Found problems: 85335
2022 CMIMC, 10
Adam places down cards one at a time from a standard 52 card deck (without replacement) in a pile. Each time he places a card, he gets points equal to the number of cards in a row immediately before his current card that are all the same suit as the current card. For instance, if there are currently two hearts on the top of the pile (and the third card in the pile is not hearts), then placing a heart would be worth 2 points, and placing a card of any other suit would be worth 0 points. What is the expected number of points Adam will have after placing all 52 cards?
[i]Proposed by Adam Bertelli[/i]
2000 Czech and Slovak Match, 2
Let ${ABC}$ be a triangle, ${k}$ its incircle and ${k_a,k_b,k_c}$ three circles orthogonal to ${k}$ passing through ${B}$ and ${C, A}$ and ${C}$ , and ${A}$ and ${B}$ respectively. The circles ${k_a,k_b}$ meet again in ${C'}$ ; in the same way we obtain the points ${B'}$ and ${A'}$ . Prove that the radius of the circumcircle of ${A'B'C'}$ is half the radius of ${k}$ .
2009 Moldova Team Selection Test, 1
Let $ m,n\in \mathbb{N}^*$. Find the least $ n$ for which exists $ m$, such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$, $ n \minus{} 1$ of length $ 2$, $ ...$, $ 1$ square of length $ n$. For the found value of $ n$ give the example of covering.
2010 Contests, 2
If, instead, the graph is a graph of VELOCITY vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)?
(A) at B
(B) at C
(C) at D
(D) at both B and D
(E) From C to D
1992 Tournament Of Towns, (322) 3
A numismatist Fred has some coins. A diameter of any coin is no more than $10$ cm. All the coins are contained in a one-layer box of dimensions $30$ cm by $70$ cm. He is presented with a new coin. Its diameter is $25$ cm. Prove that it is possible to put all the coins in a one-layer box of dimensions $55$ cm by $55$ cm.
(Fedja Nazarov, St Petersburg)
2012 HMNT, 6
Let $\pi$ be a permutation of the numbers from $1$ through $2012$. What is the maximum possible number of integers $n$ with $1 \le n \le 2011$ such that $\pi (n)$ divides $\pi (n + 1)$?
1999 Belarusian National Olympiad, 2
Let $m, n$ be positive integers. Starting with all positive integers written in a line, we can form a list of numbers in two ways:
$(1)$ Erasing every $m$-th and then, in the obtained list, erasing every $n$-th number;
$(2)$ Erasing every $n$-th number and then, in the obtained list, erasing every $m$-th number.
A pair $(m,n)$ is called [i]good[/i] if, whenever some positive integer $k$ occurs in both these lists, then it occurs in both lists on the same position.
(a) Show that the pair $(2, n)$ is good for any $n\in \mathbb{N}$.
(b) Is there a good pair $(m, n)$ with $2<m<n$?
1997 All-Russian Olympiad, 4
A polygon can be divided into 100 rectangles, but not into 99. Prove that it cannot be divided into 100 triangles.
[i]A. Shapovalov[/i]
2010 Stanford Mathematics Tournament, 6
Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, ...$ Find $n$ such that the first $n$ terms sum up to $2010.$
2024 Argentina Cono Sur TST, 1
Two players take turns playing on a $3\times1001$ board whose squares are initially all white. Each player, in his turn, paints two squares located in the same row or column black, not necessarily adjacent. The player who cannot make his move loses the game. Determine which of the two players has a strategy that allows them to win, no matter how well his opponent plays.
2011 Mongolia Team Selection Test, 2
Mongolia TST 2011 Test 1 #2
Let $p$ be a prime number. Prove that:
$\sum_{k=0}^p (-1)^k \dbinom{p}{k} \dbinom{p+k}{k} \equiv -1 (\mod p^3)$
(proposed by B. Batbayasgalan, inspired by Putnam olympiad problem)
Note: I believe they meant to say $p>2$ as well.
2016 Federal Competition For Advanced Students, P2, 4
Let $a,b,c\ge-1$ be real numbers with $a^3+b^3+c^3=1$.
Prove that $a+b+c+a^2+b^2+c^2\le4$, and determine the cases of equality.
(Proposed by Karl Czakler)
1991 IMO, 1
Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that
\[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}.
\]
2016 Mathematical Talent Reward Programme, MCQ: P 9
$f$ be a function satisfying $2f(x)+3f(-x)=x^2+5x$. Find $f(7)$
[list=1]
[*] $-\frac{105}{4}$
[*] $-\frac{126}{5}$
[*] $-\frac{120}{7}$
[*] $-\frac{132}{7}$
[/list]
2021 Israel Olympic Revenge, 1
Let $\mathbb N$ be the set of positive integers. Find all functions $f\colon\mathbb N\to\mathbb N$ such that $$\frac{f(x)-f(y)+x+y}{x-y+1}$$ is an integer, for all positive integers $x,y$ with $x>y$.
2015 South East Mathematical Olympiad, 6
Given a positive integer $n\geq 2$. Let $A=\{ (a,b)\mid a,b\in \{ 1,2,…,n\} \}$ be the set of points in Cartesian coordinate plane. How many ways to colour points in $A$, each by one of three fixed colour, such that, for any $a,b\in \{ 1,2,…,n-1\}$, if $(a,b)$ and $(a+1,b)$ have same colour, then $(a,b+1)$ and $(a+1,b+1)$ also have same colour.
2002 Belarusian National Olympiad, 8
The set of three-digit natural numbers formed from digits $1,2, 3, 4, 5, 6$ is called [i]nice [/i] if it satisfies the following condition: for any two different digits from $1, 2, 3, 4, 5, 6$ there exists a number from the set which contains both of them.
For any nice set we calculate the sum of all its elements. Determine the smallest possible value of these sums.
(E. Barabanov)
2006 Brazil National Olympiad, 5
Let $P$ be a convex $2006$-gon. The $1003$ diagonals connecting opposite vertices and the $1003$ lines connecting the midpoints of opposite sides are concurrent, that is, all $2006$ lines have a common point. Prove that the opposite sides of $P$ are parallel and congruent.
2008 China Team Selection Test, 6
Find the maximal constant $ M$, such that for arbitrary integer $ n\geq 3,$ there exist two sequences of positive real number $ a_{1},a_{2},\cdots,a_{n},$ and $ b_{1},b_{2},\cdots,b_{n},$ satisfying
(1):$ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2):$ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$.
2021 Taiwan TST Round 2, 5
Let $\|x\|_*=(|x|+|x-1|-1)/2$. Find all $f:\mathbb{N}\to\mathbb{N}$ such that
\[f^{(\|f(x)-x\|_*)}(x)=x, \quad\forall x\in\mathbb{N}.\]
Here $f^{(0)}(x)=x$ and $f^{(n)}(x)=f(f^{(n-1)}(x))$ for all $n\in\mathbb{N}$.
[i]Proposed by usjl[/i]
2011 China Western Mathematical Olympiad, 3
In triangle $ABC$ with $AB>AC$ and incenter $I$, the incircle touches $BC,CA,AB$ at $D,E,F$ respectively. $M$ is the midpoint of $BC$, and the altitude at $A$ meets $BC$ at $H$. Ray $AI$ meets lines $DE$ and $DF$ at $K$ and $L$, respectively. Prove that the points $M,L,H,K$ are concyclic.
2015 Harvard-MIT Mathematics Tournament, 1
Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$, $(2,0)$, $(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown. [asy]size(120); defaultpen(linewidth(0.7));
draw(origin--(2,0)--(2,1)--(0,1)--cycle^^(1,0)--(1,1));[/asy] Pro selects a point $P$ at random in the interior of $R$. Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares.
2014 China Team Selection Test, 4
For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that:
$y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$ .
Find the smallest positive number $\lambda$ such that for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$
(High School Affiliated to Nanjing Normal University )
2008 AMC 12/AHSME, 15
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $ R$ be the region formed by the union of the square and all the triangles, and $ S$ be the smallest convex polygon that contains $ R$. What is the area of the region that is inside $ S$ but outside $ R$?
$ \textbf{(A)} \; \frac{1}{4} \qquad \textbf{(B)} \; \frac{\sqrt{2}}{4} \qquad \textbf{(C)} \; 1 \qquad \textbf{(D)} \; \sqrt{3} \qquad \textbf{(E)} \; 2 \sqrt{3}$
2006 Turkey Team Selection Test, 3
If $x,y,z$ are positive real numbers and $xy+yz+zx=1$ prove that
\[ \frac{27}{4} (x+y)(y+z)(z+x) \geq ( \sqrt{x+y} +\sqrt{ y+z} + \sqrt{z+x} )^2 \geq 6 \sqrt 3. \]