Olympiad problems

2018 Oral Moscow Geometry Olympiad, 4

On the side $AB$ of the triangle $ABC$, point $M$ is selected. In triangle $ACM$ point $I_1$ is the center of the inscribed circle, $J_1$ is the center of excircle wrt side $CM$. In the triangle $BCM$ point $I_2$ is the center of the inscribed circle, $J_2$ is the center of excircle wrt side $CM$. Prove that the line passing through the midpoints of the segments $I_1I_2$ and $J_1J_2$ is perpendicular to $AB$.

2013 IMO Shortlist, A6

Tags: algebra , polynomial

Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that \[ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) \] for all real number $x$.

2012 Gheorghe Vranceanu, 2

Tags: vector geometry , geometry , affine geometry

$ G $ is the centroid of $ ABC. $ The incircle of $ ABC $ touches $ BC,CA,AB $ at $ D,E,F, $ respectively. Show that $ ABC $ is equilateral if and only if $ BC\cdot\overrightarrow{GD}+ AC\cdot\overrightarrow{GE} +AB\cdot\overrightarrow{GF} =0. $ [i]Marian Ursărescu[/i]

2007 Harvard-MIT Mathematics Tournament, 10

Tags: geometry

$ABCD$ is a convex quadrilateral such that $AB=2$, $BC=3$, $CD=7$, and $AD=6$. It also has an incircle. Given that $\angle ABC$ is right, determine the radius of this incircle.

1998 IberoAmerican Olympiad For University Students, 3

Tags: number theory

The positive divisors of a positive integer $n$ are written in increasing order starting with 1. \[1=d_1<d_2<d_3<\cdots<n\] Find $n$ if it is known that: [b]i[/b]. $\, n=d_{13}+d_{14}+d_{15}$ [b]ii[/b]. $\,(d_5+1)^3=d_{15}+1$

2003 Kazakhstan National Olympiad, 2

Tags: algebra , inequalities

For positive real numbers $ x, y, z $, prove the inequality: $$ \displaylines {\frac {x ^ 3} {x + y} + \frac {y ^ 3} {y + z} + \frac {z ^ 3} {z + x} \geq \frac {xy + yz + zx} {2}.} $$

2022 Canadian Junior Mathematical Olympiad, 4

Tags: number theory

I think we are allowed to discuss since its after 24 hours How do you do this Prove that $d(1)+d(3)+..+d(2n-1)\leq d(2)+d(4)+...d(2n)$ which $d(x)$ is the divisor function

2019 Brazil Undergrad MO, 6

Tags: permutation , combinatorics

In a hidden friend, suppose no one takes oneself. We say that the hidden friend has "marmalade" if there are two people $A$ and $ B$ such that A took $B$ and $ B $ took $A$. For each positive integer n, let $f (n)$ be the number of hidden friends with n people where there is no “marmalade”, i.e. $f (n)$ is equal to the number of permutations $\sigma$ of {$1, 2,. . . , n$} such that: *$\sigma (i) \neq i$ for all $i=1,2,...,n$ * there are no $ 1 \leq i <j \leq n $ such that $ \sigma (i) = j$ and $\sigma (j) = i. $ Determine the limit $\lim_{n \to + \infty} \frac{f(n)}{n!}$

1936 Eotvos Mathematical Competition, 3

Tags: number theory

Let $a$ be any positive integer. Prove that there exists a unique pair of positive integers $x$ and $y$ such that $$x +\frac12 (x + y - 1)(x + y- 2) = a.$$

2001 National Olympiad First Round, 7

Tags:

How many ordered triples of positive integers $(a,b,c)$ are there such that $(2a+b)(2b+a)=2^c$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of the preceding} $

1938 Eotvos Mathematical Competition, 1

Tags: number theory , perfect square

Prove that an integer $n$ can be expressed as the sum of two squares if and only if $2n$ can be expressed as the sum of two squares.

2017 Harvard-MIT Mathematics Tournament, 9

Tags: roots of unity , number theory

Let $n$ be an odd positive integer greater than $2$, and consider a regular $n$-gon $\mathcal{G}$ in the plane centered at the origin. Let a [i]subpolygon[/i] $\mathcal{G}'$ be a polygon with at least $3$ vertices whose vertex set is a subset of that of $\mathcal{G}$. Say $\mathcal{G}'$ is [i]well-centered[/i] if its centroid is the origin. Also, say $\mathcal{G}'$ is [i]decomposable[/i] if its vertex set can be written as the disjoint union of regular polygons with at least $3$ vertices. Show that all well-centered subpolygons are decomposable if and only if $n$ has at most two distinct prime divisors.

2012 Tournament of Towns, 2

Tags: number theory , divisor

The number $4$ has an odd number of odd positive divisors, namely $1$, and an even number of even positive divisors, namely $2$ and $4$. Is there a number with an odd number of even positive divisors and an even number of odd positive divisors?

2010 CHMMC Winter, 4

Tags: number theory

Compute the number of integer solutions $(x, y)$ to $xy - 18x - 35y = 1890$.

2020 Tournament Of Towns, 4

Tags: algebra , polynomial

We say that a nonconstant polynomial $p(x)$ with real coefficients is split into two squares if it is represented as $a(x) +b(x)$ where $a(x)$ and $b(x)$ are squares of polynomials with real coefficients. Is there such a polynomial $p(x)$ that it may be split into two squares: a) in exactly one way; b) in exactly two ways? Note: two splittings that differ only in the order of summands are considered to be the same. Sergey Markelov

Kyiv City MO 1984-93 - geometry, 1992.7.2

Tags: geometry , equilateral , right angle

Inside a right angle is given a point $A$. Construct an equilateral triangle, one of the vertices of which is point $A$, and two others lie on the sides of the angle (one on each side).

2018 Malaysia National Olympiad, B2

Tags: number theory , digit

Let $a$ and $b$ be positive integers such that (i) both $a$ and $b$ have at least two digits; (ii) $a + b$ is divisible by $10$; (iii) $a$ can be changed into $b$ by changing its last digit. Prove that the hundreds digit of the product $ab$ is even.

2015 BMT Spring, 16

Tags: combinatorics

A binary decision tree is a list of $n$ yes/no questions, together with instructions for the order in which they should be asked (without repetition). For instance, if $n = 3$, there are $12$ possible binary decision trees, one of which asks question $2$ first, then question $3$ (followed by question $ 1$) if the answer was yes or question $1$ (followed by question $3$) if the answer was no. Determine the greatest possible $k$ such that $2^k$ divides the number of binary decision trees on $n = 13$ questions.

1958 AMC 12/AHSME, 27

Tags: analytic geometry , graphing lines , slope

The points $ (2,\minus{}3)$, $ (4,3)$, and $ (5, k/2)$ are on the same straight line. The value(s) of $ k$ is (are): $ \textbf{(A)}\ 12\qquad \textbf{(B)}\ \minus{}12\qquad \textbf{(C)}\ \pm 12\qquad \textbf{(D)}\ {12}\text{ or }{6}\qquad \textbf{(E)}\ {6}\text{ or }{6\frac{2}{3}}$

2021 Indonesia TST, G

Tags: geometric transformation , geometry

Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.

1996 Cono Sur Olympiad, 3

Tags: combinatorics , algebra

A shop sells bottles with this capacity: $1L, 2L, 3L,..., 1996L$, the prices of bottles satifies this $2$ conditions: $1$. Two bottles have the same price, if and only if, your capacities satifies $m - n = 1000$ $2$. The price of bottle $m$($1001>m>0$) is $1996 - m$ dollars. Find all pair(s) $m$ and $n$ such that: a) $m + n = 1000$ b) the cost is smallest possible!!! c) with the pair, the shop can measure $k$ liters, with $0<k<1996$(for all $k$ integer) Note: The operations to measure are: i) To fill or empty any one of two bottles ii)Pass water of a bottle for other bottle We can measure $k$ liters when the capacity of one bottle plus the capacity of other bottle is equal to $k$

VI Soros Olympiad 1999 - 2000 (Russia), 10.3

Tags: geometry , euler circle

he center of the circle passing through the midpoints of all sides of triangle $ABC$ lies on the bisector of its angle $C$. Find the side $AB$ if $BC = a$, $AC = b$ ($a$ is not equal to $b$).

1998 IMO Shortlist, 1

Tags: inequalities , algebra , n-variable inequality , arithmetic mean-geometric mean

Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$. Prove that \[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]

1963 Miklós Schweitzer, 4

Tags: superior algebra , algebra , polynomial

Call a polynomial positive reducible if it can be written as a product of two nonconstant polynomials with positive real coefficients. Let $ f(x)$ be a polynomial with $ f(0)\not\equal{}0$ such that $ f(x^n)$ is positive reducible for some natural number $ n$. Prove that $ f(x)$ itself is positive reducible. [L. Redei]

2024 China Girls Math Olympiad, 3

Tags: number theory

Find the smallest real $\lambda$, such that for any positive integers $n, a, b$, such that $n \nmid a+b$, there exists a positive integer $1 \leq k \leq n-1$, satisfying $$\{\frac{ak} {n}\}+\{\frac{bk} {n}\} \leq \lambda.$$