Olympiad problems

2014 Regional Competition For Advanced Students, 2

You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\ bc + bd = 5c + 5d \\ ac + cd = 7a + 7d \\ ad + bd = 9a + 9b \end{cases} $

2017 Benelux, 4

Tags: number theory , number theory proposed , Benelux , Olympiad , math olympiad , algebra , combinatorics

A [i]Benelux n-square[/i] (with $n\geq 2$) is an $n\times n$ grid consisting of $n^2$ cells, each of them containing a positive integer, satisfying the following conditions: $\bullet$ the $n^2$ positive integers are pairwise distinct. $\bullet$ if for each row and each column we compute the greatest common divisor of the $n$ numbers in that row/column, then we obtain $2n$ different outcomes. (a) Prove that, in each Benelux n-square (with $n \geq 2$), there exists a cell containing a number which is at least $2n^2.$ (b) Call a Benelux n-square [i]minimal[/i] if all $n^2$ numbers in the cells are at most $2n^2.$ Determine all $n\geq 2$ for which there exists a minimal Benelux n-square.

2016 Irish Math Olympiad, 7

Tags: combinatorics , math olympiad , counting , Array , Arrangements

A rectangular array of positive integers has $4$ rows. The sum of the entries in each column is $20$. Within each row, all entries are distinct. What is the maximum possible number of columns?

2012 German National Olympiad, 6

Tags: algebra , math olympiad

Let $a_1$ and $a_2$ be postive real numbers. Let $a_{n+2}=1+\frac{a_{n+1}}{a_{n}}$ Prove that $|a_{2012}-2|<10^{-200}$

2004 Spain Mathematical Olympiad, Problem 2

Tags: geometry , Plane Geometry , math olympiad

${ABCD}$ is a quadrilateral, ${P}$ and ${Q}$ are midpoints of the diagonals ${BD}$ and ${AC}$, respectively. The lines parallel to the diagonals originating from ${P}$ and ${Q}$ intersect in the point ${O}$. If we join the four midpoints of the sides, ${X}$, ${Y}$, ${Z}$, and ${T}$, to ${O}$, we form four quadrilaterals: ${OXBY}$, ${OYCZ}$, ${OZDT}$, and ${OTAX}$. Prove that the four newly formed quadrilaterals have the same areas.

2018 European Mathematical Cup, 3

Tags: algebra , emc , math olympiad

For which real numbers $k > 1$ does there exist a bounded set of positive real numbers $S$ with at least $3$ elements such that $$k(a - b)\in S$$ for all $a,b\in S $ with $a > b?$ Remark: A set of positive real numbers $S$ is bounded if there exists a positive real number $M$ such that $x < M$ for all $x \in S.$

2004 Spain Mathematical Olympiad, Problem 4

Tags: number theory , math olympiad

Does there exist such a power of ${2}$, that when written in the decimal system its digits are all different than zero and it is possible to reorder the other digits to form another power of ${2}$? Justify your answer.

2017 Brazil Undergrad MO, 6

Tags: Combinatorics of words , combinatorics , math olympiad

Let's consider words over the alphabet $\{a,b\}$ to be sequences of $a$ and $b$ with finite length. We say $u \leq v$ if $u$ is a subword of $v$ if we can get $u$ erasing some letter of $v$ (for example $aba \leq abbab$). We say that $u$ differentiates the words $x$ and $y$ if $u \leq x$ but $u \not\leq y$ or vice versa. Let $m$ and $l$ be positive integers. We say that two words are $m-$equivalents if there does not exist some $u$ with length smaller than $m$ that differentiates $x$ and $y$. a) Show that, if $2m \leq l$, there exists two distinct words with length $l$ \ $m-$equivalents. b) Show that, if $2m > l$, any two distinct words with length $l$ aren't $m-$equivalent.

2004 Spain Mathematical Olympiad, Problem 1

Tags: math olympiad , Matrices , matrix , Arithmetic Progression , algebra unsolved

We have a set of ${221}$ real numbers whose sum is ${110721}$. It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to ${2004}$.

2014 Contests, 2

Tags: algebra , system of equations , math olympiad

You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\ bc + bd = 5c + 5d \\ ac + cd = 7a + 7d \\ ad + bd = 9a + 9b \end{cases} $