Olympiad problems

2008 Putnam, B6

Tags: modular arithmetic , linear algebra , matrix , induction , floor function , function

Let $ n$ and $ k$ be positive integers. Say that a permutation $ \sigma$ of $ \{1,2,\dots n\}$ is $ k$-[i]limited[/i] if $ |\sigma(i)\minus{}i|\le k$ for all $ i.$ Prove that the number of $ k$-limited permutations of $ \{1,2,\dots n\}$ is odd if and only if $ n\equiv 0$ or $ 1\pmod{2k\plus{}1}.$

2012-2013 SDML (High School), 4

Tags: geometry , 3d geometry

For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$? $\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$

2024 Korea Summer Program Practice Test, 2

Tags: number theory

Find all integer sequences $a_1 , a_2 , \ldots , a_{2024}$ such that $1\le a_i \le 2024$ for $1\le i\le 2024$ and $$i+j|ia_i-ja_j$$ for each pair $1\le i,j \le 2024$.

2020 BMT Fall, Tie 3

Tags: number theory

Three distinct integers $a_1$, $a_2$, $a_3$ between $1$ and $21$, inclusive, are selected uniformly at random. The probability that the greatest common factor of $a_i-a_j$ and $21$ is $7$ for some positive integers $i $ and $j$, where $1 \le i \ne j \le3 $, can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

2023 VIASM Summer Challenge, Problem 1

Tags: algebra , inequalities

Find the largest positive real number $k$ such that the inequality$$a^3+b^3+c^3-3\ge k(3-ab-bc-ca)$$holds for all positive real triples $(a;b;c)$ satisfying $a+b+c=3.$

2018 MIG, 19

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Rectangle $ABCD$, with integer side lengths, has equal area and perimeter. What is the positive difference between the two possible areas of $ABCD$? $\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 5\qquad\textbf{(E) } 6$

2009 Croatia Team Selection Test, 2

Tags: combinatorics

In each field of 2009*2009 table you can write either 1 or -1. Denote Ak multiple of all numbers in k-th row and Bj the multiple of all numbers in j-th column. Is it possible to write the numbers in such a way that $ \sum_{i\equal{}1}^{2009}{Ai}\plus{} \sum_{i\equal{}1}^{2009}{Bi}\equal{}0$?

2011 Mathcenter Contest + Longlist, 3 sl3

Tags: inequalities , algebra , sum

We will call the sequence of positive real numbers. $a_1,a_2,\dots ,a_n$ of [i]length [/i] $n$ when $$a_1\geq \frac{a_1+a_2}{2}\geq \dots \geq \frac{a_1+a_2+\cdots +a_n}{n}.$$ Let $x_1,x_2,\dots ,x_n$ and $y_1,y_2,\dots ,y_n$ be sequences of length $n.$ Prove that $$\sum_{i = 1}^{n}x_iy_i\geq\frac{1}{n}\left(\sum_{i = 1}^{n}x_i\right)\left(\sum_{i = 1}^{n}y_i\right).$$ [i](tatari/nightmare)[/i]

2002 AMC 12/AHSME, -1

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This test and the matching AMC 10P were developed for the use of a group of Taiwan schools, in early January of 2002. When Taiwan had taken the contests, the AMC released the questions here as a set of practice questions for the 2002 AMC 10 and AMC 12 contests.

2020 Centroamerican and Caribbean Math Olympiad, 3

Tags: algebra , functional equation , function

Find all the functions $f: \mathbb{Z}\to \mathbb{Z}$ satisfying the following property: if $a$, $b$ and $c$ are integers such that $a+b+c=0$, then $$f(a)+f(b)+f(c)=a^2+b^2+c^2.$$

2022 Thailand TST, 1

Tags: algebra , combinatorics

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

2013 National Chemistry Olympiad, 60

Tags:

Which vitamin is the most soluble in water? ${ \textbf{(A)}\ \text{A} \qquad\textbf{(B)}\ \text{K} \qquad\textbf{(C)}\ \text{C} \qquad\textbf{(D)}}\ \text{D} \qquad $

2015 Irish Math Olympiad, 7

Tags: combinatorics , subset , set

Let $n > 1$ be an integer and $\Omega=\{1,2,...,2n-1,2n\}$ the set of all positive integers that are not larger than $2n$. A nonempty subset $S$ of $\Omega$ is called [i]sum-free[/i] if, for all elements $x, y$ belonging to $S, x + y$ does not belong to $S$. We allow $x = y$ in this condition. Prove that $\Omega$ has more than $2^n$ distinct [i]sum-free[/i] subsets.

2014 All-Russian Olympiad, 2

Tags: power of a point , geometry

Let $M$ be the midpoint of the side $AC$ of $ \triangle ABC$. Let $P\in AM$ and $Q\in CM$ be such that $PQ=\frac{AC}{2}$. Let $(ABQ)$ intersect with $BC$ at $X\not= B$ and $(BCP)$ intersect with $BA$ at $Y\not= B$. Prove that the quadrilateral $BXMY$ is cyclic. [i]F. Ivlev, F. Nilov[/i]

2023 Estonia Team Selection Test, 2

Tags:

For any natural number $n{}$ and positive integer $k{}$, we say that $n{}$ is $k-good$ if there exist non-negative integers $a_1,\ldots, a_k$ such that $$n=a_1^2+a_2^4+a_3^8+\ldots+a_k^{2^k}.$$ Is there a positive integer $k{}$ for which every natural number is $k-good$?

2014 Sharygin Geometry Olympiad, 23

Tags: geometry , harmonic division , concyclic

Let $A, B, C$ and $D$ be a triharmonic quadruple of points, i.e $AB\cdot CD = AC \cdot BD = AD \cdot BC.$ Let $A_1$ be a point distinct from $A$ such that the quadruple $A_1, B, C$ and $D$ is triharmonic. Points $B_1, C_1$ and $D_1$ are defined similarly. Prove that a) $A, B, C_1, D_1$ are concyclic; b) the quadruple $A_1, B_1, C_1, D_1$ is triharmonic.

2017 Taiwan TST Round 1, 2

Tags: cyclic quadrilateral , geometry , incenter , reflection , inversion

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

2007 German National Olympiad, 1

Tags: inequalities , algebra , real number

Determine all real numbers $x$ such that for all positive integers $n$ the inequality $(1+x)^n \leq 1+(2^n -1)x$ is true.

Durer Math Competition CD Finals - geometry, 2012.D5

Tags: geometry , area of a triangle , color

The points of a circle of unit radius are colored in two colors. Prove that $3$ points of the same color can be chosen such that the area of the triangle they define is at least $\frac{9}{10}$.

2016 Kyrgyzstan National Olympiad, 1

Tags: algebra

If $a+b+c=0$ ,then find the value of $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})$

1997 Estonia National Olympiad, 2

Tags: inequalities

Let $x$ and $y$ be real numbers. Show that\[x^2+y^2+1>x\sqrt{y^2+1}+y\sqrt{x^2+1}.\]

2021 JBMO TST - Turkey, 1

Tags: geometry , circles , perpendicular lines

In an acute-angled triangle $ABC$, the circle with diameter $[AB]$ intersects the altitude drawn from vertex $C$ at a point $D$ and the circle with diameter $[AC]$ intersects the altitude drawn from vertex $B$ at a point $E$. Let the lines $BD$ and $CE$ intersect at $F$. Prove that $$AF\perp DE$$

1998 Tournament Of Towns, 1

Tags: number theory , divisible , divide

Do there exist $10$ positive integers such that each of them is divisible by none of the other numbers but the square of each of these numbers is divisible by each of the other numbers? (Folklore)

1975 Chisinau City MO, 102

Tags: combinatorics , number theory , divisible , divide , digit

Two people write a $2k$-digit number, using only the numbers $1, 2, 3, 4$ and $5$. The first number on the left is written by the first of them, the second - the second, the third - the first, etc. Can the second one achieve this so that the resulting number is divisible by $9$, if the first seeks to interfere with it? Consider the cases $k = 10$ and $k = 15$.

2010 F = Ma, 7

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Harry Potter is sitting $2.0$ meters from the center of a merry-go-round when Draco Malfoy casts a spell that glues Harry in place and then makes the merry-go-round start spinning on its axis. Harry has a mass of $\text{50.0 kg}$ and can withstand $\text{5.0} \ g\text{'s}$ of acceleration before passing out. What is the magnitude of Harry's angular momentum when he passes out? (A) $\text{200 kg} \cdot \text{m}^2\text{/s}$ (A) $\text{330 kg} \cdot \text{m}^2\text{/s}$ (A) $\text{660 kg} \cdot \text{m}^2\text{/s}$ (A) $\text{1000 kg} \cdot \text{m}^2\text{/s}$ (A) $\text{2200 kg} \cdot \text{m}^2\text{/s}$