Olympiad problems

2017 Romanian Masters In Mathematics, 1

[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]

2012 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: number theory , board , combinatorics

In each square of a $100 \times 100$ board there is written an integer. The allowed operation is to choose four squares that form the figure or any of its reflections or rotations, and add $1$ to each of the four numbers. The aim is, through operations allowed, achieving a board with the smallest possible number of different residues modulo $33$. What is the minimum number that can be achieved with certainty?

2006 Portugal MO, 4

Tags: parallelogram , isosceles , geometry

In the parallelogram $[ABCD], E$ is the midpoint of $[AD]$ and $F$ the orthogonal projection of $B$ on $[CE]$. Prove that the triangle $[ABF]$ is isosceles. [img]https://1.bp.blogspot.com/-DLmFg8ayEQ4/X4XMohA5TjI/AAAAAAAAMnk/thlIKnNUiCkuu9cg1Aq7Zltz8SenmFWuwCLcBGAsYHQ/s0/2006%2Bportugal%2Bp4.png[/img]

2017 CMIMC Algebra, 6

Tags: algebra

Suppose $P$ is a quintic polynomial with real coefficients with $P(0)=2$ and $P(1)=3$ such that $|z|=1$ whenever $z$ is a complex number satisfying $P(z) = 0$. What is the smallest possible value of $P(2)$ over all such polynomials $P$?

1928 Eotvos Mathematical Competition, 2

Tags: combinatorics

Put the numbers $1,2,3,...,n$ on the circumference of a circle in such a way that the difference between neighboring numbers is at most $2$. Prove that there is just one solution (if regard is paid only to the order in which the numbers are arranged).

2019 HMNT, 2

Tags: geometry

Sandy likes to eat waffles for breakfast. To make them, she centers a circle of wafflebatter of radius $3$ cm at the origin of the coordinate plane and her waffle iron imprints non-overlapping unit-square holes centered at each lattice point. How many of these holes are contained entirely within the area of the waffle?

1965 All Russian Mathematical Olympiad, 057

Tags: combinatorics , game strategy

Given a board $3\times3$ and $9$ cards with some numbers (known to the players). Two players, in turn, put those cards on the board. The first wins if the sum of the numbers in the first and the third row is greater than in the first and the third column. Prove that it doesn't matter what numbers are on the cards, but if the first plays the best way, the second can not win.

2024 Junior Balkan Team Selection Tests - Moldova, 12

Tags: number theory

[b]Version 1.[/b] Find all primes $p$ satisfying the following conditions: (i) $\frac{p+1}{2}$ is a prime number. (ii) There are at least three distinct positive integers $n$ for which $\frac{p^2+n}{p+n^2}$ is an integer. [b]Version 2.[/b] Let $p \neq 5$ be a prime number such that $\frac{p+1}{2}$ is also a prime. Suppose there exist positive integers $a <b$ such that $\frac{p^2+a}{p+a^2}$ and $\frac{p^2+b}{p+b^2}$ are integers. Show that $b=(a-1)^2+1$.

2007 All-Russian Olympiad, 3

Tags: rhombus , ratio , geometry

Given a rhombus $ABCD$. A point $M$ is chosen on its side $BC$. The lines, which pass through $M$ and are perpendicular to $BD$ and $AC$, meet line $AD$ in points $P$ and $Q$ respectively. Suppose that the lines $PB,QC,AM$ have a common point. Find all possible values of a ratio $\frac{BM}{MC}$. [i]S. Berlov, F. Petrov, A. Akopyan[/i]

2014 AMC 12/AHSME, 14

Tags: geometry

A rectangular box has a total surface area of $94$ square inches. The sum of the lengths of all its edges is $48$ inches. What is the sum of the lengths in inches of all of its interior diagonals? ${ \textbf{(A)}\ 8\sqrt{3}\qquad\textbf{(B)}\ 10\sqrt{2}\qquad\textbf{(C)}\ 16\sqrt{3}\qquad\textbf{(D)}}\ 20\sqrt{2}\qquad\textbf{(E)}\ 40\sqrt{2} $

2024 Greece National Olympiad, 3

Tags: combinatorics

Let $n \geq 2$ be a positive integer and let $A, B$ be two finite sets of integers such that $|A| \leq n$. Let $C$ be a subset of the set $\{(a, b) | a \in A, b \in B\}$. Achilles writes on a board all possible distinct differences $a-b$ for $(a, b) \in C$ and suppose that their count is $d$. He writes on another board all triplets $(k, l, m)$, where $(k, l), (k, m) \in C$ and suppose that their count is $p$. Show that $np \geq d^2.$

1950 AMC 12/AHSME, 18

Tags: logarithm

Of the following (1) $ a(x\minus{}y)\equal{}ax\minus{}ay$ (2) $ a^{x\minus{}y}\equal{}a^x\minus{}a^y$ (3) $ \log (x\minus{}y)\equal{}\log x\minus{}\log y$ (4) $ \frac {\log x}{\log y}\equal{} \log{x}\minus{} \log{y}$ (5) $ a(xy)\equal{}ax\times ay$ $\textbf{(A)}\ \text{Only 1 and 4 are true} \qquad\\ \textbf{(B)}\ \text{Only 1 and 5 are true} \qquad\\ \textbf{(C)}\ \text{Only 1 and 3 are true} \qquad\\ \textbf{(D)}\ \text{Only 1 and 2 are true} \qquad\\ \textbf{(E)}\ \text{Only 1 is true}$

1967 Poland - Second Round, 2

Tags: combinatorics

There are 100 persons in a hall, everyone knowing at least 66 of the others. Prove that there is a case in which among any four some two don’t know each other.

2017 CMIMC Number Theory, 1

Tags: number theory

There exist two distinct positive integers, both of which are divisors of $10^{10}$, with sum equal to $157$. What are they?