Olympiad problems

1988 IMO Shortlist, 25

A positive integer is called a [b]double number[/b] if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers which are perfect squares.

LMT Guts Rounds, 2020 F9

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If $xy:yz:zx=6:8:12,$ and $x^3+y^3+z^3:xyz$ is $m:n$ where $m$ and $n$ are relatively prime positive integers, then find $m+n.$ [i]Proposed by Ada Tsui[/i]

2008 Denmark MO - Mohr Contest, 3

Tags: combinatorics , number theory , game , game strategy , winning strategy

The numbers from $1$ to $500$ are written on the board. Two players $A$ and $B$ erase alternately one number at a time, and $A$ deletes the first number. If the sum of the last two number on the board is divisible by $3$, $B$ wins, otherwise $A$ wins. Which player can lay out a strategy that ensures this player's victory?

2014 Saudi Arabia GMO TST, 2

Tags: integer polynomial , polynomial , algebra

Let $S = \{f(a, b) | a, b = 1,2,3, 4$ and $a \ne b\}$, and consider all nonzero polynomials $p(X,Y )$ with integer coefficients such that $p(a, b) = 0$ for every element $(a,b)$ in $S$. (a) What is the minimal degree of such polynomial $p(X, Y )$ ? (b) Determine all such polynomials $p(X, Y )$ with minimal degree.

1978 Chisinau City MO, 157

Tags: geometry , geometric inequality

Prove that the side $AB$ of a convex quadrilateral $ABCD$ is less than its diagonal $AC$ if $|AB|+|BC| \le |AC| +| CD|$.

1990 Turkey Team Selection Test, 6

Tags: abstract algebra , induction , number theory

Let $k\geq 2$ and $n_1, \dots, n_k \in \mathbf{Z}^+$. If $n_2 | (2^{n_1} -1)$, $n_3 | (2^{n_2} -1)$, $\dots$, $n_k | (2^{n_{k-1}} -1)$, $n_1 | (2^{n_k} -1)$, show that $n_1 = \dots = n_k =1$.

2003 IberoAmerican, 1

Tags: combinatorics

Let $M=\{1,2,\dots,49\}$ be the set of the first $49$ positive integers. Determine the maximum integer $k$ such that the set $M$ has a subset of $k$ elements such that there is no $6$ consecutive integers in such subset. For this value of $k$, find the number of subsets of $M$ with $k$ elements with the given property.

2016 Harvard-MIT Mathematics Tournament, 28

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Among citizens of Cambridge there exist $8$ different types of blood antigens. In a crowded lecture hall are $256$ students, each of whom has a blood type corresponding to a distinct subset of the antigens; the remaining of the antigens are foreign to them. Quito the Mosquito flies around the lecture hall, picks a subset of the students uniformly at random, and bites the chosen students in a random order. After biting a student, Quito stores a bit of any antigens that student had. A student bitten while Quito had $k$ blood antigen foreign to him/her will suffer for $k$ hours. What is the expected total suffering of all $256$ students, in hours?

1990 IMO Longlists, 49

Tags: geometry

$AB, AC$ are two chords of the circle centered at $O$. The diameter, which is perpendicular to $BC$, intersects $AB, AC$ at $F, G$ respectively ($F$ is in the circle). The tangent from $G$ tangents the circle at $T$. Prove that $F$ is the projection of $T$ on $OG. $

2019 AMC 12/AHSME, 1

Tags: geometry , percent , ratio

The area of a pizza with radius $4$ inches is $N$ percent larger than the area of a pizza with radius $3$ inches. What is the integer closest to $N$? $\textbf{(A) } 25 \qquad\textbf{(B) } 33 \qquad\textbf{(C) } 44\qquad\textbf{(D) } 66 \qquad\textbf{(E) } 78$

2017 CHMMC (Fall), 10

Tags: algebra , polynomial

Let $\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$. It is known that $1<\alpha<2$. We define the sequence of polynomials $\left\{{p_n(x)}\right\}_{n\ge0}$ by taking $p_0(x)=x$ and setting \begin{align*} p_{n+1}(x)=(p_n(x))^2-\alpha \end{align*} How many distinct real roots does $p_{10}(x)$ have?

2004 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: number theory

How many integers $n>1$ are there such that $n$ divides $x^{13}-x$ for every positive integer $x$?

2009 Brazil Team Selection Test, 1

Tags: algebra , number theory , equation

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

2010 LMT, 12

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Al and Bob play Rock Paper Scissors until someone wins a game. What is the probability that this happens on the sixth game?

2016 Azerbaijan Balkan MO TST, 2

Tags: number theory

Set $A$ consists of natural numbers such that these numbers can be expressed as $2x^2+3y^2,$ where $x$ and $y$ are integers. $(x^2+y^2\not=0)$ $a)$ Prove that there is no perfect square in the set $A.$ $b)$ Prove that multiple of odd number of elements of the set $A$ cannot be a perfect square.

2024 Iran MO (3rd Round), 3

Tags: algebra

An integer number $n\geq 2$ and real numbers $x_1<x_2<\cdots < x_n$ are given. $f: \mathbb R \to \mathbb R$ is a function defined as $$ f(x) = \left | \dfrac{(x-x_2)(x-x_3)\cdots (x-x_n)}{(x_1-x_2)(x_1-x_3)\cdots (x_1-x_n)} \right | + \cdots + \left | \dfrac{(x-x_1)(x-x_2)\cdots (x-x_{n-1})}{(x_n-x_1)(x_n-x_2)\cdots (x_n-x_{n-1})} \right |. $$ Prove that there exists $i\in \{1,2,\cdots,n-1\}$ such that for all $x\in (x_i,x_{i+1})$ one has $f(x)< \sqrt n$. Proposed by [i]Navid Safaei[/i]

1999 China Team Selection Test, 3

Tags: combinatorics

Let $S = \lbrace 1, 2, \ldots, 15 \rbrace$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions: [b]I.[/b] $|A_i| = 7, i = 1, 2, \ldots, n$; [b]II.[/b] $|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n$ [b]III.[/b] For any 3-element subset $M$ of $S$, there exists $A_k$ such that $M \subset A_k$. Find the smallest possible value of $n$.

1992 AMC 12/AHSME, 12

Tags: geometry , geometric transformation , reflection

Let $y = mx + b$ be the image when the line $x - 3y + 11 = 0$ is reflected across the x-axis. The value of $m + b$ is $ \textbf{(A)}\ -6\qquad\textbf{(B)}\ -5\qquad\textbf{(C)}\ -4\qquad\textbf{(D)}\ -3\qquad\textbf{(E)}\ -2 $

2004 Harvard-MIT Mathematics Tournament, 8

Tags: geometry

A triangle has side lengths $18$, $24$, and $30$. Find the area of the triangle whose vertices are the incenter, circumcenter, and centroid of the original triangle.

2002 IMO Shortlist, 2

Tags: inequalities , algebra , sequence , bounded

Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.

1989 Flanders Math Olympiad, 3

Tags: trigonometry , absolute value

Show that:\[\alpha = \pm \frac{\pi}{12} + k\cdot \frac{\pi}2 (k\in \mathbb{Z}) \Longleftrightarrow\ |{\tan \alpha}| + |{\cot \alpha}| = 4\]

1994 AMC 8, 3

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Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at $\text{(A)}\ \text{3:40 P.M.} \qquad \text{(B)}\ \text{3:55 P.M.} \qquad \text{(C)}\ \text{4:10 P.M.} \qquad \text{(D)}\ \text{4:25 P.M.} \qquad \text{(E)}\ \text{4:40 P.M.}$

1998 Brazil Team Selection Test, Problem 4

Tags: algebra , polynomial

(a) Show that, for each positive integer $n$, the number of monic polynomials of degree $n$ with integer coefficients having all its roots on the unit circle is finite. (b) Let $P(x)$ be a monic polynomial with integer coefficients having all its roots on the unit circle. Show that there exists a positive integer $m$ such that $y^m=1$ for each root $y$ of $P(x)$.

2023 ISI Entrance UGB, 8

Tags: continuity , differentiability , function , calculus

Let $f \colon [0,1] \to \mathbb{R}$ be a continuous function which is differentiable on $(0,1)$. Prove that either $f(x) = ax + b$ for all $x \in [0,1]$ for some constants $a,b \in \mathbb{R}$ or there exists $t \in (0,1)$ such that $|f(1) - f(0)| < |f'(t)|$.

2018 Mathematical Talent Reward Programme, MCQ: P 1

Tags: counting , combinatorics

A coin is tossed 9 times. Hence $2^{9}$ different outcomes are possible. In how many cases 2 consecutive heads does not appear? [list=1] [*] 34 [*] 55 [*] 89 [*] None of these [/list]