Found problems: 85335
2022 CMIMC, 14
Let a tree on $mn + 1$ vertices be $(m,n)$-nice if the following conditions hold:
[list]
[*] $m + 1$ colors are assigned to the nodes of the tree
[*] for the first $m$ colors, there will be exactly $n$ nodes of color $i$ $(1\le i \le m)$
[*] the root node of the tree will be the unique node of color $m+1$. \item the $(m,n)$-nice trees must also satisfy the condition that for any two non-root nodes $i, j$, if the color of $i$ equals the color of $j$, then the color of the parent of $i$ equals the color of the parent of $j$.
[*] Nodes of the same color are considered indistinguishable (swapping any two of them results in the same tree).
[/list]
Let $N(u,v,l)$ denote the number of $(u,v)$-nice trees with $l$ leaves. Note that $N(2,2,2) = 2, N(2,2,3) = 4, N(2,2,4) = 6$. Compute the remainder when $\sum_{l = 123}^{789} N(8,101,l)$ is divided by $101$.
Definition: Any rooted, ordered tree consists of some set of nodes, each of which has a (possibly empty) ordered list of children. Each node is the child of exactly one other node, with the exception of the root, which has not parent. There also cannot be any cycles of nodes which are all linearly children of each other.
[i]Proposed by Advait Nene[/i]
1976 Euclid, 1
Source: 1976 Euclid Part A Problem 1
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In the diagram, $ABCD$ and $EFGH$ are similar rectangles. $DK:KC=3:2$. Then rectangle $ABCD:$ rectangle $EFGH$ is equal to
[asy]draw((75,0)--(0,0)--(0,50)--(75,50)--(75,0)--(55,0)--(55,20)--(100,20)--(100,0)--cycle);
draw((55,5)--(60,5)--(60,0));
draw((75,5)--(80,5)--(80,0));
label("A",(0,50),NW);
label("B",(0,0),SW);
label("C",(75,0),SE);
label("D",(75,50),NE);
label("E",(55,20),NW);
label("F",(55,0),SW);
label("G",(100,0),SE);
label("H",(100,20),NE);
label("K",(75,20),NE);[/asy]
$\textbf{(A) } 3:2 \qquad \textbf{(B) } 9:4 \qquad \textbf{(C) } 5:2 \qquad \textbf{(D) } 25:4 \qquad \textbf{(E) } 6:2$
2011 China Team Selection Test, 2
Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.
2021 CMIMC, 7
Let $P$ and $Q$ be fixed points in the Euclidean plane. Consider another point $O_0$. Define $O_{i+1}$ as the center of the unique circle passing through $O_i$, $P$ and $Q$. (Assume that $O_i,P,Q$ are never collinear.) How many possible positions of $O_0$ satisfy that $O_{2021}=O_{0}$?
[i]Proposed by Fei Peng[/i]
1969 IMO Shortlist, 44
$(MON 5)$ Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation $x^2 - ax + b = 0$.
1986 AMC 8, 8
In the product $ B2 \times 7B\equal{}6396$, $ B$ is a digit. The value of $ B$ is
\[ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8
\]
2012 Sharygin Geometry Olympiad, 3
A paper square was bent by a line in such way that one vertex came to a side not containing this vertex. Three circles are inscribed into three obtained triangles (see Figure). Prove that one of their radii is equal to the sum of the two remaining ones.
(L.Steingarts)
2024 Azerbaijan JBMO TST, 3
There are $n$ blocks placed on the unit squares of a $n \times n$ chessboard such that there is exactly one block in each row and each column. Find the maximum value $k$, in terms of $n$, such that however the blocks are arranged, we can place $k$ rooks on the board without any two of them threatening each other.
(Two rooks are not threatening each other if there is a block lying between them.)
2016 PUMaC Geometry B, 7
In isosceles triangle $ABC$ with base $BC$, let $M$ be the midpoint of $BC$. Let $P$ be the intersection of the circumcircle of $\vartriangle ACM$ with the circle with center $B$ passing through $M$, such that $P \ne M$. If $\angle BPC = 135^o$, then $\frac{CP}{AP}$ can be written as $a +\sqrt{b}$ for positive integers $a$ and $b$, where $b$ is not divisible by the square of any prime. Find $a + b$.
2024 Belarusian National Olympiad, 10.7
Let's call a pair of positive integers $(k,n)$ [i]interesting[/i] if $n$ is composite and for every divisor $d<n$ of $n$ at least one of $d-k$ and $d+k$ is also a divisor of $n$
Find the number of interesting pairs $(k,n)$ with $k \leq 100$
[i]M. Karpuk[/i]
2021 Science ON Seniors, 3
Let $m,n\in \mathbb{Z}_{\ge 1}$ and a rectangular board $m\times n$ sliced by parallel lines to the rectangle's sides into $mn$ unit squares. At moment $t=0$, there is an ant inside every square, positioned exactly in its centre, such that it is oriented towards one of the rectangle's sides. Every second, all the ants move exactly a unit following their current orientation; however, if two ants meet at the centre of a unit square, both of them turn $180^o$ around (the turn happens instantly, without any loss of time) and the next second they continue their motion following their new orientation. If two ants meet at the midpoint of a side of a unit square, they just continue moving, without changing their orientation.\\ \\
Prove that, after finitely many seconds, some ant must fall off the table.\\ \\
[i](Oliver Hayman)[/i]
2024 Israel TST, P1
Triangle $ABC$ with $\angle BAC=60^\circ$ is given. The circumcircle of $ABC$ is $\Omega$, and the orthocenter of $ABC$ is $H$. Let $S$ denote the midpoint of the arc $BC$ of $\Omega$ which doesn't contain $A$. Point $P$ was chosen on $\Omega$ so that $\angle HPS=90^\circ$. Prove that there exists a circle that goes through $P$ and $S$ and is tangent to lines $AB$, $AC$.
2019 Simurgh, 1
Show that there exists a $10 \times 10$ table of distinct natural numbers such that if $R_i$ is equal to the multiplication of numbers of row $i$ and $S_i$ is equal to multiplication of numbers of column $i$, then numbers $R_1$, $R_2$, ... , $R_{10}$ make a nontrivial arithmetic sequence and numbers $S_1$, $S_2$, ... , $S_{10}$ also make a nontrivial arithmetic sequence.
(A nontrivial arithmetic sequence is an arithmetic sequence with common difference between terms not equal to $0$).
2004 Junior Balkan Team Selection Tests - Romania, 2
For each positive integer $n\leq 49$ we define the numbers $a_n = 3n+\sqrt{n^2-1}$ and $b_n=2(\sqrt{n^2+n}+\sqrt{n^2-n})$. Prove that there exist two integers $A,B$ such that
\[ \sqrt{a_1-b_1}+\sqrt{a_2-b_2} + \cdots + \sqrt{a_{49}-b_{49}} = A+B\sqrt2. \]
2021 Greece JBMO TST, 1
If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$
1992 Swedish Mathematical Competition, 1
Is $\frac{19^{92} - 91^{29}}{90}$ an integer?
Maryland University HSMC part II, 2012
[b]p1.[/b] (a) Suppose $101$ Dalmatians chase $2012$ squirrels. Each squirrel gets chased by at most one Dalmatian, and each Dalmatian chases at least one squirrel. Show that two Dalmatians chase the same number of squirrels.
(b) What is the largest number of Dalmatians that can chase $2012$ squirrels in a way that each Dalmatian chases at least one squirrel and no two Dalmatians chase the same number of squirrels?
[b]p2.[/b] Lucy and Linus play the following game. They start by putting the integers $1, 2, 3, ..., 2012$ in a hat. In each round of the game, Lucy and Linus each draw a number from the hat. If the two numbers are $a$ and $b$, they throw away these numbers and put the number $|a - b|$ back into the hat. After $2011$ rounds, there is only one number in the hat. If it is even, Lucy wins. If it is odd, Linus wins.
(a) Prove that there is a sequence of drawings that makes Lucy win.
(b) Prove that Lucy always wins.
[b]p3.[/b] Suppose $x$ is a positive real number and $x^{1990}$, $x^{2001}$, and $x^{2012}$ differ by integers. Prove that $x$ is an integer.
[b]p4.[/b] Suppose that each point in three-dimensional space is colored with one of five colors and suppose that each color is used at least once. Prove that there is some plane that contains at least four of the colors.
[b]p5.[/b] Two circles, $C_1$ and $C_2$, are tangent at point $A$, with $C_1$ lying inside $C_2$ (and $C_1 \ne C_2$). The line through their centers intersects $C_1$ at $B_1$ and $C_2$ at $B_2$. A line $L$ is drawn through $A$ and it intersects $C_1$ at $P_1$ (with $P_1 \ne A$) and intersects $C_2$ at $P_2$ (with $P_2 \ne A$). The perpendicular from $P_2$ to the line $B_1B_2$ intersects the line $B_1B_2$ at $F$. Prove that if the line $P_1F$ is tangent to $C_1$ then $F$ is the midpoint of the line segment $B_1B_2$.
[img]https://cdn.artofproblemsolving.com/attachments/9/e/4db59be9fa764d3e910a828ed3296907ca5657.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2007 Tournament Of Towns, 4
From the first 64 positive integers are chosen two subsets with 16 numbers in each. The first subset contains only odd numbers while the second one contains only even numbers. Total sums of both subsets are the same. Prove that among all the chosen numbers there are two whose sum equals 65.
[i](3 points)[/i]
2019 Caucasus Mathematical Olympiad, 7
15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible
to have equal numbers of apricots in all the boxes after $k$ moves.
2023 Peru MO (ONEM), 2
For each positive real number $x$, let $f(x)=\frac{x}{1+x}$ . Prove that if $a$, $b,$ $c$ are the sidelengths of a triangle, then $f(a)$, $f(b),$ $f(c)$ are sidelengths of a triangle.
2011 May Olympiad, 5
Determine for which natural numbers $n$ it is possible to completely cover a board of $ n \times n$, divided into $1 \times 1$ squares, with pieces like the one in the figure, without gaps or overlays and without leaving the board. Each of the pieces covers exactly six boxes.
Note: Parts can be rotated.
[img]https://cdn.artofproblemsolving.com/attachments/c/2/d87d234b7f9799da873bebec845c721e4567f9.png[/img]
Kyiv City MO Juniors Round2 2010+ geometry, 2022.9.4
Let $\omega$ denote the circumscribed circle of triangle $ABC$, $I$ be its incenter, and $K$ be any point on arc $AC$ of $\omega$ not containing $B$. Point $P$ is symmetric to $I$ with respect to point $K$. Point $T$ on arc $AC$ of $\omega$ containing point $B$ is such that $\angle KCT = \angle PCI$. Show that the bisectors of angles $AKC$ and $ATC$ meet on line $CI$.
[i](Proposed by Anton Trygub)[/i]
1992 Chile National Olympiad, 1
Determine all naturals $n$ such that $2^n + 5$ is a perfect square.
2003 JHMMC 8, 14
In rectangle $ABCD$, $AB = 7$ and $AC = 25$. What is its area?
2006 MOP Homework, 3
Let $P_{n}$ denote the number of paths in the coordinate plane traveling from $(0, 0)$ to $(n, 0)$ with three kinds of moves: [i]upstep[/i] $u = [1, 1]$, [i]downstep[/i] $d = [1,-1]$, and [i]flatstep[/i] $f = [1, 0]$ with the path always staying above the line $y = 0.$ Let $C_{n}= \frac{1}{n+1}\binom{2n}{n}$ be the $n^{th}$ Catalan number. Prove that
$P_{n}= \sum_{i = 0}^\infty \binom{n}{2i}C_{i}$ and $C_{n}= \sum_{i = 0}^{2n}(-1)^{i}\binom{2n}{i}P_{2n-i}.$
[hide="Solution to Part 1"]
Let a path string, $S_{k}$, denote a string of $u, d, f$ corresponding to upsteps, downsteps, and flatsteps of length $k$ which successfully travels from $(0, 0)$ to $(n, 0)$ without passing below $y = 0.$ Also, let each entry of a path string be a slot. Lastly, denote $u_{k}, d_{k}, f_{k}$ to be the number of upsteps, downsteps, and flatsteps, respectively, in $S_{k}.$
Note that in our situation, all such path strings are in the form $S_{n},$ so all our path strings have $n$ slots. Since the starting and ending $y$ values are the same, the number of upsteps must equal the number of downsteps.
Let us observe the case when there are $2k$ downsteps and upsteps totally. Thus, there are $\binom{n}{2k}$ ways to choose the slots in which the upsteps and the downsteps appear. Now, we must arrange the downsteps and upsteps in such a way that $d_{n}= u_{n}$ and a greater number of upsteps preceed downsteps, as the path is always above $y = 0$. Note that a bijection exists between this and the number of ways to binary bracket $k$ letters. The number of binary brackets of $k$ letters is just the $k^{th}$ Catalan number. We then place the flatsteps in the rest of the slots. Thus, there are a total of $\sum_{k = 0}^\infty \binom{n}{2k}C_{k}$ ways to get an $S_{n}.$
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