Found problems: 85335
2019 China Team Selection Test, 2
Let $S$ be a set of positive integers, such that $n \in S$ if and only if $$\sum_{d|n,d<n,d \in S} d \le n$$
Find all positive integers $n=2^k \cdot p$ where $k$ is a non-negative integer and $p$ is an odd prime, such that $$\sum_{d|n,d<n,d \in S} d = n$$
2018 Junior Balkan Team Selection Tests - Moldova, 1
$a_1,a_2,...a_{2018}$ are positive numbers,and $a_{2018}^2+a_{2017}^2=a_{2016}^2-a_{2015}^2+a_{2014}^2-...+a_{2}^2-a_{1}^2.$ Prove that $A=a_1a_2...a_{2018}+2025$ is a difference of two squares
1998 Greece National Olympiad, 2
For a regular $n$-gon, let $M$ be the set of the lengths of the segments joining its vertices. Show that the sum of the squares of the elements of $M$ is greater than twice the area of the polygon.
2010 National Olympiad First Round, 12
How many integer quadruples $a,b,c,d$ are there such that $7$ divides $ab-cd$ where $0\leq a,b,c,d < 7$?
$ \textbf{(A)}\ 412
\qquad\textbf{(B)}\ 385
\qquad\textbf{(C)}\ 294
\qquad\textbf{(D)}\ 252
\qquad\textbf{(E)}\ \text{None}
$
2016 Iran Team Selection Test, 3
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
2021 Saudi Arabia BMO TST, 4
In the popular game of Minesweeper, some fields of an $a \times b$ board are marked with a mine and on all the remaining fields the number of adjacent fields that contain a mine is recorded. Two fields are considered adjacent if they share a common vertex. For which $k \in \{0, 1, 2, 3, 4, 5, 6, 7, 8\}$ is it possible for some $a$ and $b$ , $ab > 2021$, to create a board whose fields are covered in mines, except for $2021$ fields who are all marked with $k$?
2012 Belarus Team Selection Test, 1
For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$
[i]Proposed by Suhaimi Ramly, Malaysia[/i]
1992 IMO Longlists, 79
Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that
\[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\]
where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$
2009 AMC 10, 19
A particular $ 12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $ 1$, it mistakenly displays a $ 9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?
$ \textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac58\qquad \textbf{(C)}\ \frac34\qquad \textbf{(D)}\ \frac56\qquad \textbf{(E)}\ \frac {9}{10}$
2015 Belarus Team Selection Test, 1
A circle intersects a parabola at four distinct points. Let $M$ and $N$ be the midpoints of the arcs of the circle which are outside the parabola. Prove that the line $MN$ is perpendicular to the axis of the parabola.
I. Voronovich
1989 National High School Mathematics League, 15
For any positive integer $n$, $a_n>0$, and $\sum_{j=1}^{n}a_j^3=\left(\sum_{j=1}^{n}a_j\right)^2$. Prove that $a_n=n$
1998 May Olympiad, 3
Given a $4 \times 4$ grid board with each square painted a different color, you want to cut it into two pieces of equal area by making a single cut along the grid lines. In how many ways can it be done?
2012 Kazakhstan National Olympiad, 2
Let $ABCD$ be an inscribed quadrilateral, in which $\angle BAD<90$. On the rays $AB$ and $AD$ are selected points $K$ and $L$, respectively, such that$ KA = KD, LA = LB$. Let $N$ - the midpoint of $AC$.Prove that if $\angle BNC=\angle DNC $,so $\angle KNL=\angle BCD $
2016 Latvia Baltic Way TST, 1
$2016$ numbers written on the board: $\frac{1}{2016}, \frac{2}{2016}, \frac{3}{2016}, ..., \frac{2016}{2016}$. In one move, it is allowed to choose any two numbers $a$ and $b$ written on the board, delete them, and write the number $3ab - 2a - 2b + 2$ instead. Determine what number will remain written on the board after $2015$ moves.
MOAA Gunga Bowls, 2023.2
Harry wants to put $5$ identical blue books, $3$ identical red books, and $1$ white book on his bookshelf. If no two adjacent books may be the same color, how many distinct arrangements can Harry make?
[i]Proposed by Anthony Yang[/i]
Putnam 1939, B2
Evaluate $\int_{1}^{3} ( (x - 1)(3 - x) )^{\dfrac{-1}{2}} dx$ and $\int_{1}^{\infty} (e^{x+1} + e^{3-x})^{-1} dx.$
2017 Azerbaijan Junior National Olympiad, P4
A Rhombus and an Isosceles trapezoid that has same area is drawn in the same circle's outside. Compare their acute angles \\
(explain your answer)
2023 Girls in Mathematics Tournament, 2
Let $a,b,c$ real numbers such that $a^n+b^n= c^n$ for three positive integers consecutive of $n$. Prove that $abc= 0$
2021 MMATHS, 4
Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}, BC = 14,$ and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expressed as $a\sqrt{b} - c$ for positive integers $a,b,c$ with $c$ squarefree, find $a + b + c$.
[i]Proposed by Andrew Wu[/i]
2010 Today's Calculation Of Integral, 596
Find the minimum value of $\int_0^{\frac{\pi}{2}} |a\sin 2x-\cos ^ 2 x|dx\ (a>0).$
2009 Shimane University entrance exam/Medicine
1998 Tournament Of Towns, 3
On an $8 \times 8$ chessboard, $17$ cells are marked. Prove that one can always choose two cells among the marked ones so that a Knight will need at least three moves to go from one of the chosen cells to the other.
(R Zhenodarov)
2009 Brazil Team Selection Test, 3
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.
[i]Proposed by Mohsen Jamaali, Iran[/i]
2020 Saint Petersburg Mathematical Olympiad, 5.
The altitudes $BB_1$ and $CC_1$ of the acute triangle $\triangle ABC$ intersect at $H$. The circle centered at $O_b$ passes through points $A,C_1$, and the midpoint of $BH$. The circle centered at $O_c$ passes through $A,B_1$ and the midpoint of $CH$. Prove that $B_1 O_b +C_1O_c > \frac{BC}{4}$
2021 Regional Competition For Advanced Students, 3
The numbers $1, 2, ..., 2020$ and $2021$ are written on a blackboard. The following operation is executed:
Two numbers are chosen, both are erased and replaced by the absolute value of their difference.
This operation is repeated until there is only one number left on the blackboard.
(a) Show that $2021$ can be the final number on the blackboard.
(b) Show that $2020$ cannot be the final number on the blackboard.
(Karl Czakler)
2016 Dutch BxMO TST, 4
The Facebook group Olympiad training has at least five members. There is a certain integer $k$ with following property: [i]for each $k$-tuple of members there is at least one member of this $k$-tuple friends with each of the other $k - 1$.[/i]
(Friendship is mutual: if $A$ is friends with $B$, then also $B$ is friends with $A$.)
(a) Suppose $k = 4$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members?
(b) Suppose $k = 5$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members?