Olympiad problems

2010 Contests, 3

There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$. For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$. Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible). Sorry for my bad English.

2007 Iran Team Selection Test, 3

Tags: analytic geometry , graphing lines , slope , trigonometry , geometry , geometric transformation , reflection

Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence. a) Prove that $B_{n}$ does not depend on location of $P$. b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.

2015 IFYM, Sozopol, 1

Tags: parallelogram , inequalities , trigonometry , trig identities , law of cosines , geometry

Let ABCD be a convex quadrilateral such that $AB + CD = \sqrt{2}AC$ and $BC + DA = \sqrt{2}BD$. Prove that ABCD is a parallelogram.

1997 Poland - Second Round, 5

Tags: sum , probability , dice , remainder , number theory , combinatorics

We have thrown $k$ white dice and $m$ black dice. Find the probability that the remainder modulo $7$ of the sum of the numbers on the white dice is equal to the remainder modulo $7$ of the sum of the numbers on the black dice.

2021 Durer Math Competition Finals, 1

Tags: number theory , prime

Show that if the difference of two positive cube numbers is a positive prime, then this prime number has remainder $1$ after division by $6$.

2015 District Olympiad, 4

Tags: function , algebra

Find the functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation: $$ \gcd\left( x,f(y)\right)\cdot\text{lcm}\left(f(x), y\right) = \gcd (x,y)\cdot\text{lcm}\left( f(x), f(y)\right) ,\quad\forall x,y\in\mathbb{N} . $$

2017 Kosovo National Mathematical Olympiad, 4

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4. Find all triples of consecutive numbers ,whose sum of squares is equal to some fourdigit number with all four digits being equal.

1953 Moscow Mathematical Olympiad, 258

Tags: combinatorics , chessboard , infinite board

A knight stands on an infinite chess board. Find all places it can reach in exactly $2n$ moves.

1997 Tournament Of Towns, (558) 3

Tags: number theory , diophantine , diophantine equation

Prove that the equation $$xy(x -y) + yz(y-z) + zx(z-x) = 6$$ has infinitely many solutions in integers $x, y$ and $z$. (N Vassiliev)

2005 iTest, 36

Tags: linear algebra , matrix

Find the determinant of this matrix: $\begin{bmatrix} 2 & 2 & 2 & 2 & 2 & 2 \\ 4 & 2 & 2 & 2 & 2 & 2 \\ 4 & 4 & 2 & 2 & 2 & 2 \\ 4 & 4 & 4 & 2 & 2 & 2 \\ 4 & 4 & 4 & 4 & 2 & 2 \\ 4 & 4 & 4& 4 & 4 & 2 \end{bmatrix} $

2008 Balkan MO Shortlist, G8

Tags: geometric inequality , geometry , max , distance

Let $P$ be a point in the interior of a triangle $ABC$ and let $d_a,d_b,d_c$ be its distances to $BC,CA,AB$ respectively. Prove that max $(AP, BP, CP) \ge \sqrt{d_a^2+d_b^2+d_c^2}$

2022 JBMO Shortlist, A4

Tags: inequality , shortlist , algebra

Suppose that $a, b,$ and $c$ are positive real numbers such that $$a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ Find the largest possible value of the expression $$\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.$$

2020 Ukrainian Geometry Olympiad - April, 1

Tags: geometry , isosceles , angle bisector , perpendicular

In triangle $ABC$, bisectors are drawn $AA_1$ and $CC_1$. Prove that if the length of the perpendiculars drawn from the vertex $B$ on lines $AA1$ and $CC_1$ are equal, then $\vartriangle ABC$ is isosceles.

2009 Purple Comet Problems, 8

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Find the least positive integer that has exactly $20$ positive integer divisors.

2024 Mongolian Mathematical Olympiad, 3

Tags: combinatorics , number theory , complete residue system

A set $X$ consisting of $n$ positive integers is called $\textit{good}$ if the following condition holds: For any two different subsets of $X$, say $A$ and $B$, the number $s(A) - s(B)$ is not divisible by $2^n$. (Here, for a set $A$, $s(A)$ denotes the sum of the elements of $A$) Given $n$, find the number of good sets of size $n$, all of whose elements is strictly less than $2^n$.

2005 Bundeswettbewerb Mathematik, 4

Tags: invariant , induction , combinatorics

Prove that each finite set of integers can be arranged without intersection.

1978 IMO Longlists, 2

Tags: algebra , generating functions

If \[f(x) = (x + 2x^2 +\cdots+ nx^n)^2 = a_2x^2 + a_3x^3 + \cdots+ a_{2n}x^{2n},\] prove that \[a_{n+1} + a_{n+2} + \cdots + a_{2n} =\dbinom{n + 1}{2}\frac{5n^2 + 5n + 2}{12}\]

2023 Harvard-MIT Mathematics Tournament, 5

Tags: guts

If $a$ and $b$ are positive real numbers such that $a \cdot 2^b=8$ and $a^b=2,$ compute $a^{\log_2 a} 2^{b^2}.$

2015 India IMO Training Camp, 2

Tags: combinatorics

A $10$-digit number is called a $\textit{cute}$ number if its digits belong to the set $\{1,2,3\}$ and the difference of every pair of consecutive digits is $1$. a) Find the total number of cute numbers. b) Prove that the sum of all cute numbers is divisibel by $1408$.

1977 Putnam, A3

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Let $u,f$ and $g$ be functions, defined for all real numbers $x$, such that $$\frac{u(x+1)+u(x-1)}{2}=f(x) \text{ and } \frac{u(x+4)+u(x-4)}{2}=g(x).$$ Determine $u(x)$ in terms of $f$ and $g$.

2021-2022 OMMC, 3

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Parabolas $P_1, P_2$ share a focus at $(20,22)$ and their directrices are the $x$ and $y$ axes respectively. They intersect at two points $X,Y.$ Find $XY^2.$ [i]Proposed by Evan Chang[/i]

PEN A Problems, 46

Tags: divisibility theory

Let $a$ and $b$ be integers. Show that $a$ and $b$ have the same parity if and only if there exist integers $c$ and $d$ such that $a^2 +b^2 +c^2 +1 = d^2$.

1989 AMC 12/AHSME, 3

Tags: geometry , rectangle , perimeter

A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is 24, then the area of the original square is [asy] draw((0,0)--(9,0)--(9,9)--(0,9)--cycle); draw((3,0)--(3,9), dashed); draw((6,0)--(6,9), dashed);[/asy] $\text{(A)} \ 24 \qquad \text{(B)} \ 36 \qquad \text{(C)} \ 64 \qquad \text{(D)} \ 81 \qquad \text{(E)} \ 96$

1981 Austrian-Polish Competition, 4

Tags: combinatorics

Let $n \ge 3$ cells be arranged into a circle. Each cell can be occupied by $0$ or $1$. The following operation is admissible: Choose any cell $C$ occupied by a $1$, change it into a $0$ and simultaneously reverse the entries in the two cells adjacent to $C$ (so that $x,y$ become $1 - x$, $1 - y$). Initially, there is a $1$ in one cell and zeros elsewhere. For which values of $n$ is it possible to obtain zeros in all cells in a finite number of admissible steps?

2016 China Team Selection Test, 5

Tags: number theory , algebra

Does there exist two infinite positive integer sets $S,T$, such that any positive integer $n$ can be uniquely expressed in the form $$n=s_1t_1+s_2t_2+\ldots+s_kt_k$$ ,where $k$ is a positive integer dependent on $n$, $s_1<\ldots<s_k$ are elements of $S$, $t_1,\ldots, t_k$ are elements of $T$?