Olympiad problems

2020 Baltic Way, 3

Tags: sequence , inequalities

A real sequence $(a_n)_{n=0}^\infty$ is defined recursively by $a_0 = 2$ and the recursion formula $$ a_{n} = \begin{dcases} a_{n-1}^2 & \text{if $a_{n-1}<\sqrt3$} \\ \frac{a_{n-1}^2}{3} & \text{if $a_{n-1}\geq\sqrt 3$.} \end{dcases} $$ Another real sequence $(b_n)_{n=1}^\infty$ is defined in terms of the first by the formula $$ b_{n} = \begin{dcases} 0 & \text{if $a_{n-1}<\sqrt3$} \\ \frac{1}{2^{n}} & \text{if $a_{n-1}\geq\sqrt 3$,} \end{dcases} $$ valid for each $n\geq 1$. Prove that $$ b_1 + b_2 + \cdots + b_{2020} < \frac23. $$

2019 Harvard-MIT Mathematics Tournament, 3

Tags: hmmt , probability

Reimu and Sanae play a game using $4$ fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the four coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then [i]neither[/i] of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?

1992 Hungary-Israel Binational, 4

Tags: analytic geometry , parallelogram , geometry

We are given a convex pentagon $ABCDE$ in the coordinate plane such that $A$, $B$, $C$, $D$, $E$ are lattice points. Let $Q$ denote the convex pentagon bounded by the five diagonals of the pentagon $ABCDE$ (so that the vertices of $Q$ are the interior points of intersection of diagonals of the pentagon $ABCDE$). Prove that there exists a lattice point inside of $Q$ or on the boundary of $Q$.

2020 CCA Math Bonanza, I9

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A sequence $a_n$ of real numbers satisfies $a_1=1$, $a_2=0$, and $a_n=(S_{n-1}+1)S_{n-2}$ for all integers $n\geq3$, where $S_k=a_1+a_2+\dots+a_k$ for positive integers $k$. What is the smallest integer $m>2$ such that $127$ divides $a_m$? [i]2020 CCA Math Bonanza Individual Round #9[/i]

1989 China Team Selection Test, 4

Tags: geometry , circumcircle , geometric transformation , rotation , congruent triangles , cyclic quadrilateral

Given triangle $ABC$, squares $ABEF, BCGH, CAIJ$ are constructed externally on side $AB, BC, CA$, respectively. Let $AH \cap BJ = P_1$, $BJ \cap CF = Q_1$, $CF \cap AH = R_1$, $AG \cap CE = P_2$, $BI \cap AG = Q_2$, $CE \cap BI = R_2$. Prove that triangle $P_1 Q_1 R_1$ is congruent to triangle $P_2 Q_2 R_2$.

2016 Tuymaada Olympiad, 8

Tags: combinatorics , graph theory

A connected graph is given. Prove that its vertices can be coloured blue and green and some of its edges marked so that every two vertices are connected by a path of marked edges, every marked edge connects two vertices of different colour and no two green vertices are connected by an edge of the original graph.

1991 IMTS, 3

Tags: number theory , relatively prime , algebra

Prove that if $x,y$ and $z$ are pairwise relatively prime positive integers, and if $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$, then $x+y, x-z, y-z$ are perfect squares of integers.

2024 China Girls Math Olympiad, 2

Tags: combinatorics , game

There are $8$ cards on which the numbers $1$, $2$, $\dots$, $8$ are written respectively. Alice and Bob play the following game: in each turn, Alice gives two cards to Bob, who must keep one card and discard the other. The game proceeds for four turns in total; in the first two turns, Bob cannot keep both of the cards with the larger numbers, and in the last two turns, Bob also cannot keep both of the cards with the larger numbers. Let $S$ be the sum of the numbers written on the cards that Bob keeps. Find the greatest positive integer $N$ for which Bob can guarantee that $S$ is at least $N$.

1990 Tournament Of Towns, (262) 6

Tags: combinatorial geometry , geometry , combinatorics

There are some ink-blots on a white paper square with side length $a$. The area of each blot is not greater than $1$ and every line parallel to any one of the sides of the square intersects no more than one blot. Prove that the total area of the blots is not greater than $a$. (A. Razborov, Moscow)

1991 Arnold's Trivium, 98

Tags: probability , function

In the game of "Fingers", $N$ players stand in a circle and simultaneously thrust out their right hands, each with a certain number of fingers showing. The total number of fingers shown is counted out round the circle from the leader, and the player on whom the count stops is the winner. How large must $N$ be for a suitably chosen group of $N/10$ players to contain a winner with probability at least $0.9$? How does the probability that the leader wins behave as $N\to\infty$?

2011 IMC, 5

Tags: induction , perpendicular bisector , geometry

Let $F=A_0A_1...A_n$ be a convex polygon in the plane. Define for all $1 \leq k \leq n-1$ the operation $f_k$ which replaces $F$ with a new polygon $f_k(F)=A_0A_1..A_{k-1}A_k^\prime A_{k+1}...A_n$ where $A_k^\prime$ is the symmetric of $A_k$ with respect to the perpendicular bisector of $A_{k-1}A_{k+1}$. Prove that $(f_1\circ f_2 \circ f_3 \circ...\circ f_{n-1})^n(F)=F$.

2007 Sharygin Geometry Olympiad, 6

Tags: geometry , congruent triangles , equal angles

Two non-congruent triangles are called [i]analogous [/i] if they can be denoted as $ABC$ and $A'B'C'$ such that $AB = A'B', AC = A'C'$ and $\angle B = \angle B'$ . Do there exist three mutually [i]analogous[/i] triangles?

2013 Hanoi Open Mathematics Competitions, 8

Tags: geometry , pentagon , area

Let $ABCDE$ be a convex pentagon and area of $\vartriangle ABC =$ area of $\vartriangle BCD =$ area of $\vartriangle CDE=$ area of $\vartriangle DEA =$ area of $\vartriangle EAB$. Given that area of $\vartriangle ABCDE = 2$. Evaluate the area of area of $\vartriangle ABC$.

2013 Harvard-MIT Mathematics Tournament, 7

Tags: hmmt , number theory , prime factorization

Find the number of positive divisors $d$ of $15!=15\cdot 14\cdot\cdots\cdot 2\cdot 1$ such that $\gcd(d,60)=5$.

1973 Chisinau City MO, 66

Tags: symmetry , geometry , combinatorics , square

If $A$ and $B$ are points of the plane, then by $A * B$ we denote a point symmetric to $A$ with respect to $B$. Is it possible, by applying the operation $*$ several times, to obtain from the three vertices of a given square its fourth vertex?

2010 Sharygin Geometry Olympiad, 7

Tags: polygon , geometry , regular polygon , paper

Each of two regular polygons $P$ and $Q$ was divided by a line into two parts. One part of $P$ was attached to one part of $Q$ along the dividing line so that the resulting polygon was regular and not congruent to $P$ or $Q$. How many sides can it have?

2005 MOP Homework, 7

Tags: combinatorics

Let $S$ be a set of points in the plane satisfying the following conditions: (a) there are seven points in $S$ that form a convex heptagon; and (b) for any five points in $S$, if they form a convex pentagon, then there is point in $S$ lies in the interior of the pentagon. Determine the minimum value of the number of elements in $S$.

2015 BMT Spring, 12

Tags: combinatorics

How many possible arrangements of bishops are there on a $8 \times 8$ chessboard such that no bishop threatens a square on which another lies and the maximum number of bishops are used? (Note that a bishop threatens any square along a diagonal containing its square.)

2024 AMC 12/AHSME, 1

Tags:

What is the value of $9901\cdot101-99\cdot10101?$ $\textbf{(A) }2\qquad\textbf{(B) }20\qquad\textbf{(C) }21\qquad\textbf{(D) }200\qquad\textbf{(E) }2020$

2015 Greece Team Selection Test, 4

Tags: function , inequalities , algebra

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy $yf(x)+f(y) \geq f(xy)$

2023 NMTC Junior, P4

Tags: combinatorics

There are $n$ (an even number) bags. Each bag contains atleast one apple and at most $n$ apples. The total number of apples is $2n$. Prove that it is always possible to divide the bags into two parts such that the number of apples in each part is $n$.

1950 AMC 12/AHSME, 16

Tags: algebra , binomial theorem

The number of terms in the expansion of $ [(a\plus{}3b)^2(a\minus{}3b)^2]^2$ when simplified is: $\textbf{(A)}\ 4\qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

2004 Swedish Mathematical Competition, 3

Tags: algebra , functional , functional equation

A function $f$ satisfies $f(x)+x f(1-x) = x^2$ for all real $x$. Determine $f$ .

1994 AMC 8, 14

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Two children at a time can play pairball. For $90$ minutes, with only two children playing at time, five children take turns so that each one plays the same amount of time. The number of minutes each child plays is $\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 36$

2017 CCA Math Bonanza, I8

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Let $a_1,a_2,\ldots,a_{18}$ be a list of prime numbers such that $\frac{1}{64}\times a_1\times a_2\times\cdots\times a_{18}$ is one million. Determine the sum of all positive integers $n$ such that $$\sum_{i=1}^{18}\frac{1}{\log_{a_i}n}$$ is a positive integer. [i]2017 CCA Math Bonanza Individual Round #8[/i]