Found problems: 85335
Evaluate the following: [url]http://internetolympiad.org/archive/2014/AprilFools/foreign_lang.txt[/url]
Find all natural numbers $n$ with the following property:
Given the decimal writing of $n$, adding a few digits one can obtain the decimal writing of $1996n$.
Let $m\geq 4$ and $n\geq 4$. An integer is written on each cell of a $m \times n$ board. If each cell has a number equal to the arithmetic mean of some pair of numbers written on its neighbouring cells, determine the maximum amount of distinct numbers that the board may have.
Note: two neighbouring cells share a common side.
$ABC$ is an acute-angled triangle and $P$ a point inside or on the boundary. The feet of the perpendiculars from $P$ to $BC, CA, AB$ are $A', B', C'$ respectively. Show that if $ABC$ is equilateral, then $\frac{AC'+BA'+CB'}{PA'+PB'+PC'}$ is the same for all positions of $P$, but that for any other triangle it is not.
A Christmas tree must be erected inside a convex rectangular garden and attached to the posts at the corners of the garden with four ropes running at the same height from the ground. At what point should the Christmas tree be placed, so that the sum of the lengths of these four cords is as small as possible?
Prime $p$ is called [i]Prime of the Year[/i] if there exists a positive integer $n$ such that $n^2+ 1 \equiv 0$ ($mod p^{2007}$).
Prove that there are infi nite number of [i]Primes of the Year[/i].
Let $p$ be an odd prime. Show that $\frac{1}{\pi}\cdot\cos^{-1}\left(\frac{1}{p}\right)$ is irrational. (Note: $\cos^{-1}\left(x\right)$ is defined to be the unique $y$ with $0\leq y\leq\pi$ such that $\cos\left(y\right)=x$.)
Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.
The numbers $1, 2, \ldots, n$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number remains. Prove that this number is at least $\frac{4}{9}n^3$.
[i]Brian Hamrick.[/i]
Show that $\varphi(2n)\mid n!$ for all positive integer $n$.
Given that $x\ge0$, $y\ge0$, $x+2y\le6$, and $2x+y\le6$, compute the maximum possible value of $x+y$.
Find the least possible value of the following expression
$\lfloor \frac{a+b}{c+d}\rfloor +\lfloor \frac{a+c}{b+d}\rfloor +\lfloor \frac{a+d}{b+c}\rfloor + \lfloor \frac{c+d}{a+b}\rfloor +\lfloor \frac{b+d}{a+c}\rfloor +\lfloor \frac{b+c}{a+d}\rfloor$
where $a$, $b$, $c$ and $d$ are positive real numbers.
Let $A$ be a nonempty set with $n$ elements. Find the number of ways of choosing a pair of subsets $(B,C)$ of $A$ such that $B$ is a nonempty subset of $C$.
Let $x,y$ be real numbers such that $xy=1$. Let $T$ and $t$ be the largest and smallest values of the expression \\
$\hspace{2cm} \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2}$\\.
\\
If $T+t$ can be expressed in the form $\frac{m}{n}$ where $m,n$ are nonzero integers with $GCD(m,n)=1$, find the value of $m+n$.
A traveler arrived to an island where 50 natives lived. All the natives stood in a circle and each announced firstly the age of his left neighbour, then the age of his right neighbour. Each native is either a knight who told both numbers correctly or a knave who increased one of the numbers by 1 and decreased the other by 1 (on his choice). Is it always possible after that to establish which of the natives are knights and which are knaves?
[i]Alexandr Gribalko[/i]
Let $n$ a positive integer. In a $2n\times 2n$ board, $1\times n$ and $n\times 1$ pieces are arranged without overlap.
Call an arrangement [b]maximal[/b] if it is impossible to put a new piece in the board without overlapping the previous ones.
Find the least $k$ such that there is a [b]maximal[/b] arrangement that uses $k$ pieces.
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$.
Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$.
[i]Ivan Chan Kai Chin, Malaysia[/i]
Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$?
[asy]
unitsize(8);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((9,0)--(10,0)--(10,3)--(9,3)--cycle);
draw((8,1)--(11,1)--(11,2)--(8,2)--cycle);
draw((19,0)--(20,0)--(20,5)--(19,5)--cycle);
draw((18,1)--(21,1)--(21,4)--(18,4)--cycle);
draw((17,2)--(22,2)--(22,3)--(17,3)--cycle);
draw((32,0)--(33,0)--(33,7)--(32,7)--cycle);
draw((29,3)--(36,3)--(36,4)--(29,4)--cycle);
draw((31,1)--(34,1)--(34,6)--(31,6)--cycle);
draw((30,2)--(35,2)--(35,5)--(30,5)--cycle);
label("Figure",(0.5,-1),S);
label("$0$",(0.5,-2.5),S);
label("Figure",(9.5,-1),S);
label("$1$",(9.5,-2.5),S);
label("Figure",(19.5,-1),S);
label("$2$",(19.5,-2.5),S);
label("Figure",(32.5,-1),S);
label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$
Let $ ABCDEF$ be a convex hexagon that has no pair of parallel sides. It is known that, for every point $ P$ inside the hexagon, the sum: \[ \text{Area}[ABP]\plus{}\text{Area}[CDP]\plus{}\text{Area}[EFP]\] has a constant value. Prove that the triangles $ ACE$ and $ BDF$ have the same barycentre.
_____________________________________
This problem was proposed by Israel Diaz.
$ Tipe$
A natural number of five digits is called [i]Ecuadorian [/i]if it satisfies the following conditions:
$\bullet$ All its digits are different.
$\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$, but $54210$ is not since $5 \ne 4 + 2 + 1 + 0$.
Find how many Ecuadorian numbers exist.
A triangle is constructed on each side of a convex polygon in a manner that the third vertex of each triangle is the meet point of bisectors of the angles adjacent to this side. Prove that these triangles cover all the polygon.
Egor Bakaev
Let $a^3 - a- 1 = 0$. Find the exact value of the expression $$\sqrt[3]{3a^2-4a} + a\sqrt[4]{2a^2+3a+2}.$$
Two circles $\omega_1$ and $\omega_2$ are tangent externally at the point $P$. Through the point $A$ of the circle $\omega_1$ is drawn a tangent to this circle, which intersects the circle $\omega_2$ at points $B$ and $C$ (see figure). Line $CP$ intersects again the circle $\omega_1$ to $D$. Prove that the $PA$ is a bisector of the angle $DPB$.
[img]https://1.bp.blogspot.com/-nmKZGdBXfao/XOd51gRFuyI/AAAAAAAAKO0/EYo2SCW0eGcJsF64-Avo6w73ugkIIQ30ACK4BGAYYCw/s1600/Yasinsky%2B2019%2Bp2.png[/img]
Sofia and Viktor are playing the following game on a $2022 \times 2022$ board:
- Firstly, Sofia covers the table completely by dominoes, no two are overlapping and all are inside the table;
- Then Viktor without seeing the table, chooses a positive integer $n$;
- After that Viktor looks at the table covered with dominoes, chooses and fixes $n$ of them;
- Finally, Sofia removes the remaining dominoes that aren't fixed and tries to recover the table with dominoes differently from before.
If she achieves that, she wins, otherwise Viktor wins. What is the minimum number $n$ for which Viktor can always win, no matter the starting covering of dominoes.
[i]Proposed by Viktor Simjanoski[/i]
Let $A$ be the set of all permutations $a = (a_1, a_2, \ldots, a_{2003})$ of the 2003 first positive integers such that each permutation satisfies the condition: there is no proper subset $S$ of the set $\{1, 2, \ldots, 2003\}$ such that $\{a_k | k \in S\} = S.$
For each $a = (a_1, a_2, \ldots, a_{2003}) \in A$, let $d(a) = \sum^{2003}_{k=1} \left(a_k - k \right)^2.$
[b]I.[/b] Find the least value of $d(a)$. Denote this least value by $d_0$.
[b]II.[/b] Find all permutations $a \in A$ such that $d(a) = d_0$.