Olympiad problems

1978 Polish MO Finals, 3

Tags: polynomial , algebra

Prove that if $m$ is a natural number and $P,Q,R$ polynomials of degrees less than $m$ satisfying $$x^{2m}P(x,y)+y^{2m}Q(x,y) = (x+y)^{2m}R(x,y),$$ then each of the polynomials is zero.

1987 Romania Team Selection Test, 5

Tags: algebra , inequalities , rearrangement inequality

Let $A$ be the set $\{1,2,\ldots,n\}$, $n\geq 2$. Find the least number $n$ for which there exist permutations $\alpha$, $\beta$, $\gamma$, $\delta$ of the set $A$ with the property: \[ \sum_{i=1}^n \alpha(i) \beta (i) = \dfrac {19}{10} \sum^n_{i=1} \gamma(i)\delta(i) . \] [i]Marcel Chirita[/i]

2014 JHMMC 7 Contest, 19

Tags:

Lev and Alex are racing on a number line. Alex is much faster, so he goes to sleep until Lev reaches $100$. Lev runs at $5$ integers per minute and Alex runs at $7$ integers per minute (in the same direction). How many minutes from the START of the race will it take Alex to catch up to Lev (who is still running after Alex wakes up)?

2007 IMAC Arhimede, 4

Tags: divide , sum , sums and products , number theory , pigeonhole principle , combinatorics

Prove that for any given number $a_k, 1 \le k \le 5$, there are $\lambda_k \in \{-1, 0, 1\}, 1 \le k \le 5$, which are not all equal zero, such that $11 | \lambda_1a_1^2+\lambda_2a_2^2+\lambda_3a_3^2+\lambda_4a_4^2+\lambda_5a_5^2$

2018 Iran Team Selection Test, 3

Tags: number theory

Let $a_1,a_2,a_3,\cdots $ be an infinite sequence of distinct integers. Prove that there are infinitely many primes $p$ that distinct positive integers $i,j,k$ can be found such that $p\mid a_ia_ja_k-1$. [i]Proposed by Mohsen Jamali[/i]

2020 BMT Fall, 15

Tags: floor function , combinatorics

Consider a random string $s$ of $10^{2020}$ base-ten digits (there can be leading zeroes). We say a substring $s' $ (which has no leading zeroes) is self-locating if $s' $ appears in $s$ at index $s' $ where the string is indexed at $ 1$. For example the substring $11$ in the string “$122352242411$” is selflocating since the $11$th digit is $ 1$ and the $12$th digit is $ 1$. Let the expected number of self-locating substrings in s be $G$. Compute $\lfloor G \rfloor$.

2012 HMNT, 5

Tags: number theory

Given any positive integer, we can write the integer in base $12$ and add together the digits of its base $12$ representation. We perform this operation on the number $ 7^{6^{5^{4^{3^{2^{1}}}}}}$ repeatedly until a single base $12$ digit remains. Find this digit.

2010 N.N. Mihăileanu Individual, 1

Tags: conic , quadratic , algebra , polynomial

Let be two real reducible quadratic polynomials $ P,Q $ in one variable. Prove that if $ P-Q $ is irreducible, then $ P+Q $ is reducible.

2000 Korea Junior Math Olympiad, 5

Tags: digit , number theory

$a$ is a $2000$ digit natural number of the form $$a=2(A)99…99(B)(C)$$ expressed in base $10$. $a$ is not a multiple of $10$, and $2(A)+(B)(C)=99$. $a=2899..9971$ is a possible example of $a$. $b$ is a number you earn when you write the digits of $a$ in a reverse order(Writing the digits of some number in a reverse order means like reordering $1234$ into $4321$). Find every positive integer $a$ that makes $ab$ a square number.

2006 Vietnam National Olympiad, 5

Tags: polynomial , algebra

Find all polynomyals $P(x)$ with real coefficients which satisfy the following equality for all real numbers $x$: \[ P(x^2)+x(3P(x)+P(-x))=(P(x))^2+2x^2 . \]

1975 Chisinau City MO, 90

Tags: right triangle , median , triangle construction , geometry

Construct a right-angled triangle along its two medians, starting from the acute angles.

2010 Tournament Of Towns, 5

Tags: combinatorics

In a tournament with $55$ participants, one match is played at a time, with the loser dropping out. In each match, the numbers of wins so far of the two participants differ by not more than $1$. What is the maximal number of matches for the winner of the tournament?

2003 Hungary-Israel Binational, 1

Tags: inequalities

If $x_{1}, x_{2}, . . . , x_{n}$ are positive numbers, prove the inequality $\frac{x_{1}^{3}}{x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}}+\frac{x_{2}^{3}}{x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}}+...+\frac{x_{n}^{3}}{x_{n}^{2}+x_{n}x_{1}+x_{1}^{2}}\geq\frac{x_{1}+x_{2}+...+x_{n}}{3}$.

2017 Saudi Arabia BMO TST, 1

Tags: algebra , inequalities

Let $a, b, c$ be positive real numbers. Prove that $$\frac{a(b^2 + c^2)}{(b + c)(a^2 + bc)} + \frac{b(c^2 + a^2)}{(c + a)(b^2 + ca)} + \frac{c(a^2 + b^2)}{(a + b)(c^2 + ab)} \ge \frac32$$

2022 Iranian Geometry Olympiad, 3

Tags: geometry , pentagon

Let $ABCDE$ be a convex pentagon such that $AB = BC = CD$ and $\angle BDE = \angle EAC = 30 ^{\circ}$. Find the possible values of $\angle BEC$. [i]Proposed by Josef Tkadlec (Czech Republic)[/i]

Russian TST 2015, P1

Tags: divisibility , number theory

Find all pairs of natural numbers $(a,b)$ satisfying the following conditions: [list] [*]$b-1$ is divisible by $a+1$ and [*]$a^2+a+2$ is divisible by $b$. [/list]

2008 IMO Shortlist, 1

Tags: equation , number theory , algebra

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

1999 Baltic Way, 18

Tags: modular arithmetic , number theory

Let $m$ be a positive integer such that $m=2\pmod{4}$. Show that there exists at most one factorization $m=ab$ where $a$ and $b$ are positive integers satisfying \[0<a-b<\sqrt{5+4\sqrt{4m+1}}\]

1997 Romania National Olympiad, 2

Tags: fractional part , algebra

Let $n\geq 3$ be a natural number and $x\in \mathbb{R}$, for which $\{ x\} =\{ x^2\} =\{ x^n\} $ (with $\{ x\} $ we denote the fractional part of $x$). Prove that $x$ is an integer.

2020 Iran MO (3rd Round), 3

Tags: sum of digits , function , number theory

Find all functions $f$ from positive integers to themselves, such that the followings hold. $1)$.for each positive integer $n$ we have $f(n)<f(n+1)<f(n)+2020$. $2)$.for each positive integer $n$ we have $S(f(n))=f(S(n))$ where $S(n)$ is the sum of digits of $n$ in base $10$ representation.

2019 Canada National Olympiad, 1

Tags: circumcenter , geometry , distance

Points $A,B,C$ are on a plane such that $AB=BC=CA=6$. At any step, you may choose any three existing points and draw that triangle's circumcentre. Prove that you can draw a point such that its distance from an previously drawn point is: $(a)$ greater than 7 $(b)$ greater than 2019

2013 Federal Competition For Advanced Students, Part 1, 4

Tags: geometry , perpendicular bisector , ratio

Let $A$, $B$ and $C$ be three points on a line (in this order). For each circle $k$ through the points $B$ and $C$, let $D$ be one point of intersection of the perpendicular bisector of $BC$ with the circle $k$. Further, let $E$ be the second point of intersection of the line $AD$ with $k$. Show that for each circle $k$, the ratio of lengths $\overline{BE}:\overline{CE}$ is the same.