Found problems: 85335
1989 IMO Longlists, 12
Let $ P(x)$ be a polynomial such that the following inequalities are satisfied:
\[ P(0) > 0;\]\[ P(1) > P(0);\]\[ P(2) > 2P(1) \minus{} P(0);\]\[ P(3) > 3P(2) \minus{} 3P(1) \plus{} P(0);\]
and also for every natural number $ n,$ \[ P(n\plus{}4) > 4P(n\plus{}3) \minus{} 6P(n\plus{}2)\plus{}4P(n \plus{} 1) \minus{} P(n).\]
Prove that for every positive natural number $ n,$ $ P(n)$ is positive.
2024 Belarusian National Olympiad, 8.6
For each number $x$ we denote by $S(x)$ the sum of digits from its decimal representation. Find all positive integers $m$ for each of which there exists a positive integer $n$, such that $$S(n^2-2n+10)=m$$
[i]Chernov S.[/i]
2023 LMT Fall, 11
Find the number of degree $8$ polynomials $f (x)$ with nonnegative integer coefficients satisfying both $f (1) = 16$ and $f (-1) = 8$.
2014 Contests, 1.
Let $x, y$ be positive integers such that $\frac{x^2}{y}+\frac{y^2}{x}$ is an integer. Prove that $y|x^2$.
2010 China Team Selection Test, 2
Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$,
and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.
1981 Putnam, A4
A point $P$ moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it
meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection.
Let $N( t)$ be the number of starting directions from a fixed interior point $P_0$ for which $P$ escapes within $t$ units of time. Find the least constant $a$ for which constants $b$ and $c$ exist such that
$$N(t) \leq at^2 +bt+c$$
for all $t>0$ and all initial points $P_0 .$
2000 National Olympiad First Round, 36
$x_{n+1}= \left ( 1+\frac2n \right )x_n+\frac4n$, for every positive integer $n$. If $x_1=-1$, what is $x_{2000}$?
$ \textbf{(A)}\ 1999998
\qquad\textbf{(B)}\ 2000998
\qquad\textbf{(C)}\ 2009998
\qquad\textbf{(D)}\ 2000008
\qquad\textbf{(E)}\ 1999999
$
2019 Jozsef Wildt International Math Competition, W. 15
It is possible to partition the set $\{100, 101,\cdots , 1000\}$ into two subsets so that for any two distinct elements $x$ and $y$ belonging to the same subset $ \sqrt[3]{x + y}$ is irrational?
2013 Baltic Way, 13
All faces of a tetrahedron are right-angled triangles. It is known that three of its edges have the same length $s$. Find the volume of the tetrahedron.
1991 Vietnam Team Selection Test, 1
Let $T$ be an arbitrary tetrahedron satisfying the following conditions:
[b]I.[/b] Each its side has length not greater than 1,
[b]II.[/b] Each of its faces is a right triangle.
Let $s(T) = S^2_{ABC} + S^2_{BCD} + S^2_{CDA} + S^2_{DAB}$. Find the maximal possible value of $s(T)$.
2020 Harvard-MIT Mathematics Tournament, 9
Let $p > 5$ be a prime number. Show that there exists a prime number $q < p$ and a positive integer $n$ such that $p$ divides $n^2-q$.
[i]Proposed by Andrew Gu.[/i]
1993 Rioplatense Mathematical Olympiad, Level 3, 3
Given three points $A, B$ and $C$ (not collinear) construct the equilateral triangle of greater perimeter such that each of its sides passes through one of the given points.
1995 Tournament Of Towns, (480) 4
Along a track for cross-country skiing, $1000$ seats are placed in a row and numbered in order from $1$ to $1000$. By mistake, $n$ tickets were sold, $100 < n < 1000$, each with one of the numbers $1,2,..., 100$ printed on it. Also for each number $1,2,..., 100$ there exists at least one ticket with this number printed on it. Of course, there are tickets that have the same seat numbers. These $n$ spectators arrive one at a time.
Each goes to the seat shown on his ticket and occupies it if it is still empty. If not, he just says “Oh” and moves to the seat with the next number. This is repeated until he finds an empty seat and occupies it, saying “Oh” once for each occupied seat passed over but not at any other time. Prove that all the spectators will be seated and that the total number of the exclamations “Oh” that have been made before all the spectators are seated does not depend on the order in which the n spectators arrive, although it does depend on the distribution of numbers on the tickets.
(A Shen)
1981 Yugoslav Team Selection Test, Problem 1
Let $n\ge3$ be a natural number. For a set $S$ of $n$ real numbers, $A(S)$ denotes the set of all strictly increasing arithmetic sequences of three terms in $S$. At most, how many elements can the set $A(S)$ have?
2002 Iran Team Selection Test, 10
Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.
2012 Indonesia Juniors, day 2
p1. One day, a researcher placed two groups of species that were different, namely amoeba and bacteria in the same medium, each in a certain amount (in unit cells). The researcher observed that on the next day, which is the second day, it turns out that every cell species divide into two cells. On the same day every cell amoeba prey on exactly one bacterial cell. The next observation carried out every day shows the same pattern, that is, each cell species divides into two cells and then each cell amoeba prey on exactly one bacterial cell. Observation on day $100$ shows that after each species divides and then each amoeba cell preys on exactly one bacterial cell, it turns out kill bacteria. Determine the ratio of the number of amoeba to the number of bacteria on the first day.
p2. It is known that $n$ is a positive integer. Let $f(n)=\frac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}$.
Find $f(13) + f(14) + f(15) + ...+ f(112).$
p3. Budi arranges fourteen balls, each with a radius of $10$ cm. The first nine balls are placed on the table so that
form a square and touch each other. The next four balls placed on top of the first nine balls so that they touch each other. The fourteenth ball is placed on top of the four balls, so that it touches the four balls. If Bambang has fifty five balls each also has a radius of $10$ cm and all the balls are arranged following the pattern of the arrangement of the balls made by Budi, calculate the height of the center of the topmost ball is measured from the table surface in the arrangement of the balls done by Bambang.
p4. Given a triangle $ABC$ whose sides are $5$ cm, $ 8$ cm, and $\sqrt{41}$ cm. Find the maximum possible area of the rectangle can be made in the triangle $ABC$.
p5. There are $12$ people waiting in line to buy tickets to a show with the price of one ticket is $5,000.00$ Rp.. Known $5$ of them they only have $10,000$ Rp. in banknotes and the rest is only has a banknote of $5,000.00$ Rp. If the ticket seller initially only has $5,000.00$ Rp., what is the probability that the ticket seller have enough change to serve everyone according to their order in the queue?
2020 Moldova Team Selection Test, 5
Let $n$ be a natural number. Find all solutions $x$ of the system of equations $$\left\{\begin{matrix} sinx+cosx=\frac{\sqrt{n}}{2}\\tg\frac{x}{2}=\frac{\sqrt{n}-2}{3}\end{matrix}\right.$$ On interval $\left[0,\frac{\pi}{4}\right).$
2015 Online Math Open Problems, 18
Given an integer $n$, an integer $1 \le a \le n$ is called $n$-[i]well[/i] if \[ \left\lfloor\frac{n}{\left\lfloor n/a \right\rfloor}\right\rfloor = a. \] Let $f(n)$ be the number of $n$-well numbers, for each integer $n \ge 1$. Compute $f(1) + f(2) + \ldots + f(9999)$.
[i]Proposed by Ashwin Sah[/i]
2004 India Regional Mathematical Olympiad, 5
Let ABCD be a quadrilateral; X and Y be the midpoints of AC and BD respectively and lines through X and Y respectively parallel to BD, AC meet in O. Let P,Q,R,S be the midpoints of AB, BC, CD, DA respectively. Prove that
(A) APOS and APXS have the same area
(B) APOS, BQOP, CROQ, DSOR have the same area.
2024 CCA Math Bonanza, L5.4
Answer this question with a positive integer $1$ through $1000$. A positive integer ``answer" has been randomly selected from 1 to 1000, inclusive; if your selected integer is less than or equal to the ``answer", you will gain $\lfloor 20\left(\frac{x}{a}\right)^2 \rfloor$ points, where $x$ is your number and $a$ is the ``answer". If you select an integer greater than the ``answer", you will not gain any points.
[i]Lightning 5.4[/i]
2019 Singapore Junior Math Olympiad, 2
There are $315$ marbles divided into three piles of $81, 115$ and $119$. In each move Ah Meng can either merge several piles into a single pile or divide a pile with an even number of marbles into $2$ equal piles. Can Ah Meng divide the marbles into $315$ piles, each with a single marble?
2021 Thailand TST, 3
Let $P$ be a point on the circumcircle of acute triangle $ABC$. Let $D,E,F$ be the reflections of $P$ in the $A$-midline, $B$-midline, and $C$-midline. Let $\omega$ be the circumcircle of the triangle formed by the perpendicular bisectors of $AD, BE, CF$.
Show that the circumcircles of $\triangle ADP, \triangle BEP, \triangle CFP,$ and $\omega$ share a common point.
2022 Kyiv City MO Round 1, Problem 3
Let $AL$ be the inner bisector of triangle $ABC$. The circle centered at $B$ with radius $BL$ meets the ray $AL$ at points $L$ and $E$, and the circle centered at $C$ with radius $CL$ meets the ray $AL$ at points $L$ and $D$. Show that $AL^2 = AE\times AD$.
[i](Proposed by Mykola Moroz)[/i]
2001 ITAMO, 6
A panel contains $100$ light bulbs, arranged in a $10$ by $10$ square array. Some of them are on, the others are off.
The electrical system is such that when the switch corresponding to a light bulb is pressed, all the light bulbs that are on the same row or column of it (including the bulb linked to the pressed switch) change their state (that is they are turned on or off).
[list]
[*] From which starting configurations, pressing the right sequence of switches, is it possible to achieve that all bulbs are on at the same time?
[*] What is the answer to the previous question if the bulbs are $81$, arranged in a $9$ by $9$ panel?[/list]
2013 SEEMOUS, Problem 4
Let $A\in M_2(\mathbb Q)$ such that there is $n\in\mathbb N,n\ne0$, with $A^n=-I_2$. Prove that either $A^2=-I_2$ or $A^3=-I_2$.