Olympiad problems

2005 Croatia National Olympiad, 1

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 $x_{2}$.

2018 Ramnicean Hope, 1

Tags: algebra , equation , monotone

Solve in the real numbers the equation $ \sqrt[5]{2^x-2^{-1}} -\sqrt[5]{2^x+2^{-1}} =-1. $ [i]Mihai Neagu[/i]

1971 IMO Shortlist, 8

Tags: equation , root , polynomial , algebra

Determine whether there exist distinct real numbers $a, b, c, t$ for which: [i](i)[/i] the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$ [i](ii)[/i] the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$ [i](iii)[/i] the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$

2012 Centers of Excellency of Suceava, 4

Tags: cartesian circle , algebra , equation , monotone

Solve in the reals the following system. $$ \left\{ \begin{matrix} \log_2|x|\cdot\log_2|y| =3/2 \\x^2+y^2=12 \end{matrix} \right. $$ [i]Gheorghe Marchitan[/i]

2015 Germany Team Selection Test, 1

Tags: equation , number theory

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]

2014 BMT Spring, 1

Tags: algebra , equation

Find all real numbers $x$ such that $4^x-2^{x+2}+3=0$.

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]

2011 India IMO Training Camp, 2

Tags: modular arithmetic , number theory , equation , lte lemma

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2002 IMO Shortlist, 3

Tags: algebra , polynomial , equation , infinitely many solutions

Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are integers and $a\ne0$. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.

1967 IMO Shortlist, 3

Tags: algebra , polynomial , summation , equation , diophantine equation

Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$

2023 Turkey Team Selection Test, 3

Tags: number theory , equation

For all $n>1$, let $f(n)$ be the biggest divisor of $n$ except itself. Does there exists a positive integer $k$ such that the equality $n-f(n)=k$ has exactly $2023$ solutions?

1996 IMO Shortlist, 4

Tags: floor function , number theory , equation , algebra

Find all positive integers $ a$ and $ b$ for which \[ \left \lfloor \frac{a^2}{b} \right \rfloor \plus{} \left \lfloor \frac{b^2}{a} \right \rfloor \equal{} \left \lfloor \frac{a^2 \plus{} b^2}{ab} \right \rfloor \plus{} ab.\]

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 $n = 3t(n) + 5r(n)$.

2003 Gheorghe Vranceanu, 1

Tags: floor function , algebra , equation

Solve in $ \mathbb{R}^2 $ the equation $ \lfloor x/y-y/x \rfloor =x^2/y+y/x^2. $

2002 Croatia National Olympiad, Problem 1

Tags: algebra , equation

Solve the equation $$\left(x^2+3x-4\right)^3+\left(2x^2-5x+3\right)^3=\left(3x^2-2x-1\right)^3.$$

2023 Moldova EGMO TST, 7

Tags: equation

Find all triplets of integers $(a, b, c)$, that verify the equation $$|a+3|+b^2+4\cdot c^2-14\cdot b-12\cdot c+55=0.$$

1992 IMO Longlists, 14

Tags: number theory , equation , divisibility , modular arithmetic

Integers $a_1, a_2, . . . , a_n$ satisfy $|a_k| = 1$ and \[ \sum_{k=1}^{n} a_ka_{k+1}a_{k+2}a_{k+3} = 2,\] where $a_{n+j} = a_j$. Prove that $n \neq 1992.$

1997 Israel Grosman Mathematical Olympiad, 3

Tags: radical , equation , algebra

Find all real solutions of $\sqrt[4]{13+x}+ \sqrt[4]{14-x} = 3$.

2009 Kyiv Mathematical Festival, 1

Tags: trigonometry , equation , algebra , trigonometric equation

Solve the equation $\big(2cos(x-\frac{\pi}{4})+tgx\big)^3=54 sin^2x$, $x\in \big[0,\frac{\pi}{2}\big)$

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , equation , logarithm

Solve logarithmical equation $x^{\log _{3} {x-1}} + 2(x-1)^{\log _{3} {x}}=3x^2$

1974 IMO Longlists, 8

Tags: trigonometry , algebra , trigonometric identities , trigonometric substitution , equation

Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that \[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]

2015 Junior Regional Olympiad - FBH, 5

Tags: equation , number theory

In how many ways you can pay $2015\$$ using bills of $1\$$, $10\$$, $100\$$ and $200\$$

2004 Gheorghe Vranceanu, 3

Tags: function , equation , inversability , algebra

Consider the function $ f:(-\infty,1]\longrightarrow\mathbb{R} $ defined as $$ f(x)=\left\{ \begin{matrix} \frac{5}{2} +2^x-\frac{1}{2^x} ,& \quad x<-1 \\ 3^{\sqrt{1-x^2}} ,& \quad x\in [-1,1] \end{matrix} \right. . $$ [b]a)[/b] For a fixed parameter, find the roots of $ f-m. $ [b]b)[/b] Study the inversability of the restrictions of $ f $ to $ (-\infty,-1] $ and $ [-1,1] $ and find the inverses of these that admit them. [i]D. Zaharia[/i]

1989 IMO Longlists, 4

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?

1984 IMO Longlists, 12

Tags: number theory , equation , polynomial , diophantine equation

Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]