Found problems: 85335
2008 Postal Coaching, 4
Find all real numbers$p, q$ for which the polynomial equation $P(x) = x^4 - \frac{8p^2}{q}x^3 + 4qx^2 - 3px + p^2 = 0$ has four positive roots.
2013 Junior Balkan Team Selection Tests - Romania, 1
Find all pairs of integers $(x,y)$ satisfying the following condition:
[i]each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$
[/i]
Tournament of Towns
2001 Turkey MO (2nd round), 1
Let $ABCD$ be a convex quadrilateral. The perpendicular bisectors of the sides $[AD]$ and $[BC]$ intersect at a point $P$ inside the quadrilateral and the perpendicular bisectors of the sides $[AB]$ and $[CD]$ also intersect at a point $Q$ inside the quadrilateral. Show that, if $\angle APD = \angle BPC$ then $\angle AQB = \angle CQD$
2019 Saudi Arabia BMO TST, 2
Let $I $be the incenter of triangle $ABC$and $J$ the excenter of the side $BC$: Let $M$ be the midpoint of $CB$ and $N$ the midpoint of arc $BAC$ of circle $(ABC)$. If $T$ is the symmetric of the point $N$ by the point $A$, prove that the quadrilateral $JMIT$ is cyclic
2018 Thailand TST, 1
Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed:
$$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$.
The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this.
Prove that Eduardo has a winning strategy.
[i]Proposed by Amine Natik, Morocco[/i]
2017 Harvard-MIT Mathematics Tournament, 4
Let $w = w_1 w_2 \dots w_n$ be a word. Define a [i]substring[/i] of $w$ to be a word of the form $w_i w_{i + 1} \dots w_{j - 1} w_j$, for some pair of positive integers $1 \le i \le j \le n$. Show that $w$ has at most $n$ distinct palindromic substrings.
For example, $aaaaa$ has $5$ distinct palindromic substrings, and $abcata$ has $5$ ($a$, $b$, $c$, $t$, $ata$).
2012 Federal Competition For Advanced Students, Part 2, 1
Determine the maximum value of $m$, such that the inequality
\[ (a^2+4(b^2+c^2))(b^2+4(a^2+c^2))(c^2+4(a^2+b^2)) \ge m \]
holds for every $a,b,c \in \mathbb{R} \setminus \{0\}$ with $\left|\frac{1}{a}\right|+\left|\frac{1}{b}\right|+\left|\frac{1}{c}\right|\le 3$.
When does equality occur?
2013 Stars Of Mathematics, 1
Let $\mathcal{F}$ be the family of bijective increasing functions $f\colon [0,1] \to [0,1]$, and let $a \in (0,1)$. Determine the best constants $m_a$ and $M_a$, such that for all $f \in \mathcal{F}$ we have
\[m_a \leq f(a) + f^{-1}(a) \leq M_a.\]
[i](Dan Schwarz)[/i]
2021 Canadian Mathematical Olympiad Qualification, 5
Alphonse and Beryl are playing a game. The game starts with two rectangles with integer side lengths. The players alternate turns, with Alphonse going first. On their turn, a player chooses one rectangle, and makes a cut parallel to a side, cutting the rectangle into two pieces, each of which has integer side lengths. The player then discards one of the three rectangles (either the one they did not cut, or one of the two pieces they cut) leaving two rectangles for the other player. A player loses if they cannot cut a rectangle.
Determine who wins each of the following games:
(a) The starting rectangles are $1 \times 2020$ and $2 \times 4040$.
(b) The starting rectangles are $100 \times 100$ and $100 \times 500$.
1980 VTRMC, 2
The sum of the first $n$ terms of the sequence $$1,1+2,1+2+2^2,\ldots,1+2+\cdots+2^{k-1},\ldots$$ is of the form $2^{n+R}+Sn^2+Tn+U$ for all $n>0.$ Find $R,S,T,$ and $U.$
2008 Postal Coaching, 1
Define a sequence $<x_n>$ by $x_0 = 0$ and $$\large x_n = \left\{
\begin{array}{ll}
x_{n-1} + \frac{3^r-1}{2} & if \,\,n = 3^{r-1}(3k + 1)\\
& \\
x_{n-1} - \frac{3^r+1}{2} & if \,\, n = 3^{r-1}(3k + 2)\\
\end{array}
\right. $$
where $k, r$ are integers. Prove that every integer occurs exactly once in the sequence.
Novosibirsk Oral Geo Oly VII, 2019.1
Lyuba, Tanya, Lena and Ira ran across a flat field. At some point it turned out that among the pairwise distances between them there are distances of $1, 2, 3, 4$ and $5$ meters, and there are no other distances. Give an example of how this could be.
2008 May Olympiad, 3
In numbers $1010... 101$ Ones and zeros alternate, if there are $n$ ones, there are $n -1$ zeros ($n \ge 2$ ).Determine the values of $n$ for which the number $1010... 101$, which has $n$ ones, is prime.
Istek Lyceum Math Olympiad 2016, 3
Let $n$, $m$ and $k$ be positive integers satisfying $(n-1)n(n+1)=m^k.$ Prove that $k=1.$
2010 Today's Calculation Of Integral, 569
In the coordinate plane, denote by $ S(a)$ the area of the region bounded by the line passing through the point $ (1,\ 2)$ with the slope $ a$ and the parabola $ y\equal{}x^2$. When $ a$ varies in the range of $ 0\leq a\leq 6$, find the value of $ a$ such that $ S(a)$ is minimized.
2003 AMC 10, 21
Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?
$ \textbf{(A)}\ 22 \qquad
\textbf{(B)}\ 25 \qquad
\textbf{(C)}\ 27 \qquad
\textbf{(D)}\ 28 \qquad
\textbf{(E)}\ 29$
MathLinks Contest 2nd, 7.3
A convex polygon $P$ can be partitioned into $27$ parallelograms. Prove that it can also be partitioned into $21$ parallelograms.
2004 Kazakhstan National Olympiad, 7
Prove that for any $a>0,b>0,c>0$ we have
$8a^2 b^2 c^2 \geq (a^2 + ab + ac - bc)(b^2 + ba + bc - ac)(c^2 + ca + cb - ab)$.
2010 Contests, 3
A teacher wants to divide the $2010$ questions she asked in the exams during the school year into three folders of $670$ questions and give each folder to a student who solved all $670$ questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations in which there are at most two students who did not solve any given question.
2014 BAMO, 2
There are $n$ holes in a circle. The holes are numbered $1,2,3$ and so on to $n$. In the beginning, there is a peg in every hole except for hole $1$. A peg can jump in either direction over one adjacent peg to an empty hole immediately on the other side. After a peg moves, the peg it jumped over is removed. The puzzle will be solved if all pegs disappear except for one. For example, if $n=4$ the puzzle can be solved in two jumps: peg $3$ jumps peg $4$ to hole $1$, then peg $2$ jumps the peg in $1$ to hole $4$. (See illustration below, in which black circles indicate pegs and white circles are holes.)
[center][img]http://i.imgur.com/4ggOa8m.png[/img][/center]
[list=a]
[*]Can the puzzle be solved for $n=5$?
[*]Can the puzzle be solved for $n=2014$?
[/list]
In each part (a) and (b) either describe a sequence of moves to solve the puzzle or explain why it is impossible to solve the puzzle.
2019 ELMO Problems, 2
Let $m, n \ge 2$ be integers. Carl is given $n$ marked points in the plane and wishes to mark their centroid.* He has no standard compass or straightedge. Instead, he has a device which, given marked points $A$ and $B$, marks the $m-1$ points that divide segment $\overline{AB}$ into $m$ congruent parts (but does not draw the segment).
For which pairs $(m,n)$ can Carl necessarily accomplish his task, regardless of which $n$ points he is given?
*Here, the [i]centroid[/i] of $n$ points with coordinates $(x_1, y_1), \dots, (x_n, y_n)$ is the point with coordinates $\left( \frac{x_1 + \dots + x_n}{n}, \frac{y_1 + \dots + y_n}{n}\right)$.
[i]Proposed by Holden Mui and Carl Schildkraut[/i]
2018 BMT Spring, Tie 1
Compute the least positive $x$ such that $25x - 6$ is divisible by $1001$.
2022 Balkan MO Shortlist, G3
Let $ABC$ a triangle and let $\omega$ be its circumcircle. Let $E{}$ be the midpoint of the minor arc $BC$ of $\omega$, and $M{}$ the midpoint of $BC$. Let $V$ be the other point of intersection of $AM$ with $\omega$, $F{}$ the point of intersection of $AE$ with $BC$, $X{}$ the other point of intersection of the circumcircle of $FEM$ with $\omega$, $X'$ the reflection of $V{}$ with respect to $M{}$, $A'{}$ the foot of the perpendicular from $A{}$ to $BC$ and $S{}$ the other point of intersection of $XA'$ with $\omega$. If $Z \in \omega$ with $Z\neq X$ is such that $AX = AZ$, then prove that $S, X'$ and $Z{}$ are collinear.
2009 China Western Mathematical Olympiad, 3
Let $H$ be the orthocenter of acute triangle $ABC$ and $D$ the midpoint of $BC$. A line through $H$ intersects $AB,AC$ at $F,E$ respectively, such that $AE=AF$. The ray $DH$ intersects the circumcircle of $\triangle ABC$ at $P$. Prove that $P,A,E,F$ are concyclic.
2014 PUMaC Combinatorics B, 3
What is the largest $n$ such that a square cannot be partitioned into $n$ smaller, non-overlapping squares?