Olympiad problems

2000 Austrian-Polish Competition, 3

For each integer $n \ge 3$ solve in real numbers the system of equations: $$\begin{cases} x_1^3 = x_2 + x_3 + 1 \\...\\x_{n-1}^3 = x_n+ x_1 + 1\\x_{n}^3 = x_1+ x_2 + 1 \end{cases}$$

2007 Purple Comet Problems, 1

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The sum of nine consecutive odd numbers is $2007$. Find the greatest of these nine numbers.

2017 Princeton University Math Competition, 13

Tags: geometry

A point-sized cue ball is fired in a straight path from the center of a regular hexagonal billiards table of side length $1$. If it is not launched directly into a pocket but travels an integer distance before falling into one of the pockets (located in the corners), find the minimum distance that it could have traveled.

2024 Harvard-MIT Mathematics Tournament, 22

Tags: guts

Let $x<y$ be positive real numbers such that $$\sqrt{x}+\sqrt{y}=4 \quad \text{and} \quad \sqrt{x+2}+\sqrt{y+2}=5.$$ Compute $x.$

1993 APMO, 3

Tags: polynomial , ratio , algebra , inequalities

Let \begin{eqnarray*} f(x) & = & a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \ \ \mbox{and} \\ g(x) & = & c_{n+1} x^{n+1} + c_n x^n + \cdots + c_0 \end{eqnarray*} be non-zero polynomials with real coefficients such that $g(x) = (x+r)f(x)$ for some real number $r$. If $a = \max(|a_n|, \ldots, |a_0|)$ and $c = \max(|c_{n+1}|, \ldots, |c_0|)$, prove that $\frac{a}{c} \leq n+1$.

2023 Harvard-MIT Mathematics Tournament, 10

Tags: guts

The number $$316990099009901=\frac{32016000000000001}{101}$$ is the product of two distinct prime numbers. Compute the smaller of these two primes.

2020 Tournament Of Towns, 3

Tags: combinatorics

There are $41$ letters on a circle, each letter is $A$ or $B$. It is allowed to replace $ABA$ by $B$ and conversely, as well as to replace $BAB$ by $A$ and conversely. Is it necessarily true that it is possible to obtain a circle containing a single letter repeating these operations? Maxim Didin

2009 Stanford Mathematics Tournament, 1

Tags: geometry

The sum of all of the interior angles of seven polygons is $180\times17$. Find the total number of sides of the polygons.

2005 District Olympiad, 1

Tags: induction , combinatorics

Let $A_1$, $A_2$, $\ldots$, $A_n$, $n\geq 2$ be $n$ finite sets with the properties i) $|A_i| \geq 2$, for all $1\leq i \leq n$; ii) $|A_i\cap A_j| \neq 1$, for all $1\leq i<j\leq n$. Prove that the elements of the set $\displaystyle \bigcup_{i=1}^n A_i$ can be colored with 2 colors, such that all the sets $A_i$ are bi-color, for all $1\leq i \leq n$.

MOAA Accuracy Rounds, 2023.8

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Harry wants to label the points of a regular octagon with numbers $1,2,\ldots ,8$ and label the edges with $1,2,\ldots, 8$. There are special rules he must follow: If an edge is numbered even, then the sum of the numbers of its endpoints must also be even. If an edge is numbered odd, then the sum of the numbers of its endpoints must also be odd. Two octagon labelings are equivalent if they can be made equal up to rotation, but not up to reflection. If $N$ is the number of possible octagon labelings, find the remainder when $N$ is divided by $100$. [i]Proposed by Harry Kim[/i]

2013 NZMOC Camp Selection Problems, 10

Tags: algebra , inequalities

Find the largest possible real number $C$ such that for all pairs $(x, y)$ of real numbers with $x \ne y$ and $xy = 2$, $$\frac{((x + y)^2- 6))(x-y)^2 + 8))}{(x-y)^2} \ge C.$$ Also determine for which pairs $(x, y)$ equality holds.

1994 Polish MO Finals, 1

Tags: polynomial , rational root theorem , algebra

Find all triples $(x,y,z)$ of positive rationals such that $x + y + z$, $\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$ and $xyz$ are all integers.

2010 May Olympiad, 4

Tags: number theory , multiple

Find all natural numbers of $90$ digits that are multiples of $13$ and have the first $43$ digits equal to each other and nonzero, the last $43$ digits equal to each other, and the middle $4$ digits are $2, 0, 1, 0$, in that order.

2017 Romanian Masters In Mathematics, 1

Tags: algebra , additive combinatorics , representation

[b](a)[/b] Prove that every positive integer $n$ can be written uniquely in the form \[n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},\] where $k\geq 0$ and $0\le m_1<m_2\cdots <m_{2k+1}$ are integers. This number $k$ is called [i]weight[/i] of $n$. [b](b)[/b] Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.

2010 239 Open Mathematical Olympiad, 6

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We call natural numbers $n$ and $k$ are similar if they are multiples of square of a number greater than $1$. Let $f(n)$ denote the number of numbers from $1$ to $n$ similar to $n$ (for example, $f(16)=4$, since the number $16$ is similar to $4$, $8$, $12$ and $16$). What integer values can the quotient $\frac{n}{f(n)}$ take?

2004 AMC 10, 7

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On a trip from the United States to Canada, Isabella took $ d$ U.S. dollars. At the border she exchanged them all, receiving $ 10$ Canadian dollars for every $ 7$ U.S. dollars. After spending $ 60$ Canadian dollars, she had $ d$ Canadian dollars left. What is the sum of the digits of $ d$? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

2022 CMIMC, 2.7 1.3

Tags: combinatorics

For a family gathering, $8$ people order one dish each. The family sits around a circular table. Find the number of ways to place the dishes so that each person’s dish is either to the left, right, or directly in front of them. [i]Proposed by Nicole Sim[/i]

2018 ELMO Shortlist, 2

Tags: geometry

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be the midpoint of $\overline{AH}$ and let $T$ be on line $BC$ with $\angle TAO=90^{\circ}$. Let $X$ be the foot of the altitude from $O$ onto line $PT$. Prove that the midpoint of $\overline{PX}$ lies on the nine-point circle* of $\triangle ABC$. *The nine-point circle of $\triangle ABC$ is the unique circle passing through the following nine points: the midpoint of the sides, the feet of the altitudes, and the midpoints of $\overline{AH}$, $\overline{BH}$, and $\overline{CH}$. [i]Proposed by Zack Chroman[/i]

V Soros Olympiad 1998 - 99 (Russia), 10.1

Tags: analytic geometry , geometry , symmetry , symmetric

It is known that the graph of the function $y =\frac{a-6x}{2+x}$ is centrally summetric to the graph of the function $y = \frac{1}{x}$ with respect to some point. Find the value of the parameter $a$ and the coordinates of the center of symmetry.

2012 Estonia Team Selection Test, 2

Tags: number theory , sequence

For a given positive integer $n$ one has to choose positive integers $a_0, a_1,...$ so that the following conditions hold: (1) $a_i = a_{i+n}$ for any $i$, (2) $a_i$ is not divisible by $n$ for any $i$, (3) $a_{i+a_i}$ is divisible by $a_i$ for any $i$. For which positive integers $n > 1$ is this possible only if the numbers $a_0, a_1, ...$ are all equal?

1982 AMC 12/AHSME, 2

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If a number eight times as large as $x$ is increased by two, then one fourth of the result equals $\textbf{(A)} \ 2x + \frac{1}{2} \qquad \textbf{(B)} \ x + \frac{1}{2} \qquad \textbf{(C)} \ 2x+2 \qquad \textbf{(D)} \ 2x+4 \qquad \textbf{(E)} \ 2x+16$

1956 AMC 12/AHSME, 32

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George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, how many minutes after starting did they pass each other the second time? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4\frac {1}{2} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7\frac {1}{2} \qquad\textbf{(E)}\ 9$

2019 PUMaC Algebra A, 6

Tags: algebra

A [i]weak binary representation[/i] of a nonnegative integer $n$ is a representation $n=a_0+2\cdot a_1+2^2\cdot a_2+\dots$ such that $a_i\in\{0,1,2,3,4,5\}$. Determine the number of such representations for $513$.

1985 IMO Longlists, 61

Tags: combinatorics

Consider the set $A = \{0, 1, 2, \dots , 9 \}$ and let $(B_1,B_2, \dots , B_k)$ be a collection of nonempty subsets of $A$ such that $B_i \cap B_j$ has at most two elements for $i \neq j$. What is the maximal value of $k \ ?$

2022 DIME, 15

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For positive integers $n$, let $f(n)$ denote the number of integers $1 \leq a \leq 130$ for which there exists some integer $b$ such that $a^b-n$ is divisible by $131$, and let $g(n)$ denote the sum of all such $a$. Find the remainder when$$\sum_{n = 1}^{130} [f(n) \cdot g(n)]$$is divided by $131$. [i]Proposed by [b]ApraTrip[/b][/i]