Found problems: 85335
1989 Brazil National Olympiad, 3
A function $f$, defined for the set of integers, is such that $f(x)=x-10$ if $x>100$ and $f(x)=f(f(x+11))$ if $x \leq 100$.
Determine, justifying your answer, the set of all possible values for $f$.
2020 Baltic Way, 5
Find all real numbers $x,y,z$ so that
\begin{align*}
x^2 y + y^2 z + z^2 &= 0 \\
z^3 + z^2 y + z y^3 + x^2 y &= \frac{1}{4}(x^4 + y^4).
\end{align*}
1993 APMO, 3
Let
\begin{eqnarray*} f(x) & = & a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \ \ \mbox{and} \\ g(x) & = & c_{n+1} x^{n+1} + c_n x^n + \cdots + c_0 \end{eqnarray*}
be non-zero polynomials with real coefficients such that $g(x) = (x+r)f(x)$ for some real number $r$. If $a = \max(|a_n|, \ldots, |a_0|)$ and $c = \max(|c_{n+1}|, \ldots, |c_0|)$, prove that $\frac{a}{c} \leq n+1$.
2016 Romanian Master of Mathematics Shortlist, C4
Prove that a $46$-element set of integers contains two distinct doubletons $\{u, v\}$ and $\{x,y\}$ such that $u + v \equiv x + y$ (mod $2016$).
2007 Macedonia National Olympiad, 1
Let $a, b, c$ be positive real numbers. Prove that \[1+\frac{3}{ab+bc+ca}\geq\frac{6}{a+b+c}.\]
2015 Peru IMO TST, 11
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ .
[i]Proposed by Serbia[/i]
2014 USAMTS Problems, 1:
The net of 20 triangles shown below can be folded to form a regular icosahedron. Inside each of the triangular faces, write a number from 1 to 20 with each number used exactly once. Any pair of numbers that are consecutive must be written on faces sharing an edge in the folded icosahedron, and additionally, 1 and 20 must also be on faces sharing an edge. Some numbers have been given to you. (No proof is necessary.)
[asy]
unitsize(1cm);
pair c(int a, int b){return (a-b/2,sqrt(3)*b/2);}
draw(c(0,0)--c(0,1)--c(-1,1)--c(1,3)--c(1,1)--c(2,2)--c(3,2)--c(4,3)--c(4,2)--c(3,1)--c(2,1)--c(2,-1)--c(1,-1)--c(1,-2)--c(0,-3)--c(0,-2)--c(-1,-2)--c(1,0)--cycle);
draw(c(0,0)--c(1,1)--c(0,1)--c(1,2)--c(0,2)--c(0,1),linetype("4 4"));
draw(c(4,2)--c(3,2)--c(3,1),linetype("4 4"));
draw(c(3,2)--c(1,0)--c(1,1)--c(2,1)--c(2,2),linetype("4 4"));
draw(c(1,-2)--c(0,-2)--c(0,-1)--c(1,-1)--c(1,0)--c(2,0)--c(0,-2),linetype("4 4"));
label("2",(c(0,2)+c(1,2))/2,S);
label("15",(c(1,1)+c(2,1))/2,S);
label("6",(c(0,1)+c(1,1))/2,N);
label("14",(c(0,0)+c(1,0))/2,N);[/asy]
Geometry Mathley 2011-12, 10.3
Let $ABC$ be a triangle inscribed in a circle $(O)$. d is the tangent at $A$ of $(O), P$ is an arbitrary point in the plane. $D,E, F$ are the projections of $P$ on $BC,CA,AB$. Let $DE,DF$ intersect the line $d$ at $M,N$ respectively. The circumcircle of triangle $DEF$ meets $CA,AB$ at $K,L$ distinct from $E, F$. Prove that $KN$ meets $LM$ at a point on the circumcircle of triangle $DEF$.
Trần Quang Hùng
2022 LMT Spring, 5
Find the sum $$\sum^{2020}_{n=1} \gcd (n^3 -2n^2 +2021,n^2 -3n +3).$$
1977 IMO Longlists, 36
Consider a sequence of numbers $(a_1, a_2, \ldots , a_{2^n}).$ Define the operation
\[S\biggl((a_1, a_2, \ldots , a_{2^n})\biggr) = (a_1a_2, a_2a_3, \ldots , a_{2^{n-1}a_{2^n}, a_{2^n}a_1).}\]
Prove that whatever the sequence $(a_1, a_2, \ldots , a_{2^n})$ is, with $a_i \in \{-1, 1\}$ for $i = 1, 2, \ldots , 2^n,$ after finitely many applications of the operation we get the sequence $(1, 1, \ldots, 1).$
Gheorghe Țițeica 2025, P4
Let $R$ be a ring. Let $x,y\in R$ such that $x^2=y^2=0$. Prove that if $x+y-xy$ is nilpotent, so is $xy$.
[i]Janez Šter[/i]
2011 All-Russian Olympiad, 2
On side $BC$ of parallelogram $ABCD$ ($A$ is acute) lies point $T$ so that triangle $ATD$ is an acute triangle. Let $O_1$, $O_2$, and $O_3$ be the circumcenters of triangles $ABT$, $DAT$, and $CDT$ respectively. Prove that the orthocenter of triangle $O_1O_2O_3$ lies on line $AD$.
2022 Junior Balkan Team Selection Tests - Romania, P2
Find the largest positive integer $n$ such that the following is true:
There exists $n$ distinct positive integers $x_1,~x_2,\dots,x_n$ such that whatever the numbers $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$ are, not all null, the number $n^3$ do not divide $\sum_{k=1}^n a_kx_k$.
Math Hour Olympiad, Grades 8-10, 2016
[u]Round 1[/u]
[b]p1.[/b] Alice and Bob compiled a list of movies that exactly one of them saw, then Cindy and Dale did the same. To their surprise, these two lists were identical. Prove that if Alice and Cindy list all movies that exactly one of them saw, this list will be identical to the one for Bob and Dale.
[b]p2.[/b] Several whole rounds of cheese were stored in a pantry. One night some rats sneaked in and consumed $10$ of the rounds, each rat eating an equal portion. Some were satisfied, but $7$ greedy rats returned the next night to finish the remaining rounds. Their portions on the second night happened to be half as large as on the first night. How many rounds of cheese were initially in the pantry?
[b]p3.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order.
Count the number of blueberries in the top pancake, and call that number N. Pick up the stack of the top N pancakes, and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack.
[b]p4.[/b] There are two lemonade stands along the $4$-mile-long circular road that surrounds Sour Lake. $100$ children live in houses along the road. Every day, each child buys a glass of lemonade from the stand that is closest to her house, as long as she does not have to walk more than one mile along the road to get there.
A stand's [u]advantage [/u] is the difference between the number of glasses it sells and the number of glasses its competitor sells. The stands are positioned such that neither stand can increase its advantage by moving to a new location, if the other stand stays still. What is the maximum number of kids who can't buy lemonade (because both stands are too far away)?
[b]p5.[/b] Merlin uses several spells to move around his $64$-room castle. When Merlin casts a spell in a room, he ends up in a different room of the castle. Where he ends up only depends on the room where he cast the spell and which spell he cast. The castle has the following magic property: if a sequence of spells brings Merlin from some room $A$ back to room $A$, then from any other room $B$ in the castle, that same sequence brings Merlin back to room $B$. Prove that there are two different rooms $X$ and $Y$ and a sequence of spells that both takes Merlin from $X$ to $Y$ and from $Y$ to $X$.
[u]Round 2[/u]
[b]p6.[/b] Captains Hook, Line, and Sinker are deciding where to hide their treasure. It is currently buried at the $X$ in the map below, near the lairs of the three pirates. Each pirate would prefer that the treasure be located as close to his own lair as possible. You are allowed to propose a new location for the treasure to the pirates. If at least two out of the three pirates prefer the new location (because it moves closer to their own lairs), then the treasure will be moved there. Assuming the pirates’ lairs form an acute triangle, is it always possible to propose a sequence of new locations so that the treasure eventually ends up in your backyard (wherever that is)?
[img]https://cdn.artofproblemsolving.com/attachments/c/c/a9e65624d97dec612ef06f8b30be5540cfc362.png[/img]
[b]p7.[/b] Homer went on a Donut Diet for the month of May ($31$ days). He ate at least one donut every day of the month. However, over any stretch of $7$ consecutive days, he did not eat more than $13$ donuts. Prove that there was some stretch of consecutive days over which Homer ate exactly $30$ donuts.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2008 Germany Team Selection Test, 2
Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$.
[i]Author: Christopher Bradley, United Kingdom [/i]
2023 Baltic Way, 6
Let $n$ be a positive integer. Each cell of an $n \times n$ table is coloured in one of $k$ colours where every colour is used at least once. Two different colours $A$ and $B$ are said to touch each other, if there exists a cell coloured in $A$ sharing a side with a cell coloured in $B$. The table is coloured in such a way that each colour touches at most $2$ other colours. What is the maximal value of $k$ in terms of $n$?
India EGMO 2021 TST, 4
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.
Proposed by United Kingdom
1994 Tournament Of Towns, (406) 4
Prove that among any $10$ entries of the table
$$0 \,\,\,\, 1 \,\,\,\, 2 \,\,\,\, 3 \,\,\,\, ... \,\,\,\, 9$$
$$9 \,\,\,\, 0 \,\,\,\, 1 \,\,\,\, 2 \,\,\,\, ... \,\,\,\, 8$$
$$8 \,\,\,\, 9 \,\,\,\, 0 \,\,\,\, 1 \,\,\,\, ... \,\,\,\, 7$$
$$1 \,\,\,\, 2 \,\,\,\, 3 \,\,\,\, 4 \,\,\,\, ... \,\,\,\, 0$$
standing in different rows and different columns, at least two are equal.
(A Savin)
1971 IMO Longlists, 55
Prove that the polynomial $x^4+\lambda x^3+\mu x^2+\nu x+1$ has no real roots if $\lambda, \mu , \nu $ are real numbers satisfying
\[|\lambda |+|\mu |+|\nu |\le \sqrt{2} \]
2025 Poland - First Round, 2
Let $ABCD$ be a rectangle inscribed in circle $\omega$ with center $O$. Line $l$ passes trough $O$ and intersects lines $BC$ and $AD$ at points $E$ and $F$ respectively. Points $K$ and $L$ are the intersection points of $l$ and $\omega$ and points $K, E, F, L$ lie in this order on the line $l$. Lines tangent to $w$ in $K$ and $L$ intersect $CD$ at $M$ and $N$ respectively. Prove that $E, F, M, N$ lie on a common circle.
2020 Memorial "Aleksandar Blazhevski-Cane", 2
One positive integer is written in each $1 \times 1$ square of the $m \times n$ board. The following operations are allowed :
(1) In an arbitrarily selected row of the board, all numbers should be reduced by $1$.
(2) In an arbitrarily selected column of the board, double all the numbers.
Is it always possible, after a final number of steps, for all the numbers written on the board to be equal to $-1$?
(Explain the answer.)
2011 Saudi Arabia Pre-TST, 1.1
Let $a, b, c$ be positive real numbers. Prove that $$8(a+b+c) \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \right) \le 9 \left(1+\frac{a}{b} \right)\left(1+\frac{b}{c} \right)\left(1+\frac{c}{a} \right)$$
1966 IMO Shortlist, 19
Construct a triangle given the radii of the excircles.
1991 AMC 12/AHSME, 15
A circular table has exactly 60 chairs around it. There are $N$ people seated at this table in such a way that the next person to be seated must sit next to someone. The smallest possible value of $N$ is
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 58 $
2010 239 Open Mathematical Olympiad, 2
The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $K$ and $L$, respectively. the $B$-excircle touches the side $AC$ of this triangle at point $P$. Line $KL$ intersects with the line passing through $A$ and parallel to $BC$ at point $M$. Prove that $PL = PM$.