Olympiad problems

1996 Tournament Of Towns, (517) 4

Tags: combinatorics

For what integers $n > 1$ can it happen that in a group of $n +1$ girls and $n$ boys, all the girls know a different number of boys while all the boys know the same number of girls? (NB Vassiliev)

2021 Alibaba Global Math Competition, 15

Tags: calculus , integration , topology , analysis , manifold , differential geometry

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $n \ge 2$. Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$, where $\text{Ric}$ is the Ricci tensor of $(M,g)$. Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$. Prove that \[\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.\] Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$.

PEN J Problems, 13

Tags: divisor function

Determine all positive integers $k$ such that \[\frac{d(n^{2})}{d(n)}= k\] for some $n \in \mathbb{N}$.

2019 Junior Balkan MO, 1

Tags: number theory , prime number

Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number $x^p + y^p + z^p - x - y - z$ is a product of exactly three distinct prime numbers.

Novosibirsk Oral Geo Oly VIII, 2020.5

Tags: perpendicular bisector , midpoint , geometry

Line $\ell$ is perpendicular to one of the medians of the triangle. The median perpendiculars to the sides of this triangle intersect the line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the other two.

2010 LMT, 35

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Consider a set of $6$ fixed points in the plane, with no three collinear. Between some pairs of these points, we may draw one arrow from one point to the other. How many possible configurations of arrows are there such that if there is an arrow from point $A$ to point $B$ and an arrow from $B$ to $C,$ then there is an arrow from $A$ to $C?$ Your score will be $16-\frac{1}{800}|\textbf{Your Answer}-\textbf{Actual Answer}|$ rounded to the nearest integer or zero, whichever is higher.

1969 IMO Longlists, 27

Tags: geometry , 3d geometry , ratio , sphere

$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?

1991 Putnam, B3

Tags: geometry , rectangle , combinatorics

Can we find $N$ such that all $m\times n$ rectangles with $m,n>N$ can be tiled with $4\times6$ and $5\times7$ rectangles?

1993 Poland - Second Round, 3

Tags: geometry , 3d geometry , tetrahedron , inequalities , geometric inequality

A tetrahedron $OA_1B_1C_1$ is given. Let $A_2,A_3 \in OA_1, A_2,A_3 \in OA_1, A_2,A_3 \in OA_1$ be points such that the planes $A_1B_1C_1,A_2B_2C_2$ and $A_3B_3C_3$ are parallel and $OA_1 > OA_2 > OA_3 > 0$. Let $V_i$ be the volume of the tetrahedron $OA_iB_iC_i$ ($i = 1,2,3$) and $V$ be the volume of $OA_1B_2C_3$. Prove that $V_1 +V_2 +V_3 \ge 3V$.

IV Soros Olympiad 1997 - 98 (Russia), 10.8

Tags: ratio , geometry

In triangle $ABC$, angle $B$ is different from a right angle, $AB : BC = k$. Let $M$ be the midpoint of $AC$. Lines symmetric to $BM$ wrt $AB$ and $BC$ intersect line $AC$ at points $D$ and $E$. Find $BD : BE$.

2024 Korea National Olympiad, 6

Tags: number theory

For a positive integer $n$, let $g(n) = \left[ \displaystyle \frac{2024}{n} \right]$. Find the value of $$\sum_{n = 1}^{2024}\left(1 - (-1)^{g(n)}\right)\phi(n).$$

2009 Balkan MO Shortlist, N2

Tags: number theory

Solve the equation \[ 3^x \minus{} 5^y \equal{} z^2.\] in positive integers. [i]Greece[/i]

2014 Balkan MO Shortlist, A3

Tags: sequence , algebra

$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$

2021 Swedish Mathematical Competition, 2

Tags: number theory

Anna is out shopping for fruit. She observes that four oranges, three bananas and one lemon costs exactly the same as three oranges and two lemons (all prices are in whole kroner). Just then her friend Bengt calls, and Anna tells this to him. Bengt complains, that ''information is not enough for me to know how much each fruit costs''. ''No'', says Anna,' 'but three oranges and two lemons cost as many kroner as your mother is old''. Unfortunately, it's not enough either, but if she had been younger then the information would have been sufficient for you to be able to figure out what the fruits costs. How old is Bengt's mother?

1996 Iran MO (2nd round), 2

Tags: number theory

Let $a,b,c,d$ be positive integers such that $ab\equal{}cd$. Prove that $a\plus{}b\plus{}c\plus{}d$ is a composite number.

2012 Indonesia TST, 4

Tags: number theory

Find all quadruplets of positive integers $(m,n,k,l)$ such that $3^m = 2^k + 7^n$ and $m^k = 1 + k + k^2 + k^3 + \ldots + k^l$.

1999 Irish Math Olympiad, 4

Tags: number theory

Find all positive integers $ m$ with the property that the fourth power of the number of (positive) divisors of $ m$ equals $ m$.

2018 Purple Comet Problems, 10

Tags: geometry

The triangle below is divided into nine stripes of equal width each parallel to the base of the triangle. The darkened stripes have a total area of $135$. Find the total area of the light colored stripes. [img]https://cdn.artofproblemsolving.com/attachments/0/8/f34b86ccf50ef3944f5fbfd615a68607f4fadc.png[/img]

1991 Vietnam Team Selection Test, 1

Tags: 3d geometry , tetrahedron , geometry

Let $T$ be an arbitrary tetrahedron satisfying the following conditions: [b]I.[/b] Each its side has length not greater than 1, [b]II.[/b] Each of its faces is a right triangle. Let $s(T) = S^2_{ABC} + S^2_{BCD} + S^2_{CDA} + S^2_{DAB}$. Find the maximal possible value of $s(T)$.

1988 AMC 8, 21

Tags:

A fifth number,$n$ , is added to the set $ \{ 3,6,9,10\} $ to make the mean of the set of five numbers equal to its median. The number of possible values of $n$ is $ \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ \text{more than }4 $

2014 Paraguay Mathematical Olympiad, 4

Tags: combinatorics

Nair and Yuli play the following game: $1.$ There is a coin to be moved along a horizontal array with $203$ cells. $2.$ At the beginning, the coin is at the first cell, counting from left to right. $3.$ Nair plays first. $4.$ Each of the players, in their turns, can move the coin $1$, $2$, or $3$ cells to the right. $5.$ The winner is the one who reaches the last cell first. What strategy does Nair need to use in order to always win the game?

1966 IMO Shortlist, 43

Tags: combinatorics , coloring , graph theory , ramsey theory , extremal combinatorics

Given $5$ points in a plane, no three of them being collinear. Each two of these $5$ points are joined with a segment, and every of these segments is painted either red or blue; assume that there is no triangle whose sides are segments of equal color. [b]a.)[/b] Show that: [i](1)[/i] Among the four segments originating at any of the $5$ points, two are red and two are blue. [i](2)[/i] The red segments form a closed way passing through all $5$ given points. (Similarly for the blue segments.) [b]b.)[/b] Give a plan how to paint the segments either red or blue in order to have the condition (no triangle with equally colored sides) satisfied.

2018 Hong Kong TST, 2

Tags: ratio , geometry

Given triangle $ABC$, let $D$ be an inner point of segment $BC$. Let $P$ and $Q$ be distinct inner points of the segment $AD$. Let $K=BP\cap AC, L=CP\cap AB, E=BQ\cap AC, F=CQ\cap AB$. Given that $KL\parallel EF$, find all possible values of the ratio $BD:DC$.

2011 China Western Mathematical Olympiad, 3

Tags: incenter , circumcircle , geometry

In triangle $ABC$ with $AB>AC$ and incenter $I$, the incircle touches $BC,CA,AB$ at $D,E,F$ respectively. $M$ is the midpoint of $BC$, and the altitude at $A$ meets $BC$ at $H$. Ray $AI$ meets lines $DE$ and $DF$ at $K$ and $L$, respectively. Prove that the points $M,L,H,K$ are concyclic.

2005 National Olympiad First Round, 26

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For every positive integer $n$, $f(2n+1)=2f(2n)$, $f(2n)=f(2n-1)+1$, and $f(1)=0$. What is the remainder when $f(2005)$ is divided by $5$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $