Olympiad problems

2021 HMNT, 1

Tags: combinatorics

A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated.

2000 VJIMC, Problem 4

Tags: inequalities , real analysis , measure theory

Let $\mathcal B$ be a family of open balls in $\mathbb R^n$ and $c<\lambda\left(\bigcup\mathcal B\right)$ where $\lambda$ is the $n$-dimensional Lebesgue measure. Show that there exists a finite family of pairwise disjoint balls $\{U_i\}^k_{i=1}\subseteq\mathcal B$ such that $$\sum_{j=1}^k\lambda(U_j)>\frac c{3^n}.$$

2017 Online Math Open Problems, 14

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Let $S$ be the set of all points $(x_1, x_2, x_3, \dots, x_{2017})$ in $\mathbb{R}^{2017}$ satisfying $|x_i|+|x_j|\leq 1$ for any $1\leq i< j\leq 2017$. The volume of $S$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m+n$. [i]Proposed by Yannick Yao[/i]

2000 AMC 10, 6

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The Fibonacci Sequence $ 1,1,2,3,5,8,13,21,\ldots$ starts with two 1s and each term afterwards is the sum of its predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci Sequence? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 9$

2018 Romania National Olympiad, 3

Tags: real analysis , inequalities , integral , function , calculus

Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$ $\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$ $\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$ [i]Nicolae Bourbacut[/i]

2014 European Mathematical Cup, 3

Tags: incenter , geometry , circumcircle

Let ABC be a triangle. The external and internal angle bisectors of ∠CAB intersect side BC at D and E, respectively. Let F be a point on the segment BC. The circumcircle of triangle ADF intersects AB and AC at I and J, respectively. Let N be the mid-point of IJ and H the foot of E on DN. Prove that E is the incenter of triangle AHF, or the center of the excircle. [i]Proposed by Steve Dinh[/i]

2012 Indonesia TST, 3

Tags: combinatorics

Let $S$ be a subset of $\{1,2,3,4,5,6,7,8,9,10\}$. If $S$ has the property that the sums of three elements of $S$ are all different, find the maximum number of elements of $S$.

2011 Singapore Junior Math Olympiad, 3

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$\text{Let} S_1,S_2,...S_{2011}$ $\text{be nonempty sets of consecutive integers such that any}$ $2$ $\text{of them have a common element. Prove that there is a positive integer that belongs to every}$ $S_i, i=1,...,2011$ (For example, ${2,3,4,5}$ is a set of consecutive integers while ${2,3,5}$ is not.)

1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 7

Tags: graph theory

In a class, the teacher discovers that every pupil has exactly three friends in the class, that two friends never have a common friend, and that every pair of two pupils who are not friends they have exactly one common friend. How many pupils are there in the class? A. 6 B. 9 C. 10 D. 12 E. 17

2004 All-Russian Olympiad, 4

Tags: cyclic inequality , n-variable inequality , induction , inequalities

Let $n > 3$ be a natural number, and let $x_1$, $x_2$, ..., $x_n$ be $n$ positive real numbers whose product is $1$. Prove the inequality \[ \frac {1}{1 + x_1 + x_1\cdot x_2} + \frac {1}{1 + x_2 + x_2\cdot x_3} + ... + \frac {1}{1 + x_n + x_n\cdot x_1} > 1. \]

1998 Singapore Team Selection Test, 2

Tags: subset , combinatorics

Let $n \ge 2$ be an integer. Let $S$ be a set of $n$ elements and let $A_i, 1 \le i \le m$, be distinct subsets of $S$ of size at least $2$ such that $A_i \cap A_j \ne \emptyset$, $A_i \cap A_k \ne \emptyset$, $A_j \cap A_k \ne \emptyset$ imply $A_i \cap A_j \cap A_k \ne \emptyset$. Show that $m \le 2^{n-1}$ -

2014 Contests, 1

Tags: domain , function , algebra , polynomial , inequalities

Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.

2012 Czech-Polish-Slovak Junior Match, 4

Tags: polygon , combinatorial geometry , regular polygon , isosceles , combinatorics

Prove that among any $51$ vertices of the $101$-regular polygon there are three that are the vertices of an isosceles triangle.

1957 Miklós Schweitzer, 7

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[b]7.[/b] Prove that any real number x satysfying the inequalities $0<x\leq 1$ can be represented in the form $x= \sum_{k=1}^{\infty}\frac{1}{n_k}$ where $(n_k)_{k=1}^{\infty}$ is a sequence of positive integers such that $\frac{n_{k+1}}{n_k}$ assumes, for each $k$, one of the three values $2,3$ or $4$. [b](N. 14)[/b]

2024 Brazil Team Selection Test, 6

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Let $N$ be a positive integer. Prove that there exist three permutations $a_1,\dots,a_N$, $b_1,\dots,b_N$, and $c_1,\dots,c_N$ of $1,\dots,N$ such that \[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\] for every $k=1,2,\dots,N$.

2010 LMT, 3

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Start with a positive integer. Double it, subtract $4,$ halve it, then subtract the original integer to get $x.$ What is $x?$

PEN O Problems, 28

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Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x$, $y$ taken from two different subsets, the number $x^{2}-xy+y^{2}$ belongs to the third subset.

2013 Junior Balkan Team Selection Tests - Romania, 2

Tags: point , lattice , coloring , combinatorics

Let $M$ be the set of integer coordinate points situated on the line $d$ of real numbers. We color the elements of M in black or white. Show that at least one of the following statements is true: (a) there exists a finite subset $F \subset M$ and a point $M \in d$ so that the elements of the set $M - F$ that are lying on one of the rays determined by $M$ on $d$ are all white, and the elements of $M - F$ that are situated on the opposite ray are all black, (b) there exists an infinite subset $S \subset M$ and a point $T \in d$ so that for each $A \in S$ the reflection of A about $T$ belongs to $S$ and has the same color as $A$

2020 CMIMC Algebra & Number Theory, Estimation

Tags: number theory , algebra , estimation

Vijay picks two random distinct primes $1\le p, q\le 10^4$. Let $r$ be the probability that $3^{2205403200}\equiv 1\bmod pq$. Estimate $r$ in the form $0.abcdef$, where $a, b, c, d, e, f$ are decimal digits.

2011 JBMO Shortlist, 4

Tags: number theory

$\boxed{\text{N4}}$ Find all primes $p,q$ such that $2p^3-q^2=2(p+q)^2$.

1966 Putnam, A5

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Let $C$ denote the family of continuous functions on the real axis. Let $T$ be a mapping of $C$ into $C$ which has the following properties: 1. $T$ is linear, i.e. $T(c_1\psi _1+c_2\psi _2)= c_1T\psi _1+c_2T\psi _2$ for $c_1$ and $c_2$ real and $\psi_1$ and $\psi_2$ in $C$. 2. $T$ is local, i.e. if $\psi_1 \equiv \psi_2$ in some interval $I$ then also $T\psi_1 \equiv T\psi_2$ holds in $I$. Show that $T$ must necessarily be of the form $T\psi(x)=f(x)\psi(x)$ where $f(x)$ is a suitable continuous function.

2001 Croatia National Olympiad, Problem 2

Tags: geometry , 3d geometry , tetrahedron

A piece of paper in the shape of a square $FBHD$ with side $a$ is given. Points $G,A$ on $FB$ and $E,C$ on $BH$ are marked so that $FG=GA=AB$ and $BE=EC=CH$. The paper is folded along $DG,DA,DC$ and $AC$ so that $G$ overlaps with $B$, and $F$ and $H$ overlap with $E$. Compute the volume of the obtained tetrahedron $ABCD$.

2022-23 IOQM India, 15

Tags: inequalities , algebra

Let $x,y$ be real numbers such that $xy=1$. Let $T$ and $t$ be the largest and smallest values of the expression \\ $\hspace{2cm} \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2}$\\. \\ If $T+t$ can be expressed in the form $\frac{m}{n}$ where $m,n$ are nonzero integers with $GCD(m,n)=1$, find the value of $m+n$.

2005 USAMTS Problems, 2

Tags: expected value , probability

George has six ropes. He chooses two of the twelve loose ends at random (possibly from the same rope), and ties them together, leaving ten loose ends. He again chooses two loose ends at random and joins them, and so on, until there are no loose ends. Find, with proof, the expected value of the number of loops George ends up with.

2019 LIMIT Category B, Problem 11

Tags: triangle inequality , geometry , combinatorics , inequalities

Let $S=\{1,2,\ldots,10\}$. Three numbers are chosen with replacement from $S$. If the chosen numbers denote the lengths of sides of a triangle, then the probability that they will form a triangle is: $\textbf{(A)}~\frac{101}{200}$ $\textbf{(B)}~\frac{99}{200}$ $\textbf{(C)}~\frac12$ $\textbf{(D)}~\frac{110}{200}$