Found problems: 85335
2017-IMOC, N3
Find all functions $f:\mathbb N\to\mathbb N_0$ such that for all $m,n\in\mathbb N$,
\begin{align*}
f(mn)&=f(m)f(n)\\
f(m+n)&=\min(f(m),f(n))\qquad\text{if }f(m)\ne f(n)\end{align*}
2009 JBMO TST - Macedonia, 4
In every $1\times1$ cell of a rectangle board a natural number is written. In one step it is allowed the numbers written in every cell of arbitrary chosen row, to be doubled, or the numbers written in the cells of the arbitrary chosen column to be decreased by 1. Will after final number of steps all the numbers on the board be $0$?
2017 Swedish Mathematical Competition, 1
Xenia and Yagve take turns in playing the following game: A coin is placed on the first box in a row of nine cells. At each turn the player may choose to move the coin forward one step, move the coin forward four steps, or move coin back two steps. For a move to be allowed, the coin must land on one of them of nine cells. The winner is one who gets to move the coin to the last ninth cell. Who wins, given that Xenia makes the first move, and both players play optimally?
2021 Grand Duchy of Lithuania, 1
Prove that for any polynomial $f(x)$ (with real coefficients) there exist polynomials $g(x)$ and $h(x)$ (with real coefficients) such that $f(x) = g(h(x)) - h(g(x))$.
2000 Korea Junior Math Olympiad, 1
For arbitrary natural number $a$, show that $\gcd(a^3+1, a^7+1)=a+1$.
2015 Singapore MO Open, 2
A boy lives in a small island in which there are three roads at every junction. He starts
from his home and walks along the roads. At each junction he would choose to turn
to the road on his right or left alternatively, i.e., his choices would be . . ., left, right,
left,... Prove that he will eventually return to his home.
2009 India Regional Mathematical Olympiad, 4
Find the sum of all 3-digit natural numbers which contain at least one odd digit and at least one even digit.
2018 Pan African, 5
Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd.
$$
Show that
$$
\frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12
$$
and that $-12$ is the maximum.
2023 Kyiv City MO Round 1, Problem 4
Let's call a pair of positive integers $\overline{a_1a_2\ldots a_k}$ and $\overline{b_1b_2\ldots b_k}$ $k$-similar if all digits $a_1, a_2, \ldots, a_k , b_1 , b_2, \ldots, b_k$ are distinct, and there exist distinct positive integers $m, n$, for which the following equality holds:
$$a_1^m + a_2^m + \ldots + a_k^m = b_1^n + b_2^n + \ldots + b_k^n$$
For which largest $k$ do there exist $k$-similar numbers?
[i]Proposed by Oleksiy Masalitin[/i]
2024 LMT Fall, 23
Define $\overline{a}$ of a positive integer $a$ to be the number $a$ with its digits reversed. For example, $\overline{31564} = 46513.$ Find the sum of all positive integers $n \leq 100$ such that $(\overline{n})^2=\overline{n^2}.$ (Note: For a number that ends with a zero, like 450, the reverse would exclude the zero, so $\overline{450}=54$).
2011 AMC 8, 11
The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?
[asy]
size(300);
real i;
defaultpen(linewidth(0.8));
draw((0,140)--origin--(220,0));
for(i=1;i<13;i=i+1) {
draw((0,10*i)--(220,10*i));
}
label("$0$",origin,W);
label("$20$",(0,20),W);
label("$40$",(0,40),W);
label("$60$",(0,60),W);
label("$80$",(0,80),W);
label("$100$",(0,100),W);
label("$120$",(0,120),W);
path MonD=(20,0)--(20,60)--(30,60)--(30,0)--cycle,MonL=(30,0)--(30,70)--(40,70)--(40,0)--cycle,TuesD=(60,0)--(60,90)--(70,90)--(70,0)--cycle,TuesL=(70,0)--(70,80)--(80,80)--(80,0)--cycle,WedD=(100,0)--(100,100)--(110,100)--(110,0)--cycle,WedL=(110,0)--(110,120)--(120,120)--(120,0)--cycle,ThurD=(140,0)--(140,80)--(150,80)--(150,0)--cycle,ThurL=(150,0)--(150,110)--(160,110)--(160,0)--cycle,FriD=(180,0)--(180,70)--(190,70)--(190,0)--cycle,FriL=(190,0)--(190,50)--(200,50)--(200,0)--cycle;
fill(MonD,grey);
fill(MonL,lightgrey);
fill(TuesD,grey);
fill(TuesL,lightgrey);
fill(WedD,grey);
fill(WedL,lightgrey);
fill(ThurD,grey);
fill(ThurL,lightgrey);
fill(FriD,grey);
fill(FriL,lightgrey);
draw(MonD^^MonL^^TuesD^^TuesL^^WedD^^WedL^^ThurD^^ThurL^^FriD^^FriL);
label("M",(30,-5),S);
label("Tu",(70,-5),S);
label("W",(110,-5),S);
label("Th",(150,-5),S);
label("F",(190,-5),S);
label("M",(-25,85),W);
label("I",(-27,75),W);
label("N",(-25,65),W);
label("U",(-25,55),W);
label("T",(-25,45),W);
label("E",(-25,35),W);
label("S",(-26,25),W);[/asy]
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 $
1993 Dutch Mathematical Olympiad, 4
Let $ C$ be a circle with center $ M$ in a plane $ V$, and $ P$ be a point not on the circle $ C$.
$ (a)$ If $ P$ is fixed, prove that $ AP^2\plus{}BP^2$ is a constant for every diameter $ AB$ of the circle $ C$.
$ (b)$ Let $ AB$ be a fixed diameter of $ C$ and $ P$ a point on a fixed sphere $ S$ not intersecting $ V$. Determine the points $ P$ on $ S$ that minimize $ AP^2\plus{}BP^2$.
1996 AIME Problems, 4
A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x.$
2012 Lusophon Mathematical Olympiad, 6
A quadrilateral $ABCD$ is inscribed in a circle of center $O$. It is known that the diagonals $AC$ and $BD$ are perpendicular. On each side we build semicircles, externally, as shown in the figure.
a) Show that the triangles $AOB$ and $COD$ have the equal areas.
b) If $AC=8$ cm and $BD= 6$ cm, determine the area of the shaded region.
2019 South East Mathematical Olympiad, 4
Let $X$ be a $5\times 5$ matrix with each entry be $0$ or $1$. Let $x_{i,j}$ be the $(i,j)$-th entry of $X$ ($i,j=1,2,\hdots,5$). Consider all the $24$ ordered sequence in the rows, columns and diagonals of $X$ in the following:
\begin{align*}
&(x_{i,1}, x_{i,2},\hdots,x_{i,5}),\ (x_{i,5},x_{i,4},\hdots,x_{i,1}),\ (i=1,2,\hdots,5) \\
&(x_{1,j}, x_{2,j},\hdots,x_{5,j}),\ (x_{5,j},x_{4,j},\hdots,x_{1,j}),\ (j=1,2,\hdots,5) \\
&(x_{1,1},x_{2,2},\hdots,x_{5,5}),\ (x_{5,5},x_{4,4},\hdots,x_{1,1}) \\
&(x_{1,5},x_{2,4},\hdots,x_{5,1}),\ (x_{5,1},x_{4,2},\hdots,x_{1,5})
\end{align*}
Suppose that all of the sequences are different. Find all the possible values of the sum of all entries in $X$.
2025 NCJMO, 5
Each element of set $\mathcal{S}$ is colored with multiple colors. A $\textit{rainbow}$ is a subset of $\mathcal{S}$ which has amongst its elements at least $1$ color from each element of $\mathcal{S}$. A $\textit{minimal rainbow}$ is a rainbow where removing any single element gives a non-rainbow.
Prove that the union of all minimal rainbows is $\mathcal{S}$.
[i]Grisham Paimagam[/i]
2021 BMT, 6
Three distinct integers are chosen uniformly at random from the set
$$\{2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030\}.$$
Compute the probability that their arithmetic mean is an integer.
2023 Kyiv City MO Round 1, Problem 2
You are given $n\geq 4$ positive real numbers. Consider all $\frac{n(n-1)}{2}$ pairwise sums of these numbers. Show that some two of these sums differ in at most $\sqrt[n-2]{2}$ times.
[i]Proposed by Anton Trygub[/i]
2010 Tournament Of Towns, 1
Alex has a piece of cheese. He chooses a positive number $a\neq 1$ and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio $1:a.$ His goal is to divide the cheese into two piles of equal masses. Can he do it?
2015 Balkan MO Shortlist, G2
Let $ABC$ be a triangle with circumcircle $\omega$ . Point $D$ lies on the arc $BC$ of $\omega$ and is different than $B,C$ and the midpoint of arc $BC$. Tangent of $\Gamma$ at $D$ intersects lines $BC$, $CA$, $AB$ at $A',B',C'$, respectively. Lines $BB'$ and $CC'$ intersect at $E$. Line $AA'$ intersects the circle $\omega$ again at $F$. Prove that points $D,E,F$ are collinear.
(Saudi Arabia)
1980 Bulgaria National Olympiad, Problem 5
Prove that the number of ways of choosing $6$ among the first $49$ positive integers, at least two of which are consecutive, is equal to $\binom{49}6-\binom{44}6$.
1982 IMO Longlists, 8
A box contains $p$ white balls and $q$ black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated until the last two balls are removed from the box and one last ball is put in. What is the probability that this last ball is white?
2025 Bulgarian Winter Tournament, 10.2
Let $D$ be an arbitrary point on the side $BC$ of the non-isosceles acute triangle $ABC$. The circle with center $D$ and radius $DA$ intersects the rays $AB^\to$ (after $B$) and $AC^\to$ (after $C$) at $M$ and $N$. Prove that the orthocenter of triangle $AMN$ lies on a fixed line, independent of the choice of $D$.
2021 Olimphíada, 3
Let $n$ be a positive integer. In the $\mathit{philand}$ language, words are all finite sequences formed by the letters "$P$", "$H$" and "$I$". Philipe, who speaks only the $\mathit{philand}$ language, writes the word $PHIPHI\ldots PHI$ on a piece of paper, where $PHI$ is repeated $n$ times. He can do the following operations:
• Erase two identical letters and write in their place two different letters from the original and from each other;
(Ex: $PP\rightarrow HI$)
• Erase two distinct letters and rewrite them changing the order in which they appear;
(Ex: $PI\rightarrow IP$)
• Erase two distinct letters and write the letter distinct from the two he erased.
(Ex: $PH\rightarrow I$)
Find the largest integer $C$ such that any Philandese word of up to $C$ letters can be written by Philip through the above operations.
Note: Operations are taken on adjacent letters.
1995 Iran MO (2nd round), 1
Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another.
(For example, $23=9+8+6.)$