Found problems: 85335
2001 Moldova Team Selection Test, 12
Let $n{}$ $(n\geq 1)$ be an integer and a set $A=\{1,2,\ldots,n\}$. The set $A{}$ is $k-partitionable$ if it can be partitioned in $k{}$ disjoint sets with the same sum of elements. Show that $A{}$ is $k-partitionable$ if and only if $2k$ divides $n(n+1)$ and $2k\leq n+1$.
2022 Princeton University Math Competition, A3 / B5
Randy has a deck of $29$ distinct cards. He chooses one of the $29!$ permutations of the deck and then repeatedly rearranges the deck using that permutation until the deck returns to its original order for the first time. What is the maximum number of times Randy may need to rearrange the deck?
2017 Lusophon Mathematical Olympiad, 6
Let ABC be a scalene triangle. Consider points D, E, F on segments AB, BC, CA, respectively, such that $\overline{AF}$=$\overline{DF}$ and $\overline{BE}$=$\overline{DE}$.
Show that the circumcenter of ABC lies on the circumcircle of CEF.
2017 Tournament Of Towns, 5
There is a set of control weights, each of them weighs a non-integer number of grams. Any
integer weight from $1$ g to $40$ g can be balanced by some of these weights (the control
weights are on one balance pan, and the measured weight on the other pan).What is the
least possible number of the control weights?
[i](Alexandr Shapovalov)[/i]
1994 Tournament Of Towns, (399) 1
Construct a convex quadrilateral given the lengths of all its sides and the length of the segment between the midpoints of its diagonals.
(Folklore)
1962 IMO, 4
Solve the equation $\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1$
1989 Dutch Mathematical Olympiad, 4
Given is a regular $n$-sided pyramid with top $T$ and base $A_1A_2A_3... A_n$. The line perpendicular to the ground plane through a point $B$ of the ground plane within $A_1A_2A_3... A_n$ intersects the plane $TA_1A_2$ at $C_1$, the plane $TA_2A_3$ at $C_2$, and so on, and finally the plane $TA_nA_1$ at $C_n$. Prove that $BC_1 + BC_2 + ... + BC_n$ is independent of choice of $B$'s.
KoMaL A Problems 2022/2023, A.836
For every \(i \in \mathbb{N}\) let \(A_i\), \(B_i\) and \(C_i\) be three finite and pairwise disjoint subsets of \(\mathbb{N}\). Suppose that for every pairwise disjoint sets \(A\), \(B\) and \( C\) with union \(\mathbb N\) there exists \(i\in \mathbb{N}\) such that \(A_i \subset A\), \(B_i \subset B\) and \(C_i \subset C\). Prove that there also exists a finite \(S\subset \mathbb{N}\) such that for every pairwise disjoint sets \(A\), \(B\) and \(C\) with union $\mathbb N$ there exists \(i\in S\) such that \(A_i \subset A\), \(B_i \subset B\) and \(C_i \subset C\).
[i]Submitted by András Imolay, Budapest[/i]
2008 AIME Problems, 10
Let $ ABCD$ be an isosceles trapezoid with $ \overline{AD}\parallel{}\overline{BC}$ whose angle at the longer base $ \overline{AD}$ is $ \dfrac{\pi}{3}$. The diagonals have length $ 10\sqrt {21}$, and point $ E$ is at distances $ 10\sqrt {7}$ and $ 30\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \overline{AD}$. The distance $ EF$ can be expressed in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
MOAA Team Rounds, 2019.9
Jonathan finds all ordered triples $(a, b, c)$ of positive integers such that $abc = 720$. For each ordered triple, he writes their sum $a + b + c$ on the board. (Numbers may appear more than once.) What is the sum of all the numbers written on the board?
2017 IFYM, Sozopol, 5
$f: \mathbb{R} \rightarrow \mathbb{R}$ is a function such that for $\forall x,y\in \mathbb{R}$ the equation
$f(xy+x+y)=f(xy)+f(x)+f(y)$
is true. Prove that $f(x+y)=f(x)+f(y)$ for $\forall$ $x,y\in \mathbb{R}$.
1989 AMC 8, 8
$(2\times 3\times 4)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right) =$
$\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 26$
2021 Malaysia IMONST 1, 2
If $x +\frac{1}{x} = 5$, what is the value of $x^3 +\frac{1}{x^3} $ ?
2024 Romanian Master of Mathematics, 2
Consider an odd prime $p$ and a positive integer $N < 50p$. Let $a_1, a_2, \ldots , a_N$ be a list of positive integers less than $p$ such that any specific value occurs at most $\frac{51}{100}N$ times and $a_1 + a_2 + \cdots· + a_N$ is not divisible by $p$. Prove that there exists a permutation $b_1, b_2, \ldots , b_N$ of the $a_i$ such that, for all $k = 1, 2, \ldots , N$, the sum $b_1 + b_2 + \cdots + b_k$ is not divisible by $p$.
[i]Will Steinberg, United Kingdom[/i]
2022 AMC 12/AHSME, 8
What is the graph of $y^4+1=x^4+2y^2$ in the coordinate plane?
$ \textbf{(A)}\ \textbf{Two intersecting parabolas} \qquad
\textbf{(B)}\ \textbf{Two nonintersecting parabolas} \qquad
\textbf{(C)}\ \textbf{Two intersecting circles} \qquad
\textbf{(D)}\ \textbf{A circle and a hyperbola} \qquad
\textbf{(E)}\ \textbf{A circle and two parabolas}$
2009 QEDMO 6th, 9
For every natural $n$ let $\phi (n)$ be the number of coprime numbers $k \in \{1,2,...,n\}$. (Example: $\phi (12) = 4$, because among the numbers $1, 2, ..., 12$ there are only the$ 4$ numbers, $1, 5, 7$ and $11$ coprime to$12.$)
If $k$ is a natural number, then one defines $\phi^k (n)=\underbrace{\strut \phi (\phi ...(\phi (n)) ...)}_{(k \, times \phi)}$ (Example: $\phi^3 (n)=\phi (\phi (\phi (n))) $)
For every whole $n> 2$ let $c(n)$ be the smallest natural number $k$ with $\phi^k (n)= 2$.
Prove that $c (ab) = c (a) + c (b)$ for odd integers $a$ and $b$, both of which are greater than $2$, .
1990 IMO Longlists, 50
During the class interval, $n$ children sit in a circle and play the game described below. The teacher goes around the children clockwisely and hands out candies to them according to the following regulations: Select a child, give him a candy; and give the child next to the first child a candy too; then skip over one child and give next child a candy; then skip over two children; give the next child a candy; then skip over three children; give the next child a candy;...
Find the value of $n$ for which the teacher can ensure that every child get at least one candy eventually (maybe after many circles).
2020 Baltic Way, 8
Let $n$ be a given positive integer.
A restaurant offers a choice of $n$ starters, $n$ main dishes, $n$ desserts and $n$ wines.
A merry company dines at the restaurant, with each guest choosing a starter, a main dish, a dessert and a wine.
No two people place exactly the same order.
It turns out that there is no collection of $n$ guests such that their orders coincide in three of these aspects,
but in the fourth one they all differ. (For example, there are no $n$ people that order exactly the same three courses of food, but $n$ different wines.) What is the maximal number of guests?
2011 ELMO Shortlist, 7
Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$.
[i]Alex Zhu.[/i]
Novosibirsk Oral Geo Oly IX, 2019.1
The circle is inscribed in a triangle, inscribed in a semicircle. Find the marked angle $a$.
[img]https://cdn.artofproblemsolving.com/attachments/8/e/334c8662377155086e9211da3589145f460b52.png[/img]
1955 Moscow Mathematical Olympiad, 294
a) A square table with $49$ small squares is filled with numbers $1$ to $7$ so that in each row and in each column all numbers from $1$ to $7$ are present. Let the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , 7$ are present.
b) A square table with $n^2$ small squares is filled with numbers $1$ to $n$ so that in each row and in each column all numbers from $1$ to $n$ are present. Let $n$ be odd and the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , n$ are present.
2023 Belarusian National Olympiad, 8.3
In the triangle $ABC$ points $M$ and $N$ are the midpoints of sides $AC$ and $AB$ respectively. $I$ is the incenter of the triangle. It is known that the angle $MIC$ is a right angle.
Find the angle $NIB$.
2015 Romania Team Selection Test, 3
A Pythagorean triple is a solution of the equation $x^2 + y^2 = z^2$ in positive integers such that $x < y$. Given any non-negative integer $n$ , show that some positive integer appears in precisely $n$ distinct Pythagorean triples.
2024 Korea Winter Program Practice Test, Q5
For each positive integer $n>1$, if $n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$($p_i$ are pairwise different prime numbers and $\alpha_i$ are positive integers), define $f(n)$ as $\alpha_1+\alpha_2+\cdots+\alpha_k$. For $n=1$, let $f(1)=0$. Find all pairs of integer polynomials $P(x)$ and $Q(x)$ such that for any positive integer $m$, $f(P(m))=Q(f(m))$ holds.
2010 Contests, 3
Prove that there exists a set $S$ of lines in the three dimensional space satisfying the following conditions:
$i)$ For each point $P$ in the space, there exist a unique line of $S$ containing $P$.
$ii)$ There are no two lines of $S$ which are parallel.