Found problems: 85335
2014 Finnish National High School Mathematics, 2
The center of the circumcircle of the acute triangle $ABC$ is $M$, and the circumcircle of $ABM$ meets $BC$ and $AC$ at $P$ and $Q$ ($P\ne B$). Show that the extension of the line segment $CM$ is perpendicular to $PQ$.
2023 LMT Fall, 3C
Determine the least integer $n$ such that for any set of $n$ lines in the 2D plane, there exists either a subset of $1001$ lines that are all parallel, or a subset of $1001$ lines that are pairwise nonparallel.
[i]Proposed by Samuel Wang[/i]
[hide=Solution][i]Solution.[/i] $\boxed{1000001}$
Since being parallel is a transitive property, we note that in order for this to not exist, there must exist at most $1001$ groups of lines, all pairwise intersecting, with each group containing at most $1001$ lines. Thus, $n = 1000^2 + 1 =
\boxed{1000001}$.[/hide]
2020 Online Math Open Problems, 26
The bivariate functions $f_0, f_1, f_2, f_3, \dots$ are sequentially defined by the relations $f_0(x,y) = 0$ and $f_{n+1}(x,y) = \bigl|x+|y+f_n(x,y)|\bigr|$ for all integers $n \geq 0$. For independently and randomly selected values $x_0, y_0 \in [-2, 2]$, let $p_n$ be the probability that $f_n(x_0, y_0) < 1$. Let $a,b,c,$ and $d$ be positive integers such that the limit of the sequence $p_1,p_3,p_5,p_7,\dots$ is $\frac{\pi^2+a}{b}$ and the limit of the sequence $p_0,p_2,p_4,p_6,p_8, \dots$ is $\frac{\pi^2+c}{d}$. Compute $1000a+100b+10c+d$.
[i]Proposed by Sean Li[/i]
2024 Belarusian National Olympiad, 8.2
Let $S$ be the set of all non-increasing sequences of numbers $a_1 \geq a_2 \geq \ldots \geq a_{101}$ such that $a_i \in \{ 0,1,\ldots ,101 \}$ for all $1 \leq i \leq 101$
For every sequence $s \in S$ let $$f(s)=\lceil \frac{a_1}{2} \rceil+\lfloor \frac{a_2}{2} \rfloor + \lceil \frac{a_3}{2} \rceil + \ldots + \lfloor \frac{a_{100}}{2} \rfloor + \lceil \frac{a_{101}}{2} \rceil$$
where $\lfloor x \rfloor$ is the greatest integer, not exceeding $x$, and $\lceil x \rceil$ is the least integer at least $x$.
Prove that the number of sequences $s \in S$ for which $f(s)$ is even is the same, as the number of sequences $s$ for which $f(s)$ is odd
[i]M. Zorka[/i]
STEMS 2023 Math Cat A, 2
Consider the set $S$ of permutations of $1, 2, \dots, 2022$ such that for all numbers $k$ in the
permutation, the number of numbers less than $k$ that follow $k$ is even.
For example, for $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$
If $|S| = (a!)^b$ where $a, b \in \mathbb{N}$, then find the product $ab$.
1985 IMO Longlists, 23
Let $\mathbb N = {1, 2, 3, . . .}$. For real $x, y$, set $S(x, y) = \{s | s = [nx+y], n \in \mathbb N\}$. Prove that if $r > 1$ is a rational number, there exist real numbers $u$ and $v$ such that
\[S(r, 0) \cap S(u, v) = \emptyset, S(r, 0) \cup S(u, v) = \mathbb N.\]
2024 USA IMO Team Selection Test, 6
Find all functions $f\colon\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$,
\[f(xf(y))+f(y)=f(x+y)+f(xy).\]
[i]Milan Haiman[/i]
2008 Greece JBMO TST, 3
Let $x_1,x_2,x_3,...,x_{102}$ be natural numbers such that $x_1<x_2<x_3<...<x_{102}<255$.
Prove that among the numbers $d_1=x_2-x_1, d_2=x_3-x_2, ..., d_{101}=x_{102}-x_{101}$ there are at least $26$ equal.
2012 Bogdan Stan, 4
Let be three real positive numbers $ \alpha ,\beta ,\gamma $ and let $ M,N $ be points on the sides $ AB,BC, $ respectively, of a triangle $ ABC, $ such that $ \frac{MA}{MB} =\frac{\alpha }{\beta } $ and $ \frac{NB}{NC} =\frac{\beta }{\gamma } . $ Also, let $ P $ be the intersection of $ CM $ with $ AN. $ Show that:
$$ \frac{1}{\alpha }\overrightarrow{PA} +\frac{1}{\beta }\overrightarrow{PB} +\frac{1}{\gamma }\overrightarrow{PC} =0 $$
1962 All-Soviet Union Olympiad, 4
Prove that there are no integers $a, b, c, d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$ and $2$ at $x=62$.
2009 Romania Team Selection Test, 1
We call Golomb ruler a ruler of length $l$, bearing $k+1\geq 2$ marks $0<a_1<\ldots <a_{k-1}<l$, such that the lengths that can be measured using marks on the ruler are consecutive integers starting with $1$, and each such length be measurable between just two of the gradations of the ruler. Find all Golomb rulers.
2013 ELMO Shortlist, 5
There is a $2012\times 2012$ grid with rows numbered $1,2,\dots 2012$ and columns numbered $1,2,\dots, 2012$, and we place some rectangular napkins on it such that the sides of the napkins all lie on grid lines. Each napkin has a positive integer thickness. (in micrometers!)
(a) Show that there exist $2012^2$ unique integers $a_{i,j}$ where $i,j \in [1,2012]$ such that for all $x,y\in [1,2012]$, the sum \[ \sum _{i=1}^{x} \sum_{j=1}^{y} a_{i,j} \] is equal to the sum of the thicknesses of all the napkins that cover the grid square in row $x$ and column $y$.
(b) Show that if we use at most $500,000$ napkins, at least half of the $a_{i,j}$ will be $0$.
[i]Proposed by Ray Li[/i]
2006 MOP Homework, 7
Let $A_{n,k}$ denote the set of lattice paths in the coordinate plane of upsteps $u=[1,1]$, downsteps $d=[1,-1]$, and flatsteps $f=[1,0]$ that contain $n$ steps, $k$ of which are slanted ($u$ or $d$). A sharp turn is a consecutive pair of $ud$ or $du$. Let $B_{n,k}$ denote the set of paths in $A_{n,k}$ with no upsteps among the first $k-1$ steps, and let $C_{n,k}$ denote the set of paths in $A_{n,k}$ with no sharps anywhere. For example, $fdu$ is in $B_{3,2}$ but not in $C_{3,2}$, while $ufd$ is in $C_{3,2}$ but not $B_{3,2}$. For $1 \le k \le n$, prove that the sets $B_{n,k}$ and $C_{n,k}$ contains the same number of elements.
2022 JHMT HS, 3
Triangle $WSE$ has side lengths $WS=13$, $SE=15$, and $WE=14$. Points $J$ and $H$ lie on $\overline{SE}$ such that $SJ=JH=HE=5$. Let the angle bisector of $\angle{WES}$ intersect $\overline{WH}$ and $\overline{WJ}$ at points $M$ and $T$, respectively. Find the area of quadrilateral $JHMT$.
2019 Junior Balkan Team Selection Tests - Romania, 3
Real numbers $a,b,c,d$ such that $|a|>1$ , $|b|>1$ , $|c|>1$ , $|d|>1$ and $ab(c+d)+dc(a+b)+a+b+c+d=0$ then prove that $\frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}+\frac{1}{d-1} >0$
PEN A Problems, 110
For each positive integer $n$, write the sum $\sum_{m=1}^n 1/m$ in the form $p_n/q_n$, where $p_n$ and $q_n$ are relatively prime positive integers. Determine all $n$ such that 5 does not divide $q_n$.
2015 Turkey MO (2nd round), 5
In a cyclic quadrilateral $ABCD$ whose largest interior angle is $D$, lines $BC$ and $AD$ intersect at point $E$, while lines $AB$ and $CD$ intersect at point $F$. A point $P$ is taken in the interior of quadrilateral $ABCD$ for which $\angle EPD=\angle FPD=\angle BAD$. $O$ is the circumcenter of quadrilateral $ABCD$. Line $FO$ intersects the lines $AD$, $EP$, $BC$ at $X$, $Q$, $Y$, respectively. If $\angle DQX = \angle CQY$, show that $\angle AEB=90^\circ$.
II Soros Olympiad 1995 - 96 (Russia), 11.4
Consider the graph of the function $y = (1 -x^2)^3$. Find the set of points $M(x,y)$ through which you can draw at least $6$ lines touching this graph.
2010 National Chemistry Olympiad, 16
Moist air is less dense than dry air at the same temperature and barometric pressure. Which is the best explanation for this observation?
$ \textbf{(A)}\hspace{.05in}\ce{H2O} \text{ is a polar molecule but } \ce{N2} \text{ and } \ce{O2} \text{ are not} \qquad$
$\textbf{(B)}\hspace{.05in} \ce{H2O} \text{has a higher boiling point than } \ce{N2} \text{or} \ce{O2}\qquad$
$\textbf{(C)}\hspace{.05in}\ce{H2O} \text{has a lower molar mass than} \ce{N2} \text{or} \ce{O2}\qquad$
$\textbf{(D)}\hspace{.05in}\ce{H2O} \text{has a higher heat capacity than} \ce{N2} \text{or} \ce{O2}\qquad$
2013 Philippine MO, 3
3. Let n be a positive integer. The numbers 1, 2, 3,....., 2n are randomly assigned to 2n distinct points on a circle. To each chord joining two of these points, a value is assigned equal to the absolute value of the difference between the assigned numbers at its endpoints.
Show that one can choose n pairwise non-intersecting chords such that the sum of the values assigned to them is $n^2$ .
2012 Poland - Second Round, 2
Let $ABC$ be a triangle with $\angle A=60^{\circ}$ and $AB\neq AC$, $I$-incenter, $O$-circumcenter. Prove that perpendicular bisector of $AI$, line $OI$ and line $BC$ have a common point.
LMT Speed Rounds, 2011.7
A triangle $ABC$ has side lengths $AB=8$ and $BC=10.$ Given that the altitude to side $BC$ has length $4,$ what is the length of the altitude to side $AB?$
2017 Yasinsky Geometry Olympiad, 6
Given a circle $\omega$ of radius $r$ and a point $A$, which is far from the center of the circle at a distance $d<r$. Find the geometric locus of vertices $C$ of all possible $ABCD$ rectangles, where points $B$ and $D$ lie on the circle $\omega$.
2022 MMATHS, 12
Let triangle $ABC$ with incenter $I$ satisfy $AB = 3$, $AC = 4$, and $BC = 5$. Suppose that $D$ and $E$ lie on $AB$ and $AC$, respectively, such that $D$, $I$, and $E$ are collinear and $DE \perp AI$. Points $P$ and $Q$ lie on side $BC$ such that $IP = BP$ and $IQ = CQ$, and lines $DP$ and $EQ$ meet at $S$. Compute $SI^2$.
2019 European Mathematical Cup, 2
Let $(x_n)_{n\in \mathbb{N}}$ be a sequence defined recursively such that $x_1=\sqrt{2}$ and
$$x_{n+1}=x_n+\frac{1}{x_n}\text{ for }n\in \mathbb{N}.$$
Prove that the following inequality holds:
$$\frac{x_1^2}{2x_1x_2-1}+\frac{x_2^2}{2x_2x_3-1}+\dotsc +\frac{x_{2018}^2}{2x_{2018}x_{2019}-1}+\frac{x_{2019}^2}{2x_{2019}x_{2020}-1}>\frac{2019^2}{x_{2019}^2+\frac{1}{x_{2019}^2}}.$$
[i]Proposed by Ivan Novak[/i]