Found problems: 85335
1953 Moscow Mathematical Olympiad, 250
Somebody wrote $1953$ digits on a circle. The $1953$-digit number obtained by reading these figures clockwise, beginning at a certain point, is divisible by $27$. Prove that if one begins reading the figures at any other place, one gets another $1953$-digit number also divisible by $27$.
1989 Putnam, B4
Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?
1988 Poland - Second Round, 5
Decide whether any rectangle that can be covered by 25 circles of radius 2 can also be covered by 100 circles of radius 1.
Russian TST 2019, P3
Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.
1992 Poland - Second Round, 5
Determine the upper limit of the volume of spheres contained in tetrahedra of all heights not longer than $ 1 $.
2019 District Olympiad, 3
Consider the rectangular parallelepiped $ABCDA'B'C'D' $ as such the measure of the dihedral angle formed by the planes $(A'BD)$ and $(C'BD)$ is $90^o$ and the measure of the dihedral angle formed by the planes $(AB'C)$ and $(D'B'C)$ is $60^o$. Determine and measure the dihedral angle formed by the planes $(BC'D)$ and $(A'C'D)$.
1957 AMC 12/AHSME, 7
The area of a circle inscribed in an equilateral triangle is $ 48\pi$. The perimeter of this triangle is:
$ \textbf{(A)}\ 72\sqrt{3} \qquad
\textbf{(B)}\ 48\sqrt{3}\qquad
\textbf{(C)}\ 36\qquad
\textbf{(D)}\ 24\qquad
\textbf{(E)}\ 72$
2010 Bundeswettbewerb Mathematik, 2
There are $9999$ rods with lengths $1, 2, ..., 9998, 9999$. The players Anja and Bernd alternately remove one of the sticks, with Anja starting. The game ends when there are only three bars left. If from those three bars, a not degenerate triangle can be constructed then Anja wins, otherwise Bernd.
Who has a winning strategy?
2004 District Olympiad, 1
Let $(x_n)_{n\ge 0}$ a sequence of real numbers defined by $x_0>0$ and $x_{n+1}=x_n+\frac{1}{\sqrt{x_n}}$. Compute $\lim_{n\to \infty}x_n$ and $\lim_{n\to \infty} \frac{x_n^3}{n^2}$.
1983 Bulgaria National Olympiad, Problem 1
Determine all natural numbers $n$ for which there exists a permutation $(a_1,a_2,\ldots,a_n)$ of the numbers $0,1,\ldots,n-1$ such that, if $b_i$ is the remainder of $a_1a_2\cdots a_i$ upon division by $n$ for $i=1,\ldots,n$, then $(b_1,b_2,\ldots,b_n)$ is also a permutation of $0,1,\ldots,n-1$.
2017 AMC 10, 5
The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$
2023 USAJMO Solutions by peace09, 5
A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.
After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
1990 Austrian-Polish Competition, 2
Find all solutions in positive integers to $a^A = b^B = c^C = 1990^{1990}abc$, where $A = b^c, B = c^a, C = a^b$.
2014 Contests, 3
Let $p$ be a fixed odd prime. A $p$-tuple $(a_1,a_2,a_3,\ldots,a_p)$ of integers is said to be [i]good[/i] if
[list]
[*] [b](i)[/b] $0\le a_i\le p-1$ for all $i$, and
[*] [b](ii)[/b] $a_1+a_2+a_3+\cdots+a_p$ is not divisible by $p$, and
[*] [b](iii)[/b] $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$ is divisible by $p$.[/list]
Determine the number of good $p$-tuples.
1976 IMO Longlists, 13
A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where $[x]$ denotes the smallest integer $\leq x)$
1989 China Team Selection Test, 4
$\forall n \in \mathbb{N}$, $P(n)$ denotes the number of the partition of $n$ as the sum of positive integers (disregarding the order of the parts), e.g. since $4 = 1+1+1+1 = 1+1+2 = 1+3 = 2+2 = 4$, so $P(4)=5$. "Dispersion" of a partition denotes the number of different parts in that partitation. And denote $q(n)$ is the sum of all the dispersions, e.g. $q(4)=1+2+2+1+1=7$. $n \geq 1$. Prove that
(1) $q(n) = 1 + \sum^{n-1}_{i=1} P(i).$
(2) $1 + \sum^{n-1}_{i=1} P(i) \leq \sqrt{2} \cdot n \cdot P(n)$.
2004 Flanders Junior Olympiad, 2
How can you go from the number 11 to 25 by only multiplying with 2 or decreasing with 3 in a minimum number of steps?
2014 Junior Regional Olympiad - FBH, 5
How many are there $4$ digit numbers such that they have two odd digits and two even digits
2011 Princeton University Math Competition, B1
If the plane is partitioned into a grid of congruent equilateral triangles, prove that there does not exist a square with vertices at the vertices of this grid.
2022 Brazil National Olympiad, 3
Let $ABC$ be a triangle with incenter $I$ and let $\Gamma$ be its circumcircle. Let $M$ be the midpoint of $BC$, $K$ the midpoint of the arc $BC$ which does not contain $A$, $L$ the midpoint of the arc $BC$ which contains $A$ and $J$ the reflection of $I$ by the line $KL$. The line $LJ$ intersects $\Gamma$ again at the point $T\neq L$. The line $TM$ intersects $\Gamma$ again at the point $S\neq T$. Prove that $S, I, M, K$ lie on the same circle.
2023 Silk Road, 1
Let $ABCD$ be a trapezoid with $AD\parallel BC$. A point $M $ is chosen inside the trapezoid, and a point $N$ is chosen inside the triangle $BMC$ such that $AM\parallel CN$, $BM\parallel DN$. Prove that triangles $ABN$ and $CDM$ have equal areas.
2005 China Team Selection Test, 2
Given positive integer $n (n \geq 2)$, find the largest positive integer $\lambda$ satisfying :
For $n$ bags, if every bag contains some balls whose weights are all integer powers of $2$ (the weights of balls in a bag may not be distinct), and the total weights of balls in every bag are equal, then there exists a weight among these balls such that the total number of balls with this weight is at least $\lambda$.
2019 Mathematical Talent Reward Programme, MCQ: P 4
Suppose $\triangle ABC$ is a triangle. From the vertex $A$ draw the altitude $AH$, angle bisector (of $\angle BAC$) $AP$, median $AD$ and these intersect the side $BC$ at the points (from left in order) $H$, $P$, $D$ respectively. Let $\angle CAH = \angle HAP = \angle PAD = \angle DAB$. Then $\angle ACH =$
[list=1]
[*] $22.5^{\circ}$
[*] $45^{\circ}$
[*] $67.5^{\circ}$
[*] None of the above
[/list]
2018 Vietnam National Olympiad, 3
An investor has two rectangular pieces of land of size $120\times 100$.
a. On the first land, she want to build a house with a rectangular base of size $25\times 35$ and nines circular flower pots with diameter $5$ outside the house. Prove that even the flower pots positions are chosen arbitrary on the land, the remaining land is still sufficient to build the desired house.
b. On the second land, she want to construct a polygonal fish pond such that the distance from an arbitrary point on the land, outside the pond, to the nearest pond edge is not over $5$. Prove that the perimeter of the pond is not smaller than $440-20\sqrt{2}$.
2023 Stanford Mathematics Tournament, 10
Suppose that $p(x),q(x)$ are monic polynomials with nonnegative integer coefficients such that
\[\frac{1}{5x}\ge\frac{1}{q(x)}-\frac{1}{p(x)}\ge\frac{1}{3x^2}\]
for all integers $x\ge2$. Compute the minimum possible value of $p(1)\cdot q(1)$.