding number of parentheses: $1 - (2 + 3 - (5 + 6 - (7 + 8 - (10 +...))))$. Find all numbers $n$ such that the value of this expression is $13$.

2015 Hanoi Open Mathematics Competitions, 8

Tags: algebra , equation

Solve the equation $(x + 1)^3(x - 2)^3 + (x -1)^3(x + 2)^3 = 8(x^2 -2)^3.$

1998 Junior Balkan Team Selection Tests - Romania, 1

Tags: equation , algebra , diophantine equation

Solve in $ \mathbb{Z}^2 $ the following equation: $$ (x+1)(x+2)(x+3) +x(x+2)(x+3)+x(x+1)(x+3)+x(x+1)(x+2)=y^{2^x} . $$ [i]Adrian Zanoschi[/i]

2015 IFYM, Sozopol, 8

Tags: algebra , equation

The sequence of real numbers $a_1,a_2,...,a_{2015}$ is such that the 2015 equations: $a_1^3=a_1^2;a_1^3+a_2^3=(a_1+a_2 )^2;...;a_1^3+a_2^3+...+a_{2015}^3=(a_1+a_2+...+a_{2015} )^2$ are true. Prove that $a_1,a_2,…,a_{2015}$ are integers.

1953 Moscow Mathematical Olympiad, 256

Tags: equation , factorial , algebra

Find roots of the equation $$1 -\frac{x}{1}+ \frac{x(x - 1)}{2!} -... +\frac{ (-1)^nx(x-1)...(x - n + 1)}{n!}= 0$$

1980 IMO, 2

Tags: algebra , sequence , equation

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

2011 Hanoi Open Mathematics Competitions, 9

Tags: algebra , equation

Solve the equation $1 + x + x^2 + x^3 + ... + x^{2011} = 0$.

VII Soros Olympiad 2000 - 01, 9.5

Tags: parameterization , algebra , cubic , equation

For all valid values of $a$ and $b$, solve the equation $$\frac{x^3}{(x-a) (x-b)} +\frac{a^3}{(a-b) (a-x)} + \frac{b^3}{ (b-x) (b-a)}= x^2 + a + b$$

1966 IMO Shortlist, 40

Tags: parametric equation , equation , algebra

For a positive real number $p$, find all real solutions to the equation \[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]

2016 Dutch BxMO TST, 1

Tags: number theory , divisor , odd , equation

For a positive integer $n$ that is not a power of two, we define $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.