Olympiad problems

2017 Pan-African Shortlist, C?

The numbers from $1$ to $2017$ are written on a board. Deka and Farid play the following game : each of them, on his turn, erases one of the numbers. Anyone who erases a multiple of $2, 3$ or $5$ loses and the game is over. Is there a winning strategy for Deka ?

1999 All-Russian Olympiad, 8

Tags: combinatorics

There are $2000$ components in a circuit, every two of which were initially joined by a wire. The hooligans Vasya and Petya cut the wires one after another. Vasya, who starts, cuts one wire on his turn, while Petya cuts two or three. The hooligan who cuts the last wire from some component loses. Who has the winning strategy?

1989 Irish Math Olympiad, 2

Tags: combinatorics

2. Each of $n$ members of a club is given a different item of information. The members are allowed to share the information, but, for security reasons, only in the following way: A pair may communicate by telephone. During a telephone call only one member may speak. The member who speaks may tell the other member all the information (s)he knows. Determine the minimal number of phone calls that are required to convey all the information to each of the members. Hi, from my sketches I'm thinking the answer is $2n-2$ but I dont know how to prove that this number of calls is the smallest. Can anyone enlighten me? Thanks

2013 Tournament of Towns, 2

Tags: combinatorics

A boy and a girl were sitting on a long bench. Then twenty more children one after another came to sit on the bench, each taking a place between already sitting children. Let us call a girl brave if she sat down between two boys, and let us call a boy brave if he sat down between two girls. It happened, that in the end all girls and boys were sitting in the alternating order. Is it possible to uniquely determine the number of brave children?

2012 Chile National Olympiad, 4

Tags: ratio , isosceles triangle , isosceles , geometry , angle

Consider an isosceles triangle $ABC$, where $AB = AC$. $D$ is a point on the $AC$ side and $P$ a point on the segment $BD$ so that the angle $\angle APC = 90^o$ and $ \angle ABP = \angle BCP $. Determine the ratio $AD: DC$.

1971 IMO Longlists, 32

Tags: geometry

Two half-lines $a$ and $b$, with the common endpoint $O$, make an acute angle $\alpha$. Let $A$ on $a$ and $B$ on $b$ be points such that $OA=OB$, and let $b$ be the line through $A$ parallel to $b$. Let $\beta$ be the circle with centre $B$ and radius $BO$. We construct a sequence of half-lines $c_1,c_2,c_3,\ldots $, all lying inside the angle $\alpha$, in the following manner: (i) $c_i$ is given arbitrarily; (ii) for every natural number $k$, the circle $\beta$ intercepts on $c_k$ a segment that is of the same length as the segment cut on $b'$ by $a$ and $c_{k+1}$. Prove that the angle determined by the lines $c_k$ and $b$ has a limit as $k$ tends to infinity and find that limit.

2008 AMC 12/AHSME, 20

Tags: geometry , ratio , trigonometry , incenter , inradius , analytic geometry , angle bisector

Triangle $ ABC$ has $ AC\equal{}3$, $ BC\equal{}4$, and $ AB\equal{}5$. Point $ D$ is on $ \overline{AB}$, and $ \overline{CD}$ bisects the right angle. The inscribed circles of $ \triangle ADC$ and $ \triangle BCD$ have radii $ r_a$ and $ r_b$, respectively. What is $ r_a/r_b$? $ \textbf{(A)}\ \frac{1}{28}\left(10\minus{}\sqrt{2}\right) \qquad \textbf{(B)}\ \frac{3}{56}\left(10\minus{}\sqrt{2}\right) \qquad \textbf{(C)}\ \frac{1}{14}\left(10\minus{}\sqrt{2}\right) \qquad \textbf{(D)}\ \frac{5}{56}\left(10\minus{}\sqrt{2}\right) \\ \textbf{(E)}\ \frac{3}{28}\left(10\minus{}\sqrt{2}\right)$

2015 NIMO Problems, 3

Tags:

Let $ABCD$ be a rectangle with $AB = 6$ and $BC = 6 \sqrt 3$. We construct four semicircles $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ whose diameters are the segments $AB$, $BC$, $CD$, $DA$. It is given that $\omega_i$ and $\omega_{i+1}$ intersect at some point $X_i$ in the interior of $ABCD$ for every $i=1,2,3,4$ (indices taken modulo $4$). Compute the square of the area of $X_1X_2X_3X_4$. [i]Proposed by Evan Chen[/i]

2019 Turkey MO (2nd round), 3

Tags: combinatorics

There are 2019 students in a school, and some of these students are members of different student clubs. Each student club has an advisory board consisting of 12 students who are members of that particular club. An {\em advisory meeting} (for a particular club) can be realized only when each participant is a member of that club, and moreover, each of the 12 students forming the advisory board are present among the participants. It is known that each subset of at least 12 students in this school can realize an advisory meeting for exactly one student club. Determine all possible numbers of different student clubs with exactly 27 members.

1997 Korea National Olympiad, 6

Tags: algebra , polynomial

Find all polynomial $P(x,y)$ for any reals $x,y$ such that (i) $x^{100}+y^{100}\le P(x,y)\le 101(x^{100}+y^{100})$ (ii) $(x-y)P(x,y)=(x-1)P(x,1)+(1-y)P(1,y).$

2014 Contests, 1

Tags: number theory

Determine all pairs $(a,b)$ of positive integers satisfying \[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]

2023 Brazil EGMO Team Selection Test, 1

Tags: increasing function , strictly increasing , functional equation , number theory , algebra , function

Let $\mathbb{Z}_{>0} = \{1, 2, 3, \ldots \}$ be the set of all positive integers. Find all strictly increasing functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that $f(f(n)) = 3n$.

2008 Princeton University Math Competition, A2/B3

Tags: perimeter , geometry

Consider a convex polygon $\mathcal{P}$ in space with perimeter $20$ and area $30$. What is the volume of the locus of points that are at most $1$ unit away from some point in the interior of $\mathcal{P}$?

2005 May Olympiad, 2

Tags: number theory , consecutive

An integer is called [i]autodivi [/i] if it is divisible by the two-digit number formed by its last two digits (tens and units). For example, $78013$ is autodivi as it is divisible by $13$, $8517$ is autodivi since it is divisible by $17$. Find $6$ consecutive integers that are autodivi and that have the digits of the units, tens and hundreds other than $0$.

2020 SIME, 2

Tags:

Andrew rolls two fair six sided die each numbered from $1$ to $6$, and Brian rolls one fair $12$ sided die numbered from $1$ to $12$. The probability that the sum of the numbers obtained from Andrew's two rolls is less than the number obtained from Brian's roll can be expressed as a common fraction in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2024 Thailand Mathematical Olympiad, 1

Tags: geometry

Let $ABCD$ be a convex quadrilateral. Construct $S$ and $T$ on the side $AD$ and $AB$ respectively such that $AS=AT$. Construct $U$ and $V$ on the side $BC$ and $CD$ respectively such that $CU=CV$. Assume that $BT=BU$ and $ST, UV, BD$ are concurrent, prove that $AB+CD=BC+AD$.

1983 IMO Longlists, 68

Tags: trigonometry , function , algebra

Three of the roots of the equation $x^4 -px^3 +qx^2 -rx+s = 0$ are $\tan A, \tan B$, and $\tan C$, where $A, B$, and $C$ are angles of a triangle. Determine the fourth root as a function only of $p, q, r$, and $s.$

2017 Hanoi Open Mathematics Competitions, 9

Tags: combinatorics , combinatorial geometry

Cut off a square carton by a straight line into two pieces, then cut one of two pieces into two small pieces by a straight line, ect. By cutting $2017$ times we obtain $2018$ pieces. We write number $2$ in every triangle, number 1 in every quadrilateral, and $0$ in the polygons. Is the sum of all inserted numbers always greater than $2017$?

2022 Stanford Mathematics Tournament, 1

Tags:

Compute \[\int_0^{10}(x-5)+(x-5)^2+(x-3)^2dx.\]

2012 India Regional Mathematical Olympiad, 6

Tags: induction , polynomial , number theory , algebra

Find all positive integers such that $3^{2n}+3n^2+7$ is a perfect square.

2021 May Olympiad, 2

Tags: combinatorics , coloring

In a $2 \times 8$ squared board, you want to color each square red or blue in such a way that on each $2 \times 2$ sub-board there are at least $3$ boxes painted blue. In how many ways can this coloring be done? Note. A $2 \times 2$ board is a square made up of $4$ squares that have a common vertex.

2017 Oral Moscow Geometry Olympiad, 2

Tags: geometry , 3d geometry , pyramid

Given pyramid with base $n-gon$. How many maximum number of edges can be perpendicular to base?

1998 IMO Shortlist, 7

Tags: number theory , sum of digits , digit , divisibility

Prove that for each positive integer $n$, there exists a positive integer with the following properties: It has exactly $n$ digits. None of the digits is 0. It is divisible by the sum of its digits.

1987 IMO Longlists, 52

Tags: circumcircle , incenter , collinearity , geometry

Given a nonequilateral triangle $ABC$, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles $A'B'C'$ (the vertices listed counterclockwise) for which the triples of points $A,B', C'; A',B, C';$ and $A',B', C$ are collinear. [i]Proposed by Poland.[/i]

1989 IMO Longlists, 16

Tags: geometry , angle , geometric inequality , polygon

Show that any two points lying inside a regular $ n\minus{}$gon $ E$ can be joined by two circular arcs lying inside $ E$ and meeting at an angle of at least $ \left(1 \minus{} \frac{2}{n} \right) \cdot \pi.$