Olympiad problems

1999 Kazakhstan National Olympiad, 6

In a sequence of natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_ {1999} $, $ a_n-a_ {n-1} -a_ {n-2} $ is divisible by $ 100 (3 \leq n \leq 1999) $. It is known that $ a_1 = 19$ and $ a_2 = 99$. Find the remainder of $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $ by $8$.

1999 Romania National Olympiad, 2a

Tags: inequalities

let $x_i,y_i 1 \le i \le n$ be positive numbers such that : $\displaystyle \sum_{i=1}^n x_i \ge \sum_{i=1}^n x_iy_i$ Prove : $\displaystyle \sum_{i=1}^n x_i \le \sum _{i=1}^n \frac{x_i}{y_i}$

2019 Portugal MO, 2

Tags: number theory , digit

A five-digit integer is said to be [i]balanced [/i]i f the sum of any three of its digits is divisible by any of the other two. How many [i]balanced [/i] numbers are there?

2010 N.N. Mihăileanu Individual, 4

Tags: modular arithmetic , discrete mathematics , square grid , locusts , pigeonhole principle , combinatorics

A square grid is composed of $ n^2\equiv 1\pmod 4 $ unit cells that contained each a locust that jumped the same amount of cells in the direccion of columns or lines, without leaving the grid. Prove that, as a result of this, at least two locusts landed on the same cell. [i]Marius Cavachi[/i]

2015 Baltic Way, 12

Tags: geometry

A circle passes through vertex $B$ of the triangle $ABC$, intersects its sides $ AB $and $BC$ at points $K$ and $L$, respectively, and touches the side $ AC$ at its midpoint $M$. The point $N$ on the arc $BL$ (which does not contain $K$) is such that $\angle LKN = \angle ACB$. Find $\angle BAC $ given that the triangle $CKN$ is equilateral.

2002 Singapore Team Selection Test, 2

Tags: floor function , number theory , integer , divide , divisible

For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.

2021 JBMO Shortlist, C3

Tags: shortlist , combinatorics , process

We have a set of $343$ closed jars, each containing blue, yellow and red marbles with the number of marbles from each color being at least $1$ and at most $7$. No two jars have exactly the same contents. Initially all jars are with the caps up. To flip a jar will mean to change its position from cap-up to cap-down or vice versa. It is allowed to choose a triple of positive integers $(b; y; r) \in \{1; 2; ...; 7\}^3$ and flip all the jars whose number of blue, yellow and red marbles differ by not more than $1$ from $b, y, r$, respectively. After $n$ moves all the jars turned out to be with the caps down. Find the number of all possible values of $n$, if $n \le 2021$.

2021 Kyiv City MO Round 1, 10.5

Tags: number theory , sequence

The sequence $(a_n)$ is such that $a_{n+1} = (a_n)^n + n + 1$ for all positive integers $n$, where $a_1$ is some positive integer. Let $k$ be the greatest power of $3$ by which $a_{101}$ is divisible. Find all possible values of $k$. [i]Proposed by Kyrylo Holodnov[/i]

2010 Federal Competition For Advanced Students, P2, 1

Tags: algebra , inequalities

Show that $\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)} {(x - y)^5 + (y - z)^5 + (z - x)^5} \ge 3$ holds for all triples of distinct integers $x, y, z$. When does equality hold?

2010 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: table , combinatorics

In table of dimensions $2n \times 2n$ there are positive integers not greater than $10$, such that numbers lying in unit squares with common vertex are coprime. Prove that there exist at least one number which occurs in table at least $\frac{2n^2}{3}$ times

2011 Postal Coaching, 3

Tags: geometry

Let $ABC$ be a scalene triangle. Let $l_A$ be the tangent to the nine-point circle at the foot of the perpendicular from $A$ to $BC$, and let $l_A'$ be the tangent to the nine-point circle from the mid-point of $BC$. The lines $l_A$ and $l_A'$ intersect at $A'$ . Deﬁne $B'$ and $C'$ similarly. Show that the lines $AA' , BB'$ and $CC'$ are concurrent.

1990 Baltic Way, 20

Tags:

A creative task: propose an original competition problem together with its solution.

2015 CCA Math Bonanza, T9

Tags:

The Fibonacci numbers are defined as the sequence $F_n$ with $F_0=1$, $F_1=1$ and $F_{n+2} = F_{n+1} +F_n$. How many ways can $10$ be written as an ordered sum of numbers found in the Fibonacci sequence? For example, $3$ can be written as $1 + 1 + 1$, $2 + 1$, $1 + 2$, and $3$, for a total of $4$ ways. [i]2015 CCA Math Bonanza Team Round #9[/i]

2000 Bundeswettbewerb Mathematik, 1b

Tags: number theory , sum , digit

Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$. Is this possible when each of the numbers consists of $2000$ digits?

1968 German National Olympiad, 1

Tags: algebra , system of equations , system

Determine all ordered quadruples of real numbers $(x_1, x_2, x_3, x_4)$ for which the following system of equations exists, is fulfilled: $$x_1 + ax_2 + x_3 = b $$ $$x_2 + ax_3 + x_4 = b $$ $$x_3 + ax_4 + x_1 = b $$ $$x_4 + ax_1 + x_2 = b$$ Here $a$ and $b$ are real numbers (case distinction!).

2019 Iran Team Selection Test, 6

Tags: number theory

$\{a_{n}\}_{n\geq 0}$ and $\{b_{n}\}_{n\geq 0}$ are two sequences of positive integers that $a_{i},b_{i}\in \{0,1,2,\cdots,9\}$. There is an integer number $M$ such that $a_{n},b_{n}\neq 0$ for all $n\geq M$ and for each $n\geq 0$ $$(\overline{a_{n}\cdots a_{1}a_{0}})^{2}+999 \mid(\overline{b_{n}\cdots b_{1}b_{0}})^{2}+999 $$ prove that $a_{n}=b_{n}$ for $n\geq 0$.\\ (Note that $(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0$.) [i]Proposed by Yahya Motevassel[/i]

Kyiv City MO 1984-93 - geometry, 1986.7.5

Tags: geometry , geometric inequality

Prove that the sum of the lengths of the diagonals of an arbitrary quadrilateral is less than the sum of the lengths of its sides.

2006 District Olympiad, 3

Tags: parallelogram , geometry

Let $ABCD$ be a convex quadrilateral, $M$ the midpoint of $AB$, $N$ the midpoint of $BC$, $E$ the intersection of the segments $AN$ and $BD$, $F$ the intersection of the segments $DM$ and $AC$. Prove that if $BE = \frac 13 BD$ and $AF = \frac 13 AC$, then $ABCD$ is a parallelogram.

1977 Germany Team Selection Test, 2

Tags: polynomial , functional equation , algebra

Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$ [b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ [b]c.)[/b] $P(1,0) =1.$

2022-23 IOQM India, 12

Tags: geometry , trapezoid

Given $\triangle{ABC}$ with $\angle{B}=60^{\circ}$ and $\angle{C}=30^{\circ}$, let $P,Q,R$ be points on the sides $BA,AC,CB$ respectively such that $BPQR$ is an isosceles trapezium with $PQ \parallel BR$ and $BP=QR$.\\ Find the maximum possible value of $\frac{2[ABC]}{[BPQR]}$ where $[S]$ denotes the area of any polygon $S$.

2002 India IMO Training Camp, 15

Tags: inequalities , trigonometry , algebra

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

2017 Cono Sur Olympiad, 1

Tags: number theory

A positive integer $n$ is called [i]guayaquilean[/i] if the sum of the digits of $n$ is equal to the sum of the digits of $n^2$. Find all the possible values that the sum of the digits of a guayaquilean number can take.

2022 Iran Team Selection Test, 2

Tags: number theory

For a positive integer $n$, let $\tau(n)$ and $\sigma(n)$ be the number of positive divisors of $n$ and the sum of positive divisors of $n$, respectively. let $a$ and $b$ be positive integers such that $\sigma(a^n)$ divides $\sigma(b^n)$ for all $n\in \mathbb{N}$. Prove that each prime factor of $\tau(a)$ divides $\tau(b)$. Proposed by MohammadAmin Sharifi

2005 Bosnia and Herzegovina Team Selection Test, 5

Tags: permutation , identical , set theory

If for an arbitrary permutation $(a_1,a_2,...,a_n)$ of set ${1,2,...,n}$ holds $\frac{{a_k}^2}{a_{k+1}}\leq k+2$, $k=1,2,...,n-1$, prove that $a_k=k$ for $k=1,2,...,n$

1983 Dutch Mathematical Olympiad, 2

Tags: modular arithmetic , inequalities

Prove that if $ n$ is an odd positive integer, then the last two digits of $ 2^{2n}(2^{2n\plus{}1}\minus{}1)$ in base $ 10$ are $ 28$.