Olympiad problems

1995 Miklós Schweitzer, 9

Tags: real analysis

A serpentine is a sequence of points $P_1 , ..., P_m$ in a plane, not necessarily all different, such that the distance between $P_i$ and $P_{i+1}$ is at least 1, and the segments $P_i P_{i +1}$ are alternately horizontal and vertical. Construct a compact set in which there is a sequence of serpentines with arbitrary long lengths but there is no closed serpentine ($P_m = P_i$ for some i < m).

2016 Regional Olympiad of Mexico Southeast, 4

Tags: geometry , quadrilateral , algebra

The diagonals of a convex quadrilateral $ABCD$ intersect in $E$. Let $S_1, S_2, S_3$ and $S_4$ the areas of the triangles $AEB, BEC, CED, DEA$ respectively. Prove that, if exists real numbers $w, x, y$ and $z$ such that $$S_1=x+y+xy, S_2=y+z+yz, S_3=w+z+wz, S_4=w+x+wx,$$ then $E$ is the midpoint of $AC$ or $E$ is the midpoint of $BD$.

Kvant 2023, M2744

Tags: geometry

A regular $100$-gon was cut into several parallelograms and two triangles. Prove that these triangles are congruent.

2020 Moldova Team Selection Test, 1

Tags: progression , arithmetic progression , geometric progression , number theory

All members of geometrical progression $(b_n)_{n\geq1}$ are members of some arithmetical progression. It is known that $b_1$ is an integer. Prove that all members of this geometrical progression are integers. (progression is infinite)

2025 CMIMC Algebra/NT, 9

Tags: algebra , number theory

Find the largest prime factor of $45^5-1.$

2010 Baltic Way, 1

Tags: system of equations , algebra

Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations \[\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}\]

1964 IMO, 4

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

Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

2011 Pre-Preparation Course Examination, 4

Tags: combinatorics

A star $K_{1,3}$ is called a paw. suppose that $G$ is a graph without any induced paws. prove that $\chi(G)\le(\omega(G))^2$. (15 points)

2012 AIME Problems, 14

Tags: hmmt

In a group of nine people each person shakes hands with exactly two of the other people from the group. Let N be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when N is divided by 1000.

2017 All-Russian Olympiad, 1

Tags: combinatorics

In country some cities are connected by oneway flights( There are no more then one flight between two cities). City $A$ called "available" for city $B$, if there is flight from $B$ to $A$, maybe with some transfers. It is known, that for every 2 cities $P$ and $Q$ exist city $R$, such that $P$ and $Q$ are available from $R$. Prove, that exist city $A$, such that every city is available for $A$.

2011 AIME Problems, 12

Tags: probability , geometry , geometric transformation , rotation , number theory , relatively prime , complementary counting

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

1979 Canada National Olympiad, 4

Tags: calculus

A dog standing at the centre of a circular arena sees a rabbit at the wall. The rabbit runs round the wall and the dog pursues it along a unique path which is determined by running at the same speed and staying on the radial line joining the centre of the arena to the rabbit. Show that the dog overtakes the rabbit just as it reaches a point one-quarter of the way around the arena.

2003 Putnam, 2

Tags: inequalities , putnam inequalities

Let $a_1, a_2, \cdots , a_n$ and $b_1, b_2,\cdots, b_n$ be nonnegative real numbers. Show that \[(a_1a_2 \cdots a_n)^{1/n}+ (b_1b_2 \cdots b_n)^{1/n} \le ((a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n}\]

2004 AMC 10, 8

Tags:

A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a discard pile. The game ends when some player runs out of tokens. Players $ A$, $ B$, and $ C$ start with $ 15$, $ 14$, and $ 13$ tokens, respectively. How many rounds will there be in the game? $ \textbf{(A)}\ 36 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 38 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ 40$

2017 BMT Spring, 9

Tags: number theory , algebra

Let $a_d$ be the number of non-negative integer solutions $(a, b)$ to $a + b = d$ where $a \equiv b$ (mod $n$) for a fixed $n \in Z^+$. Consider the generating function $M(t) = a_0 + a_1t + a_2t^2 + ...$ Consider $$P(n) = \lim_{t\to 1} \left( nM(t) - \frac{1}{(1 - t)^2} \right).$$ Then $P(n)$, $n \in Z^+$ is a polynomial in $n$, so we can extend its domain to include all real numbers while having it remain a polynomial. Find $P(0)$.

2010 Contests, 4

Tags: parallelogram , incenter , perimeter , geometry

The the parallel lines through an inner point $P$ of triangle $\triangle ABC$ split the triangle into three parallelograms and three triangles adjacent to the sides of $\triangle ABC$. (a) Show that if $P$ is the incenter, the perimeter of each of the three small triangles equals the length of the adjacent side. (b) For a given triangle $\triangle ABC$, determine all inner points $P$ such that the perimeter of each of the three small triangles equals the length of the adjacent side. (c) For which inner point does the sum of the areas of the three small triangles attain a minimum? [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 4)[/i]

2012 Uzbekistan National Olympiad, 4

Tags: inequalities , linear algebra , matrix

Given $a,b$ and $c$ positive real numbers with $ab+bc+ca=1$. Then prove that $\frac{a^3}{1+9b^2ac}+\frac{b^3}{1+9c^2ab}+\frac{c^3}{1+9a^2bc} \geq \frac{(a+b+c)^3}{18}$

2024 Philippine Math Olympiad, P3

Tags: geometry , orthocenter

Given triangle $ABC$ with orthocenter $H$, the lines through $B$ and $C$ perpendicular to $AB$ and $AC$, respectively, intersect line $AH$ at $X$ and $Y$, respectively. The circle with diameter $XY$ intersects lines $BX$ and $CY$ a second time at $K$ and $L$, respectively. Prove that points $H, K$ and $L$ are collinear.

2021 Olimphíada, 2

Tags: geometry

Let $P$, $A$, $B$ and $C$ be points on a line $r$, in that order, so that $AB = BC$. Let $H$ be a point that does not belong to this line and let $S$ be the other intersection of the circles $(HPB)$ and $(HAC)$. Let $I$ be the second intersection of the circle with diameter $HB$ and $(HAC)$. Show that the points $P$, $H$, $I$ lie on the same line if and only if $HS$ is perpendicular to $r$.

2002 IMO Shortlist, 7

Tags: incenter , radical axis , polars , iran lemma , geometry

The incircle $ \Omega$ of the acute-angled triangle $ ABC$ is tangent to its side $ BC$ at a point $ K$. Let $ AD$ be an altitude of triangle $ ABC$, and let $ M$ be the midpoint of the segment $ AD$. If $ N$ is the common point of the circle $ \Omega$ and the line $ KM$ (distinct from $ K$), then prove that the incircle $ \Omega$ and the circumcircle of triangle $ BCN$ are tangent to each other at the point $ N$.

2012 Irish Math Olympiad, 5

Tags: inequalities

(a) Show that if $x$ and $y$ are positive real numbers, then $$(x+y)^5\ge 12xy(x^3+y^3)$$ (b) Prove that the constant $12$ is the best possible. In other words, prove that for any $K>12$ there exist positive real numbers $x$ and $y$ such that $$(x+y)^5<Kxy(x^3+y^3)$$

2023 Novosibirsk Oral Olympiad in Geometry, 7

Tags: geometry , square , parallel lines

A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$. [img]https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png[/img]

1985 Kurschak Competition, 1

Tags: combinatorics

We have triangulated a convex $(n+1)$-gon $P_0P_1\dots P_n$ (i.e., divided it into $n-1$ triangles with $n-2$ non-intersecting diagonals). Prove that the resulting triangles can be labelled with the numbers $1,2,\dots,n-1$ such that for any $i\in\{1,2,\dots,n-1\}$, $P_i$ is a vertex of the triangle with label $i$.

Kvant 2023, M2761

Tags: length , geometry

Is it possible to fit a regular polygon into a circle of radius one so that among the lengths of its diagonals there are 2023 different values whose product is equal to one? [i]Proposed by A. Kuznetsov[/i]

1998 All-Russian Olympiad Regional Round, 10.2

Tags: circumcircle , geometry , parallelogram

In an acute triangle $ABC$, a circle $S$ is drawn through the center $O$ of the circumcircle and the vertices $B$ and $C$. Let $OK$ be the diameter of the circle $S$, $D$ and $E$, be it's intersection points with the straight lines $AB$ and $AC$ respectively. Prove that $ADKE$ is a parallelogram.