Found problems: 85335
1997 Taiwan National Olympiad, 1
Let $a$ be rational and $b,c,d$ are real numbers, and let $f: \mathbb{R}\to [-1.1]$ be a function satisfying $f(x+a+b)-f(x+b)=c[x+2a+[x]-2[x+a]-[b]]+d$ for all $x$. Show that $f$ is periodic.
1968 IMO, 4
Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.
2011 AMC 10, 23
Seven students count from $1$ to $1000$ as follows:
[list]
[*]Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1, 3, 4, 6, 7, 9, \cdots, 997, 999, 1000.$
[*]Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
[*]Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
[*]Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
[*]Finally, George says the only number that no one else says.
[/list]
What number does George say?
$ \textbf{(A)}\ 37\qquad\textbf{(B)}\ 242\qquad\textbf{(C)}\ 365\qquad\textbf{(D)}\ 728\qquad\textbf{(E)}\ 998 $
2018 Romanian Masters in Mathematics, 1
Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .
2010 Contests, 3
Each of the small squares of a $50\times 50$ table is coloured in red or blue. Initially all squares are red. A [i]step[/i] means changing the colour of all squares on a row or on a column.
a) Prove that there exists no sequence of steps, such that at the end there are exactly $2011$ blue squares.
b) Describe a sequence of steps, such that at the end exactly $2010$ squares are blue.
[i]Adriana & Lucian Dragomir[/i]
2018 Iran Team Selection Test, 1
Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$:
$$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$
[i]Proposed by Morteza Saghafian, Mahyar Sefidgaran[/i]
1951 Poland - Second Round, 5
Prove that if the relationship between the sides and opposite angles $ A $ and $ B $ of the triangle $ ABC $ is
$$ (a^2 + b^2) \sin (A - B) = (a^2 - b^2) \sin (A + B)$$
then such a triangle is right-angled or isosceles.
Durer Math Competition CD Finals - geometry, 2016.C2
Show that in a triangle the altitude of the longest side is at most as long as it the the sum of the lengths of the perpendicular segments drawn from any point on the longest side on the other two sides.
2010 IMC, 5
Suppose that for a function $f: \mathbb{R}\to \mathbb{R}$ and real numbers $a<b$ one has $f(x)=0$ for all $x\in (a,b).$ Prove that $f(x)=0$ for all $x\in \mathbb{R}$ if
\[\sum^{p-1}_{k=0}f\left(y+\frac{k}{p}\right)=0\]
for every prime number $p$ and every real number $y.$
1990 India Regional Mathematical Olympiad, 7
A census man on duty visited a house in which the lady inmates declined to reveal their individual ages, but said "We do not mind giving you the sum of the ages of any two ladies you may choose". Thereupon, the census man said, "In that case, please give me the sum of the ages of every possible pair of you". They gave the sums as: 30, 33, 41, 58, 66, 69. The census man took these figures and happily went away.
How did he calculate the individual ages?
2021 BMT, 8
Let $\overline{AB}$ be a line segment with length $10$. Let $P$ be a point on this segment with $AP = 2$. Let $\omega_1$ and $\omega_2$ be the circles with diameters $\overline{AP}$ and $\overline{P B}$, respectively. Let $XY$ be a line externally tangent to $\omega_1$ and $\omega_2$ at distinct points $X$ and $Y$ , respectively. Compute the area of $\vartriangle XP Y$ .
2017 Princeton University Math Competition, A5/B7
[i]Greedy Algorithms, Inc.[/i] offers the following string-processing service. Each string submitted for processing has a starting price of $1$ dollar. The customer can then ask for any two adjacent characters in the string to be swapped. This may be done an arbitrary number of times, but each swap doubles the price for processing the string. Then the company returns the modified string and charges the customer $2^S$ dollars, where $S$ is the number of swaps executed. If a customer submits [b]all [/b]permutations of the string $\text{PUMAC}$ for processing and wants all of the strings to be identical after processing, what is the lowest price, in dollars, she could pay?
2006 Vietnam National Olympiad, 5
Find all polynomyals $P(x)$ with real coefficients which satisfy the following equality for all real numbers $x$: \[ P(x^2)+x(3P(x)+P(-x))=(P(x))^2+2x^2 . \]
2018 ELMO Shortlist, 4
Elmo calls a monic polynomial with real coefficients [i]tasty[/i] if all of its coefficients are in the range $[-1,1]$. A monic polynomial $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$.
[i]Proposed by Carl Schildkraut[/i]
2016 Balkan MO Shortlist, N1
Find all natural numbers $n$ for which $1^{\phi (n)} + 2^{\phi (n)} +... + n^{\phi (n)}$ is coprime with $n$.
2007 Princeton University Math Competition, 3
Points $P_1, P_2, P_3,$ and $P_4$ are $(0,0), (10, 20), (5, 15),$ and $(12, -6)$, respectively. For what point $P \in \mathbb{R}^2$ is the sum of the distances from $P$ to the other $4$ points minimal?
Kvant 2020, M2593
Each vertex of a regular polygon is colored in one of three colors so that an odd number of vertices are colored in each of the three colors. Prove that the number of isosceles triangles whose vertices are colored in three different colors is odd.
[i]From foreign Olympiads[/i]
2012 Tournament of Towns, 4
Alex marked one point on each of the six interior faces of a hollow unit cube. Then he connected by strings any two marked points on adjacent faces. Prove that the total length of these strings is at least $6\sqrt2$.
2019 Moldova EGMO TST, 4
Let $x,y>0$ be real numbers.Prove that: $$\frac{1}{x^2+y^2} +\frac{1}{x^2}+\frac{1}{y^2}\ge\frac{10}{(x+y)^2}$$
I tried CBS, but it doesn't work... Can you give an idea, please?
2012 Baltic Way, 17
Let $d(n)$ denote the number of positive divisors of $n$. Find all triples $(n,k,p)$, where $n$ and $k$ are positive integers and $p$ is a prime number, such that
\[n^{d(n)} - 1 = p^k.\]
2016 Nigerian Senior MO Round 2, Problem 7
Prove that $(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n}$ is an even integer and that $(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=w\sqrt{3}$ for some positive integer $w$, for all integers $n \geq 1$.
2019 Germany Team Selection Test, 3
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.
Prove that Sisyphus cannot reach the aim in less than
\[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]
turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )
2008 National Olympiad First Round, 12
In how many ways a cube can be painted using seven different colors in such a way that no two faces are in same color?
$
\textbf{(A)}\ 154
\qquad\textbf{(B)}\ 203
\qquad\textbf{(C)}\ 210
\qquad\textbf{(D)}\ 240
\qquad\textbf{(E)}\ \text{None of the above}
$
2021 Thailand TST, 1
For a positive integer $n$, consider a square cake which is divided into $n \times n$ pieces with at most one strawberry on each piece. We say that such a cake is [i]delicious[/i] if both diagonals are fully occupied, and each row and each column has an odd number of strawberries.
Find all positive integers $n$ such that there is an $n \times n$ delicious cake with exactly $\left\lceil\frac{n^2}{2}\right\rceil$ strawberries on it.
2018 Costa Rica - Final Round, LRP1
Arnulfo and Berenice play the following game: One of the two starts by writing a number from $ 1$ to $30$, the other chooses a number from $ 1$ to $30$ and adds it to the initial number, the first player chooses a number from $ 1$ to $30$ and adds it to the previous result, they continue doing the same until someone manages to add $2018$. When Arnulfo was about to start, Berenice told him that it was unfair, because whoever started had a winning strategy, so the numbers had better change. So they asked the following question:
Adding chosen numbers from $1 $ to $a$, until reaching the number $ b$, what conditions must meet $a$ and $ b$ so that the first player does not have a winning strategy?
Indicate if Arnulfo and Berenice are right and answer the question asked by them.