Found problems: 85335
2007 Romania National Olympiad, 4
Let $ ABCD$ be a tetrahedron.Prove that if a point $ M$ in a space satisfies the relation:
\begin{align*} MA^2 + MB^2 + CD^2 &= MB^2 + MC^2 + DA^2 \\ &= MC^2 + MD^2 + AB^2 \\ &= MD^2 + MA^2 + BC^2 . \end{align*}
then it is found on the common perpendicular of the lines $ AC$ and $ BD$.
2000 Tournament Of Towns, 4
Among a set of $2N$ coins, all identical in appearance, $2N - 2$ are real and $2$ are fake. Any two real coins have the same weight . The fake coins have the same weight, which is different from the weight of a real coin. How can one divide the coins into two groups of equal total weight by using a balance at most $4$ times, if
(a) $N = 16$,
( b ) $N = 11$ ?
(A Shapovalov)
1997 All-Russian Olympiad Regional Round, 11.7
Are there convex $n$-gonal ($n \ge 4$) and triangular pyramids such that the four trihedral angles of the $n$-gonal pyramid are equal trihedral angles of a triangular pyramid?
[hide=original wording] Существуют ли выпуклая n-угольная (n>= 4) и треугольная пирамиды такие, что четыре трехгранных угла n-угольной пирамиды равны трехгранным углам треугольной пирамиды?[/hide]
2024 pOMA, 2
Marc has an $n\times n$ board, where $n\ge 3$ is an integer, and an unlimited supply of green and red apples. Marc wants to place some apples on the board, so that the following conditions hold.
[list]
[*] Every cell of the board has exactly one apple, be it red or green.
[*] All rows and columns of the board have at least one red apple.
[*] No two rows or columns have the same apple color sequence. Note that rows are read from left to right, and columns are read from top to bottom. Also note that we [b]do not[/b] allow a row and a column to have the same color sequence.
[/list]
Find, in terms of $n$, the minimal number of red apples that Marc needs in order to fill the board in this way.
2021 MOAA, 14
Evaluate
\[\left\lfloor\frac{1\times 5}{7}\right\rfloor + \left\lfloor\frac{2\times 5}{7}\right\rfloor + \left\lfloor\frac{3\times 5}{7}\right\rfloor+\cdots+\left\lfloor\frac{100\times 5}{7}\right\rfloor.\]
[i]Proposed by Nathan Xiong[/i]
2022 Taiwan TST Round 2, A
Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]
1979 IMO Longlists, 50
Let $m$ positive integers $a_1, \dots , a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \dots , b_n$ such that all sums of distinct $b_k$’s are distinct and all $a_i \ (i \leq m)$ occur among them.
IMSC 2023, 1
Find all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(1) \neq f(-1)$ and $$f(m+n)^2 \mid f(m)-f(n)$$ for all integers $m, n$.
[i]Proposed by Liam Baker, South Africa[/i]
2022 Iranian Geometry Olympiad, 2
An isosceles trapezoid $ABCD$ $(AB \parallel CD)$ is given. Points $E$ and $F$ lie on the
sides $BC$ and $AD$, and the points $M$ and $N$ lie on the segment $EF$ such that $DF = BE$ and
$FM = NE$. Let $K$ and $L$ be the foot of perpendicular lines from $M$ and $N$ to $AB$ and $CD$, respectively. Prove that $EKFL$ is a parallelogram.
[i]Proposed by Mahdi Etesamifard[/i]
2007 AIME Problems, 12
In isosceles triangle $ABC$, $A$ is located at the origin and $B$ is located at $(20, 0)$. Point $C$ is in the first quadrant with $AC = BC$ and $\angle BAC = 75^\circ$. If $\triangle ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive y-axis, the area of the region common to the original and the rotated triangle is in the form $p\sqrt{2}+q\sqrt{3}+r\sqrt{6}+s$ where $p$, $q$, $r$, $s$ are integers. Find $(p-q+r-s)/2$.
2022 Middle European Mathematical Olympiad, 7
Determine all functions $f : \mathbb {N} \rightarrow \mathbb {N}$ such that $f$ is increasing (not necessarily strictly) and the numbers $f(n)+n+1$ and $f(f(n))-f(n)$ are both perfect squares for every positive integer $n$.
MathLinks Contest 5th, 5.1
Find all real numbers $a > 1$ such that there exists an integer $k \ge 1$ such that the sequence $\{x_n\}_{n\ge 1}$ formed with the first $k$ digits of the number $\lfloor a^n\rfloor$ is periodical.
1995 AMC 12/AHSME, 13
The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.
\[\begin{tabular}{ccccccc}& 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0\\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}\]
$\textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 8 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ \text{More than } 10$
1996 All-Russian Olympiad, 2
Several hikers travel at fixed speeds along a straight road. It is known that over some period of time, the sum of their pairwise distances is monotonically decreasing. Show that there is a hiker, the sum of whose distances to the other hikers is monotonically decreasing over the same period.
[i]A. Shapovalov[/i]
2011 Math Hour Olympiad, 6-7
[u]Round 1[/u]
[b]p1.[/b] In a chemical lab there are three vials: one that can hold $1$ oz of fluid, another that can hold $2$ oz, and a third that can hold $3$ oz. The first is filled with grape juice, the second with sulfuric acid, and the third with water. There are also $3$ empty vials in the cupboard, also of sizes $1$ oz, $2$ oz, and $3$ oz. In order to save the world with grape-flavored acid, James Bond must make three full bottles, one of each size, filled with a mixture of all three liquids so that each bottle has the same ratio of juice to acid to water. How can he do this, considering he was silly enough not to bring any equipment?
[b]p2.[/b] Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the $12$th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer.
[b]p3.[/b] Aquaman has a barrel divided up into six sections, and he has placed a red herring in each. Aquaman can command any fish of his choice to either ‘jump counterclockwise to the next sector’ or ‘jump clockwise to the next sector.’ Using a sequence of exactly $30$ of these commands, can he relocate all the red herrings to one sector? If yes, show how. If no, explain why not.
[img]https://cdn.artofproblemsolving.com/attachments/0/f/956f64e346bae82dee5cbd1326b0d1789100f3.png[/img]
[b]p4.[/b] Is it possible to place $13$ integers around a circle so that the sum of any $3$ adjacent numbers is exactly $13$?
[b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.)
[u]Round 2[/u]
[b]p6.[/b] Eight students participated in a math competition. There were eight problems to solve. Each problem was solved by exactly five people. Show that there are two students who solved all eight problems between them.
[b]p7.[/b] There are $3n$ checkers of three different colors: $n$ red, $n$ green and $n$ blue. They were used to randomly fill a board with $3$ rows and $n$ columns so that each square of the board has one checker on it. Prove that it is possible to reshuffle the checkers within each row so that in each column there are checkers of all three colors. Moving checkers to a different row is not allowed.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
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.
2017 NZMOC Camp Selection Problems, 2
Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be the point on the line $AB$, distinct from $B$, such that $CG = CB$. Let $H$ be the point on the line $BC$, distinct from $B$, such that $AB = AH$. Prove that triangle $DGH$ is isosceles.
2001 Tournament Of Towns, 3
Let $n\ge3$ be an integer. Each row in an $(n-2)\times n$ array consists of the numbers 1,2,...,$n$ in some order, and the numbers in each column are all different. Prove that this array can be expanded into an $n\times n$ array such that each row and each column consists of the numbers 1,2,...,$n$.
2009 Romania Team Selection Test, 3
Let $ ABC$ be a non-isosceles triangle, in which $ X,Y,$ and $ Z$ are the tangency points of the incircle of center $ I$ with sides $ BC,CA$ and $ AB$ respectively. Denoting by $ O$ the circumcircle of $ \triangle{ABC}$, line $ OI$ meets $ BC$ at a point $ D.$ The perpendicular dropped from $ X$ to $ YZ$ intersects $ AD$ at $ E$. Prove that $ YZ$ is the perpendicular bisector of $ [EX]$.
2008 International Zhautykov Olympiad, 1
Points $ K,L,M,N$ are repectively the midpoints of sides $ AB,BC,CD,DA$ in a convex quadrliateral $ ABCD$.Line $ KM$ meets dioganals $ AC$ and $ BD$ at points $ P$ and $ Q$,respectively.Line $ LN$ meets dioganals $ AC$ and $ BD$ at points $ R$ and $ S$,respectively.
Prove that if $ AP\cdot PC\equal{}BQ\cdot QD$,then $ AR\cdot RC\equal{}BS\cdot SD$.
2020-IMOC, A1
$\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.}$ Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$\definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z)$$holds for all $a,b,c\in\mathbb{R}$.
[i]Proposed by [/i][b][color=#FFFF00]usjl[/color][/b].
[color=#B6D7A8]#1733[/color]
PEN A Problems, 117
Find the smallest positive integer $n$ such that \[2^{1989}\; \vert \; m^{n}-1\] for all odd positive integers $m>1$.
2017 USAMO, 1
Prove that there are infinitely many distinct pairs $(a, b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.
2024 Ukraine National Mathematical Olympiad, Problem 6
The points $A, B, C, D$ lie on the line $\ell$ in this order. The points $P$ and $Q$ are chosen on one side of the line $\ell$, and the point $R$ is chosen on the other side so that:
$$\angle APB = \angle CPD = \angle QBC = \angle QCB = \angle RAD = \angle RDA$$
Prove that the points $P, Q, R$ lie on the same line.
[i]Proposed by Mykhailo Shtandenko, Fedir Yudin[/i]
2021 Argentina National Olympiad, 2
Let $m$ be a positive integer for which there exists a positive integer $n$ such that the multiplication $mn$ is a perfect square and $m- n$ is prime. Find all $m$ for $1000\leq m \leq 2021.$