Olympiad problems

2013 Argentina Cono Sur TST, 2

If $ x\neq1$, $ y\neq1$, $ x\neq y$ and \[ \frac{yz\minus{}x^{2}}{1\minus{}x}\equal{}\frac{xz\minus{}y^{2}}{1\minus{}y}\] show that both fractions are equal to $ x\plus{}y\plus{}z$.

1985 IberoAmerican, 1

Tags: Columbia , algebra unsolved , algebra

If $ x\neq1$, $ y\neq1$, $ x\neq y$ and \[ \frac{yz\minus{}x^{2}}{1\minus{}x}\equal{}\frac{xz\minus{}y^{2}}{1\minus{}y}\] show that both fractions are equal to $ x\plus{}y\plus{}z$.

1985 IberoAmerican, 3

Tags: Columbia , geometry , circumcircle , trigonometry , geometry unsolved

Given an acute triangle $ABC$, let $D$, $E$ and $F$ be points in the lines $BC$, $AC$ and $AB$ respectively. If the lines $AD$, $BE$ and $CF$ pass through $O$ the centre of the circumcircle of the triangle $ABC$, whose radius is $R$, show that: \[\frac{1}{AD}\plus{}\frac{1}{BE}\plus{}\frac{1}{CF}\equal{}\frac{2}{R}\]

2005 Colombia Team Selection Test, 1

Tags: Columbia , geometry , 3D geometry , number theory proposed , number theory

Let $a,b,c$ be integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$ prove that $abc$ is a perfect cube!

1985 IberoAmerican, 2

Tags: Columbia , function , pigeonhole principle , number theory , relatively prime , number theory unsolved

To each positive integer $ n$ it is assigned a non-negative integer $f(n)$ such that the following conditions are satisfied: (1) $ f(rs) \equal{} f(r)\plus{}f(s)$ (2) $ f(n) \equal{} 0$, if the first digit (from right to left) of $ n$ is 3. (3) $ f(10) \equal{} 0$. Find $f(1985)$. Justify your answer.

1985 IberoAmerican, 1

Tags: Columbia , algebra , polynomial , Vieta , modular arithmetic , symmetry , algebra unsolved

Find all the triples of integers $ (a, b,c)$ such that: \[ \begin{array}{ccc}a\plus{}b\plus{}c &\equal{}& 24\\ a^{2}\plus{}b^{2}\plus{}c^{2}&\equal{}& 210\\ abc &\equal{}& 440\end{array}\]

1985 IberoAmerican, 3

Tags: Columbia , Vieta , algebra unsolved , algebra

Find all the roots $ r_{1}$, $ r_{2}$, $ r_{3}$ y $ r_{4}$ of the equation $ 4x^{4}\minus{}ax^{3}\plus{}bx^{2}\minus{}cx\plus{}5 \equal{} 0$, knowing that they are real, positive and that: \[ \frac{r_{1}}{2}\plus{}\frac{r_{2}}{4}\plus{}\frac{r_{3}}{5}\plus{}\frac{r_{4}}{8}\equal{} 1.\]

2020 Colombia National Olympiad, 5

Tags: Columbia , geometry

Given an acute-angled triangle $ABC$ with $D$ is the foot of the altitude from $A.$ The perpendicular lines to $BC$ through $B$ and $C$ intersect the altitudes from $C$ and $B$ at points $M$ and $N$, respectively. Show that $AD$ $=$ $BC$ if and only if $A,M,N$ and $D$ lie on the same circle.

1985 IberoAmerican, 2

Tags: Columbia , rotation , trigonometry , geometry solved , geometry

Let $ P$ be a point in the interior of the equilateral triangle $ \triangle{}ABC$ such that $ PA \equal{} 5$, $ PB \equal{} 7$, $ PC \equal{} 8$. Find the length of the side of the triangle $ ABC$.

2014 PUMaC Individual Finals A, 2

Tags: algebra , polynomial , function , inequalities , Columbia , HMMT

Given $a,b,c \in\mathbb{R}^+$, and that $a^2+b^2+c^2=3$. Prove that \[ \frac{1}{a^3+2}+\frac{1}{b^3+2}+\frac{1}{c^3+2}\ge 1 \]