Found problems: 85335
2017 Korea - Final Round, 6
A room has $2017$ boxes in a circle. A set of boxes is [i]friendly[/i] if there are at least two boxes in the set, and for each boxes in the set, if we go clockwise starting from the box, we would pass either $0$ or odd number of boxes before encountering a new box in the set. $30$ students enter the room and picks a set of boxes so that the set is friendly, and each students puts a letter inside all of the boxes that he/she chose. If the set of the boxes which have $30$ letters inside is not friendly, show that there exists two students $A, B$ and boxes $a, b$ satisfying the following condition.
(i). $A$ chose $a$ but not $b$, and $B$ chose $b$ but not $a$.
(ii). Starting from $a$ and going clockwise to $b$, the number of boxes that we pass through, not including $a$ and $b$, is not an odd number, and none of $A$ or $B$ chose such boxes that we passed.
2005 International Zhautykov Olympiad, 1
The 40 unit squares of the 9 9-table (see below) are labeled. The horizontal or vertical row of 9 unit squares is good if it has more labeled unit squares than unlabeled ones. How many good (horizontal and vertical) rows totally could have the table?
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.
2015 IMO Shortlist, N8
For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define
\[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\]
That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity.
Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that
\[\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.\]
[i]Proposed by Rodrigo Sanches Angelo, Brazil[/i]
1970 IMO Longlists, 42
We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.
2011 AMC 8, 15
How many digits are in the product $4^5 \cdot 5^{10}$?
$ \textbf{(A)} 8 \qquad\textbf{(B)} 9 \qquad\textbf{(C)} 10 \qquad\textbf{(D)} 11 \qquad\textbf{(E)} 12 $
1987 Swedish Mathematical Competition, 6
A baker with access to a number of different spices bakes ten cakes. He uses more than half of the different kinds of spices in each cake, but no two of the combinations of spices are exactly the same. Show that there exist three spices $a,b,c$ such that every cake contains at least one of these.
2014 AMC 8, 13
If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?
$\textbf{(A) }n$ and $m$ are even $\qquad\textbf{(B) }n$ and $m$ are odd $\qquad\textbf{(C) }n+m$ is even $\qquad\textbf{(D) }n+m$ is odd $\qquad \textbf{(E) }$ none of these are impossible
2015 Thailand TSTST, 3
The circles $S_{1}$ and $S_{2}$ intersect at $M$ and $N$.Show that if vertices $A$ and $C$ of a rectangle $ABCD$ lie on $S_{1}$ while vertices $B$ and $D$ lie on $S_{2}$,then the intersection of the diagonals of the rectangle lies on the line $MN$.
2014 SEEMOUS, Problem 1
Let $n$ be a nonzero natural number and $f:\mathbb R\to\mathbb R\setminus\{0\}$ be a function such that $f(2014)=1-f(2013)$. Let $x_1,x_2,x_3,\ldots,x_n$ be real numbers not equal to each other. If
$$\begin{vmatrix}1+f(x_1)&f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&1+f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&f(x_2)&1+f(x_3)&\cdots&f(x_n)\\\vdots&\vdots&\vdots&\ddots&\vdots\\f(x_1)&f(x_2)&f(x_3)&\cdots&1+f(x_n)\end{vmatrix}=0,$$prove that $f$ is not continuous.
2022 Olimphíada, 3
Let $m$ and $n$ be positive integers. In Philand, the Kingdom of Olymphics, with $m$ cities, and the Kingdom of Mathematicians for Fun, with $n$ cities, fight a battle in rounds. Some cities in the country are connected by roads, so that it is possible to travel through all the cities via the roads. In each round of the battle, if all cities neighboring, that is, connected directly by a road, a city in one of the kingdoms are from the other kingdom, that city is conquered in the next round and switches to the other kingdom. Knowing that between the first and second round, at least one city is not conquered, show that at some point the battle must end, i.e., no city can be captured by another kingdom.
2001 JBMO ShortLists, 10
A triangle $ABC$ is inscribed in the circle $\mathcal{C}(O,R)$. Let $\alpha <1$ be the ratio of the radii of the circles tangent to $\mathcal{C}$, and both of the rays $(AB$ and $(AC$. The numbers $\beta <1$ and $\gamma <1$ are defined analogously. Prove that $\alpha + \beta + \gamma =1$.
2008 AMC 12/AHSME, 12
A function $ f$ has domain $ [0,2]$ and range $ [0,1]$. (The notation $ [a,b]$ denotes $ \{x: a\le x\le b\}$.) What are the domain and range, respectively, of the function $ g$ defined by $ g(x)\equal{}1\minus{}f(x\plus{}1)$?
$ \textbf{(A)}\ [\minus{}1,1],[\minus{}1,0] \qquad
\textbf{(B)}\ [\minus{}1,1],[0,1] \qquad
\textbf{(C)}\ [0,2],[\minus{}1,0] \qquad
\textbf{(D)}\ [1,3],[\minus{}1,0] \qquad
\textbf{(E)}\ [1,3],[0,1]$
2018 Online Math Open Problems, 6
Let $f(x)=x^2+x$ for all real $x$. There exist positive integers $m$ and $n$, and distinct nonzero real numbers $y$ and $z$, such that $f(y)=f(z)=m+\sqrt{n}$ and $f(\frac{1}{y})+f(\frac{1}{z})=\frac{1}{10}$. Compute $100m+n$.
[i]Proposed by Luke Robitaille[/i]
2015 Caucasus Mathematical Olympiad, 3
The workers laid a floor of size $n \times n$ with tiles of two types: $2 \times 2$ and $3 \times 1$.
It turned out that they were able to completely lay the floor in such a way that the same number of tiles of each type was used. Under what conditions could this happen?
(You can’t cut tiles and also put them on top of each other.)
2000 Kazakhstan National Olympiad, 6
For positive numbers $ a $, $ b $ and $ c $ satisfying the equality $ a + b + c = 1 $, prove the inequality
$$
\frac {a ^ 7 + b ^ 7} {a ^ 5 + b ^ 5} + \frac {b ^ 7 + c ^ 7} {b ^ 5 + c ^ 5} + \frac {c ^ 7 + a ^ 7} {c ^ 5 + a ^ 5} \geq \frac {1} {3}.
$$
2017 CMIMC Individual Finals, 2
Points $A$, $B$, and $C$ lie on a circle $\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\ell$ tangent to the incircle of $\triangle ABC$ at $T$ intersects $\Omega$ at points $X$ and $Y$. Suppose that $AB=30$, $BC=40$, and $XY=48$. Compute $TX\cdot TY$.
2022 Girls in Math at Yale, R6
[b]p16[/b] Madelyn is being paid $\$50$/hour to find useful [i]Non-Functional Trios[/i], where a Non-Functional Trio is defined as an ordered triple of distinct real numbers $(a, b, c)$, and a Non- Functional Trio is [i]useful [/i] if $(a, b)$, $(b, c)$, and $(c, a)$ are collinear in the Cartesian plane. Currently, she’s working on the case $a+b+c = 2022$. Find the number of useful Non-Functional Trios $(a, b, c)$ such that $a + b + c = 2022$.
[b]p17[/b] Let $p(x) = x^2 - k$, where $k$ is an integer strictly less than $250$. Find the largest possible value of k such that there exist distinct integers $a, b$ with $p(a) = b$ and $p(b) = a$.
[b]p18[/b] Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$ such that $AB = 13$, $BC = 14$, and $CA = 15$. $BH$ and $CH$ meet $\Gamma$ again at points $D$ and $E$, respectively, and $DE$ meets $AB$ and $AC$ at $F$ and $G$, respectively. The circumcircles of triangles $ABG$ and $ACF$ meet BC again at points $P$ and $Q$. If $PQ$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd (a, b) = 1$, find $a + b$.
1988 Romania Team Selection Test, 9
Prove that for all positive integers $n\geq 1$ the number $\prod^n_{k=1} k^{2k-n-1}$ is also an integer number.
[i]Laurentiu Panaitopol[/i].
1999 All-Russian Olympiad Regional Round, 11.5
Are there real numbers $a, b$ and $c$ such that for all real $x$ and $y$ the following inequality holds:
$$|x + a| + |x + y + b| + |y + c| > |x| + |x + y| + |y|?$$
2014 Federal Competition For Advanced Students, 1
Determine all real numbers $x$ and $y$ such that
$x^2 + x = y^3 - y$,
$y^2 + y = x^3 - x$
2015 IFYM, Sozopol, 4
In how many ways can $n$ rooks be placed on a $2n$ x $2n$ chessboard, so that they cover all the white fields?
2011 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle and let $BB_1,CC_1$ be respectively the bisectors of $\angle{B},\angle{C}$ with $B_1$ on $AC$ and $C_1$ on $AB$, Let $E,F$ be the feet of perpendiculars drawn from $A$ onto $BB_1,CC_1$ respectively. Suppose $D$ is the point at which the incircle of $ABC$ touches $AB$. Prove that $AD=EF$
2017 AMC 10, 7
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
$\textbf{(A)}\ 30 \%\qquad\textbf{(B)}\ 40 \%\qquad\textbf{(C)}\ 50 \%\qquad\textbf{(D)}\ 60 \%\qquad\textbf{(E)}\ 70 \%$
2003 Romania Team Selection Test, 13
A parliament has $n$ senators. The senators form 10 parties and 10 committees, such that any senator belongs to exactly one party and one committee. Find the least possible $n$ for which it is possible to label the parties and the committees with numbers from 1 to 10, such that there are at least 11 senators for which the numbers of the corresponding party and committee are equal.