Olympiad problems

2016 AMC 10, 22

For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have? $\textbf{(A) }110 \qquad \textbf{(B) } 191 \qquad \textbf{(C) } 261 \qquad \textbf{(D) } 325 \qquad \textbf{(E) } 425$

2007 China Team Selection Test, 2

Tags: modular arithmetic , algebra

Find all positive integers $ n$ such that there exists sequence consisting of $ 1$ and $ - 1: a_{1},a_{2},\cdots,a_{n}$ satisfying $ a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.$

1990 All Soviet Union Mathematical Olympiad, 512

Tags: midpoint , equal segments , equal angles , geometry , diagonal

The line joining the midpoints of two opposite sides of a convex quadrilateral makes equal angles with the diagonals. Show that the diagonals are equal.

1966 Miklós Schweitzer, 10

Tags: integration , probability and stats

For a real number $ x$ in the interval $ (0,1)$ with decimal representation \[ 0.a_1(x)a_2(x)...a_n(x)...,\] denote by $ n(x)$ the smallest nonnegative integer such that \[ \overline{a_{n(x)\plus{}1}a_{n(x)\plus{}2}a_{n(x)\plus{}3}a_{n(x)\plus{}4}}\equal{}1966 .\] Determine $ \int_0^1n(x)dx$. ($ \overline{abcd}$ denotes the decimal number with digits $ a,b,c,d .$) [i]A. Renyi[/i]

VMEO III 2006 Shortlist, G4

Tags: angle bisector , geometry , isogonal , isogonal conjugate

Let $ABC$ be a triangle with circumscribed and inscribed circles $(O)$ and $(I)$ respectively. $AA'$,$BB'$,$CC'$ are the bisectors of triangle $ABC$. Prove that $OI$ passes through the the isogonal conjugate of point $I$ with respect to triangle $A'B'C'$.

2006 Moldova National Olympiad, 10.6

Tags: circumcircle , geometric transformation , cyclic quadrilateral , analytic geometry , trigonometry , geometry , reflection

Let a triangle $ABC$ satisfy $AC = BC$; in other words, let $ABC$ be an isosceles triangle with base $AB$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Denote by $M$ the midpoint of the segment $AB$. Show that $\angle APM + \angle BPC = 180^{\circ}$.

1990 Chile National Olympiad, 2

Tags: number theory

Find all the odd naturals whose indicator is the same as $1990$. We clarify that, if a natural decomposes into prime factors in the form $\Pi_{j=1}^r p_j^{a_j}$, define the [i]indicator [/i] as : $\phi (n) = r\Pi_{j=1}^r p_j^{a_j-1} (p_j + 1)$. [hide=official wording for first sentence]Encuentre todos los naturales impares cuyo indicador es el mismo que el de 1990.[/hide]

2013 Today's Calculation Of Integral, 865

Tags: rectangle , calculus , geometric transformation , integration , analytic geometry , rotation , geometry

Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.

2004 Singapore MO Open, 2

Tags: number theory , diophantine , diophantine equation

Find the number of ordered pairs $(a, b)$ of integers, where $1 \le a, b \le 2004$, such that $x^2 + ax + b = 167 y$ has integer solutions in $x$ and $y$. Justify your answer.

2012 NIMO Problems, 1

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Dan the dog spots Cate the cat 50m away. At that instant, Cate begins running away from Dan at 6 m/s, and Dan begins running toward Cate at 8 m/s. Both of them accelerate instantaneously and run in straight lines. Compute the number of seconds it takes for Dan to reach Cate. [i]Proposed by Eugene Chen[/i]

2014 Canadian Mathematical Olympiad Qualification, 7

Tags: combinatorics , geometry , hexagon

A bug is standing at each of the vertices of a regular hexagon $ABCDEF$. At the same time each bug picks one of the vertices of the hexagon, which it is not currently in, and immediately starts moving towards that vertex. Each bug travels in a straight line from the vertex it was in originally to the vertex it picked. All bugs travel at the same speed and are of negligible size. Once a bug arrives at a vertex it picked, it stays there. In how many ways can the bugs move to the vertices so that no two bugs are ever in the same spot at the same time?

2016 BMT Spring, 2

Tags: geometry , octagon

Jennifer wants to do origami, and she has a square of side length $ 1$. However, she would prefer to use a regular octagon for her origami, so she decides to cut the four corners of the square to get a regular octagon. Once she does so, what will be the side length of the octagon Jennifer obtains?

2020 USMCA, 25

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Let $S = \{1, \cdots, 6\}$ and $\mathcal{P}$ be the set of all nonempty subsets of $S$. Let $N$ equal the number of functions $f:\mathcal P \to S$ such that if $A,B\in \mathcal P$ are disjoint, then $f(A)\neq f(B)$. Determine the number of positive integer divisors of $N$.

2001 District Olympiad, 2

Tags: number theory

Let $x,y,z\in \mathbb{R}^*$ such that $xy,yz,zx\in \mathbb{Q}$. a) Prove that $x^2+y^2+z^2$ is rational; b) If $x^3+y^3+z^3$ is rational, prove that $x,y,z$ are rational. [i]Marius Ghergu[/i]

2006 Germany Team Selection Test, 2

Tags: combinatorics , modular arithmetic , invariant

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

TNO 2024 Junior, 3

Tags: combinatorics

Antonia and Benjamin play the following game: First, Antonia writes an integer from 1 to 2024. Then, Benjamin writes a different integer from 1 to 2024. They alternate turns, each writing a new integer different from the ones previously written, until no more numbers are left. Each time Antonia writes a number, she gains a point for each digit '2' in the number and loses a point for each digit '5'. Benjamin, on the other hand, gains a point for each digit '5' in his number and loses a point for each digit '2'. Who can guarantee victory in this game?

2020 IberoAmerican, 4

Tags: number theory

Show that there exists a set $\mathcal{C}$ of $2020$ distinct, positive integers that satisfies simultaneously the following properties: $\bullet$ When one computes the greatest common divisor of each pair of elements of $\mathcal{C}$, one gets a list of numbers that are all distinct. $\bullet$ When one computes the least common multiple of each pair of elements of $\mathcal{C}$, one gets a list of numbers that are all distinct.

2019 Denmark MO - Mohr Contest, 3

Tags: number theory , divide

Seven positive integers are written on a piece of paper. No matter which five numbers one chooses, each of the remaining two numbers divides the sum of the five chosen numbers. How many distinct numbers can there be among the seven?

2008 National Olympiad First Round, 15

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Let the sequence $(a_n)$ be defined as $a_1=\frac 13$ and $a_{n+1}=\frac{a_n}{\sqrt{1+13a_n^2}}$ for every $n\geq 1$. If $a_k$ is the largest term of the sequence satisfying $a_k < \frac 1{50}$, what is $k$? $ \textbf{(A)}\ 194 \qquad\textbf{(B)}\ 193 \qquad\textbf{(C)}\ 192 \qquad\textbf{(D)}\ 191 \qquad\textbf{(E)}\ \text{None of the above} $

2011 Purple Comet Problems, 13

Tags: matrix , linear algebra

A $3$ by $3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.

2021 Purple Comet Problems, 27

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Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\tfrac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.

Ukrainian TYM Qualifying - geometry, 2013.6

Tags: geometry , locus

Given a circle $\omega$, on which marks the points $A,B,C$. Let $BF$ and $CE$ be the altitudes of the triangle $ABC$, $M$ be the midpoint of the side $AC$. Find a the locus of the intersection points of the lines $BF$ and E$M$ for all positions of point $A$ , as $A$ moves on $\omega$.

2004 Nicolae Coculescu, 2

Tags: algebra , floor function , equation

Let be a natural number $ n\ge 2. $ Find the real numbers $ a $ that satisfy the equation $$ \lfloor nx \rfloor =\sum_{k=1}^{n} \lfloor x+(k-1)a \rfloor , $$ for any real numbers $ x. $ [i]Marius Perianu[/i]

2016 Dutch IMO TST, 2

Tags: number theory , divisible , divisor , exponential

Determine all pairs $(a, b)$ of integers having the following property: there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.

2024 Canadian Open Math Challenge, A4

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Consider the sequence of consecutive even numbers starting from 0, arranged in a staggered format, where each row contains one more number than the previous row. The beginning of this arrangement is shown below: $0$ $2\; 4$ $6\;\underline{8}\;10$ $12\: 14\: 16\: 18$ $20\: 22 \: 24 \: 26\: 28 $ The number in the middle of the third row is 8. What is the number in the middle of the 101-st row?