Found problems: 85335
2023 Estonia Team Selection Test, 6
In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:
[list]
[*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller.
[*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter.
[/list]
We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.
TNO 2008 Junior, 10
A jeweler makes necklaces with round stones, four emeralds (green) and four rubies (red), arranged at equal distances from each other. One day, they decide to give away some necklaces. How many necklaces can they give away without the risk of two friends ending up with the same necklace?
(*Observation: The necklace is completely symmetrical except for the type of stone, meaning there is not a unique way to form it. Consider this while solving the problem.*)
2005 IMO Shortlist, 1
A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off.
[i]Proposed by Australia[/i]
2018 All-Russian Olympiad, 8
The board used for playing a game consists of the left and right parts. In each part there are several fields and there’re several segments connecting two fields from different parts (all the fields are connected.) Initially, there is a violet counter on a field in the left part, and a purple counter on a field in the right part. Lyosha and Pasha alternatively play their turn, starting from Pasha, by moving their chip (Lyosha-violet, and Pasha-purple) over a segment to other field that has no chip. It’s prohibited to repeat a position twice, i.e. can’t move to position that already been occupied by some earlier turns in the game. A player losses if he can’t make a move. Is there a board and an initial positions of counters that Pasha has a winning strategy?
2013 Sharygin Geometry Olympiad, 16
The incircle of triangle $ABC$ touches $BC$, $CA$, $AB$ at points $A_1$, $B_1$, $C_1$, respectively. The perpendicular from the incenter $I$ to the median from vertex $C$ meets the line $A_1B_1$ in point $K$. Prove that $CK$ is parallel to $AB$.
2017 Korea USCM, 7
Prove the following inequality holds if $\{a_n\}$ is a deceasing sequence of positive reals, and $0<\theta<\frac{\pi}{2}$.
$$\left|\sum_{n=1}^{2017} a_n \cos n\theta \right| \leq \frac{\pi a_1}{\theta}$$
1949-56 Chisinau City MO, 13
Factor the polynomial $(a+b+c)^3- a^3 -b^3 -c^3$
2020 Bulgaria Team Selection Test, 2
Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$
2004 Estonia National Olympiad, 2
Draw a line passing through a point $M$ on the angle bisector of the angle $\angle AOB$, that intersects $OA$ and $OB$ at points $K$ and $L$ respectively. Prove that the valus of the sum $\frac{1}{|OK|}+\frac{1}{|OL|}$ does not depend on the choice of the straight line passing through $M$, i.e. is defined by the size of the angle AOB and the selection of the point $M$ only.
1950 AMC 12/AHSME, 11
If in the formula $ C \equal{} \frac {en}{R\plus{}nr}$, $n$ is increased while $ e$, $R$ and $r$ are kept constant, then $C$:
$\textbf{(A)}\ \text{Increases} \qquad
\textbf{(B)}\ \text{Decreases} \qquad
\textbf{(C)}\ \text{Remains constant} \qquad
\textbf{(D)}\ \text{Increases and then decreases} \qquad\\
\textbf{(E)}\ \text{Decreases and then increases}$
Bangladesh Mathematical Olympiad 2020 Final, #9
You have 2020 piles of coins in front Of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins and so on. So, the 2020th pile contains 2020 coins. Guess a positive integer[b] k[/b], in which piles contain at least[b] k [/b]coins, take away exact[b] k[/b] coins from these piles. Find the [b]minimum number of turns[/b] you need to take way all of these coins?
1989 AIME Problems, 1
Compute $\sqrt{(31)(30)(29)(28)+1}$.
2017 ASDAN Math Tournament, 3
Line segment $AB$ has length $10$. A circle centered at $A$ has radius $5$, and a circle centered at $B$ has radius $5\sqrt{3}$. What is the area of the intersection of the two circles?
1961 Kurschak Competition, 3
Two circles centers $O$ and $O'$ are disjoint. $PP'$ is an outer tangent (with $P$ on the circle center O, and P' on the circle center $O'$). Similarly, $QQ'$ is an inner tangent (with $Q$ on the circle center $O$, and $Q'$ on the circle center $O'$). Show that the lines $PQ$ and $P'Q'$ meet on the line $OO'$.
[img]https://cdn.artofproblemsolving.com/attachments/b/d/bad305631571323a61b097f149a1bb6855cdc5.png[/img]
Estonia Open Senior - geometry, 2019.1.1
Juri and Mari play the following game. Juri starts by drawing a random triangle on a piece of paper. Mari then draws a line on the same paper that goes through the midpoint of one of the midsegments of the triangle. Then Juri adds another line that also goes through the midpoint of the same midsegment. These two lines divide the triangle into four pieces. Juri gets the piece with maximum area (or one of those with maximum area) and the piece with minimum area (or one of those with minimum area), while Mari gets the other two pieces. The player whose total area is bigger wins. Does either of the players have a winning strategy, and if so, who has it?
1997 IMO Shortlist, 14
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.
1949 Moscow Mathematical Olympiad, 158
a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$.
b) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$.
1996 Romania Team Selection Test, 15
Let $ S $ be a set of $ n $ concentric circles in the plane. Prove that if a function $ f: S\to S $ satisfies the property
\[ d( f(A),f(B)) \geq d(A,B) \] for all $ A,B \in S $, then $ d(f(A),f(B)) = d(A,B) $, where $ d $ is the euclidean distance function.
EMCC Speed Rounds, 2016
[i]20 problems for 25 minutes.[/i]
[b]p1.[/b] Compute the value of $2 + 20 + 201 + 2016$.
[b]p2.[/b] Gleb is making a doll, whose prototype is a cube with side length $5$ centimeters. If the density of the toy is $4$ grams per cubic centimeter, compute its mass in grams.
[b]p3.[/b] Find the sum of $20\%$ of $16$ and $16\%$ of $20$.
[b]p4.[/b] How many times does Akmal need to roll a standard six-sided die in order to guarantee that two of the rolled values sum to an even number?
[b]p5.[/b] During a period of one month, there are ten days without rain and twenty days without snow. What is the positive difference between the number of rainy days and the number of snowy days?
[b]p6.[/b] Joanna has a fully charged phone. After using it for $30$ minutes, she notices that $20$ percent of the battery has been consumed. Assuming a constant battery consumption rate, for how many additional minutes can she use the phone until $20$ percent of the battery remains?
[b]p7.[/b] In a square $ABCD$, points $P$, $Q$, $R$, and $S$ are chosen on sides $AB$, $BC$, $CD$, and $DA$ respectively, such that $AP = 2PB$, $BQ = 2QC$, $CR = 2RD$, and $DS = 2SA$. What fraction of square $ABCD$ is contained within square $PQRS$?
[b]p8.[/b] The sum of the reciprocals of two not necessarily distinct positive integers is $1$. Compute the sum of these two positive integers.
[b]p9.[/b] In a room of government officials, two-thirds of the men are standing and $8$ women are standing. There are twice as many standing men as standing women and twice as many women in total as men in total. Find the total number of government ocials in the room.
[b]p10.[/b] A string of lowercase English letters is called pseudo-Japanese if it begins with a consonant and alternates between consonants and vowels. (Here the letter "y" is considered neither a consonant nor vowel.) How many $4$-letter pseudo-Japanese strings are there?
[b]p11.[/b] In a wooden box, there are $2$ identical black balls, $2$ identical grey balls, and $1$ white ball. Yuka randomly draws two balls in succession without replacement. What is the probability that the first ball is strictly darker than the second one?
[b]p12.[/b] Compute the real number $x$ for which $(x + 1)^2 + (x + 2)^2 + (x + 3)^2 = (x + 4)^2 + (x + 5)^2 + (x + 6)^2$.
[b]p13.[/b] Let $ABC$ be an isosceles right triangle with $\angle C = 90^o$ and $AB = 2$. Let $D$, $E$, and $F$ be points outside $ABC$ in the same plane such that the triangles $DBC$, $AEC$, and $ABF$ are isosceles right triangles with hypotenuses $BC$, $AC$, and $AB$, respectively. Find the area of triangle $DEF$.
[b]p14.[/b] Salma is thinking of a six-digit positive integer $n$ divisible by $90$. If the sum of the digits of n is divisible by $5$, find $n$.
[b]p15.[/b] Kiady ate a total of $100$ bananas over five days. On the ($i + 1$)-th day ($1 \le i \le 4$), he ate i more bananas than he did on the $i$-th day. How many bananas did he eat on the fifth day?
[b]p16.[/b] In a unit equilateral triangle $ABC$; points $D$,$E$, and $F$ are chosen on sides $BC$, $CA$, and $AB$, respectively. If lines $DE$, $EF$, and $FD$ are perpendicular to $CA$, $AB$ and $BC$, respectively, compute the area of triangle $DEF$.
[b]p17.[/b] Carlos rolls three standard six-sided dice. What is the probability that the product of the three numbers on the top faces has units digit 5?
[b]p18.[/b] Find the positive integer $n$ for which $n^{n^n}= 3^{3^{82}}$.
[b]p19.[/b] John folds a rope in half five times then cuts the folded rope with four knife cuts, leaving five stacks of rope segments. How many pieces of rope does he now have?
[b]p20.[/b] An integer $n > 1$ is conglomerate if all positive integers less than n and relatively prime to $n$ are not composite. For example, $3$ is conglomerate since $1$ and $2$ are not composite. Find the sum of all conglomerate integers less than or equal to $200$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2019 CMIMC, 7
Suppose you start at $0$, a friend starts at $6$, and another friend starts at $8$ on the number line. Every second, the leftmost person moves left with probability $\tfrac14$, the middle person with probability $\tfrac13$, and the rightmost person with probability $\tfrac12$. If a person does not move left, they move right, and if two people are on the same spot, they are randomly assigned which one of the positions they are. Determine the expected time until you all meet in one point.
1985 Tournament Of Towns, (088) 4
A square is divided into $5$ rectangles in such a way that its $4$ vertices belong to $4$ of the rectangles , whose areas are equal , and the fifth rectangle has no points in common with the side of the square (see diagram) . Prove that the fifth rectangle is a square.
[img]https://3.bp.blogspot.com/-TQc1v_NODek/XWHHgmONboI/AAAAAAAAKi4/XES55OJS5jY9QpNmoURp4y80EkanNzmMwCK4BGAYYCw/s1600/TOT%2B1985%2BSpring%2BJ4.png[/img]
2016 Irish Math Olympiad, 9
Show that the number $a^3$ where $a=\frac{251}{ \frac{1}{\sqrt[3]{252}-5\sqrt[3]{2}}-10\sqrt[3]{63}}+\frac{1}{\frac{251}{\sqrt[3]{252}+5\sqrt[3]{2}}+10\sqrt[3]{63}}$
is an integer and find its value
2015 ASDAN Math Tournament, 9
Compute all pairs of nonzero real numbers $(x,y)$ such that
$$\frac{x}{x^2+y}+\frac{y}{x+y^2}=-1\qquad\text{and}\qquad\frac{1}{x}+\frac{1}{y}=1.$$
2003 AMC 10, 23
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $ 3$ rows of small congruent equilateral triangles, with $ 5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $ 2003$ small equilateral triangles?
[asy]unitsize(15mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
pair Ap=(0,0), Bp=(1,0), Cp=(2,0), Dp=(3,0), Gp=dir(60);
pair Fp=shift(Gp)*Bp, Ep=shift(Gp)*Cp;
pair Hp=shift(Gp)*Gp, Ip=shift(Gp)*Fp;
pair Jp=shift(Gp)*Hp;
pair[] points={Ap,Bp,Cp,Dp,Ep,Fp,Gp,Hp,Ip,Jp};
draw(Ap--Dp--Jp--cycle);
draw(Gp--Bp--Ip--Hp--Cp--Ep--cycle);
for(pair p : points)
{
fill(circle(p, 0.07),white);
}
pair[] Cn=new pair[5];
Cn[0]=centroid(Ap,Bp,Gp);
Cn[1]=centroid(Gp,Bp,Fp);
Cn[2]=centroid(Bp,Fp,Cp);
Cn[3]=centroid(Cp,Fp,Ep);
Cn[4]=centroid(Cp,Ep,Dp);
label("$1$",Cn[0]);
label("$2$",Cn[1]);
label("$3$",Cn[2]);
label("$4$",Cn[3]);
label("$5$",Cn[4]);
for (pair p : Cn)
{
draw(circle(p,0.1));
}[/asy]
$ \textbf{(A)}\ 1,\!004,\!004 \qquad
\textbf{(B)}\ 1,\!005,\!006 \qquad
\textbf{(C)}\ 1,\!507,\!509 \qquad
\textbf{(D)}\ 3,\!015,\!018 \qquad
\textbf{(E)}\ 6,\!021,\!018$
2011 IMO Shortlist, 6
Let $n$ be a positive integer, and let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word, consisting of just letters $a$ and/or $b$. Suppose that the minimal period $N$ of $W$ is greater than $2^n$.
A finite nonempty word $U$ is said to [i]appear[/i] in $W$ if there exist indices $k \leq \ell$ such that $U=x_k x_{k+1} \ldots x_{\ell}$. A finite word $U$ is called [i]ubiquitous[/i] if the four words $Ua$, $Ub$, $aU$, and $bU$ all appear in $W$. Prove that there are at least $n$ ubiquitous finite nonempty words.
[i]Proposed by Grigory Chelnokov, Russia[/i]