Found problems: 85335
1998 Bulgaria National Olympiad, 3
The sides and diagonals of a regular $n$-gon $R$ are colored in $k$ colors so that:
(i) For each color $a$ and any two vertices $A$,$B$ of $R$ , the segment $AB$ is of color $a$ or there is a vertex $C$ such that $AC$ and $BC$ are of color $a$.
(ii) The sides of any triangle with vertices at vertices of $R$ are colored in at most two colors.
Prove that $k\leq 2$.
2002 Croatia Team Selection Test, 1
In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.
2024 Thailand TST, 3
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
2013 Saint Petersburg Mathematical Olympiad, 2
in a convex quadrilateral $ABCD$ , $M,N$ are midpoints of $BC,AD$ respectively. If $AM=BN$ and $DM=CN$ then prove that $AC=BD$.
S. Berlov
1995 Tournament Of Towns, (445) 1
Prove that if $a$, $b$ and $c$ are integers and the sums
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \,\,\,\, and \,\,\,\, \frac{a}{c}+\frac{c}{b}+\frac{b}{a}$$
are also integers, then we have $|a| = |v| = |c|$.
(A Gribalko)
1982 All Soviet Union Mathematical Olympiad, 345
Given the square table $n\times n$ with $(n-1)$ marked fields. Prove that it is possible to move all the marked fields below the diagonal by moving rows and columns.
1960 AMC 12/AHSME, 24
If $\log_{2x}216 = x$, where $x$ is real, then $x$ is:
$ \textbf{(A)}\ \text{A non-square, non-cube integer} \qquad$
$\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad$
$\textbf{(C)}\ \text{An irrational number} \qquad$
$\textbf{(D)}\ \text{A perfect square}\qquad$
$\textbf{(E)}\ \text{A perfect cube} $
2022 Romania Team Selection Test, 2
Fix a nonnegative integer $a_0$ to define a sequence of integers $a_0,a_1,\ldots$ by letting $a_k,k\geq 1$ be the smallest integer (strictly) greater than $a_{k-1}$ making $a_{k-1}+a_k{}$ into a perfect square. Let $S{}$ be the set of positive integers not expressible as the difference of two terms of the sequence $(a_k)_{k\geq 0}.$ Prove that $S$ is finite and determine its size in terms of $a_0.$
1985 Miklós Schweitzer, 1
[b]1.[/b] Some proper partitions $P_1, \dots , P_n$ of a finite set $S$ (that is, partitions containing at least two parts) are called [i]independent[/i] if no matter how we choose one class from each partition, the intersection of the chosen classes is nonempty. Show that if the inequality
$\frac{\left | S \right | }{2} < \left |P_1 \right | \dots \left |P_n \right |\qquad \quad (*)$
holds for some independent partitions, then $P_1, \dots , P_n$ is maximal in the sense that there is no partition $P$ such that $P,P_1, \dots , P_n$ are independent. On the other hand, show that inequality $(*)$ is not necessary for this maximality. ([b]C.20[/b])
[E. Gesztelyi]
2014 Dutch IMO TST, 4
Determine all pairs $(p, q)$ of primes for which $p^{q+1}+q^{p+1}$ is a perfect square.
2008 Korean National Olympiad, 2
We have $x_i >i$ for all $1 \le i \le n$.
Find the minimum value of $\frac{(\sum_{i=1}^n x_i)^2}{\sum_{i=1}^n \sqrt{x^2_i - i^2}}$
2007 China Team Selection Test, 1
Let $ ABC$ be a triangle. Circle $ \omega$ passes through points $ B$ and $ C.$ Circle $ \omega_{1}$ is tangent internally to $ \omega$ and also to sides $ AB$ and $ AC$ at $ T,\, P,$ and $ Q,$ respectively. Let $ M$ be midpoint of arc $ BC\, ($containing $ T)$ of $ \omega.$ Prove that lines $ PQ,\,BC,$ and $ MT$ are concurrent.
2014 Cuba MO, 1
We have two $20 \times 13$ rectangular grids with $260$ unit cells. each one. We insert in the boxes of each of the grids the numbers $1, 2, ..., 260$ as follows:
$\bullet$ For the first grid, we start by inserting the numbers $1, 2, ..., 13$ in the boxes in the top row from left to right. We continue inserting numbers $14$, $ 15$, $...$, $26$ in the second row from left to right. We maintain the same procedure until in the last row, $20$, the numbers are placed $248$, $249$, $...$, $260$ from left to right.
$\bullet$ For the second grid we start by inserting the numbers $1$, $2$,$ ..$., $20$ from top to bottom in the farthest column right. We continue inserting the numbers $21$, $22$,$ ...$, $40$ in the second column from the right also from top to bottom.
We maintain that same procedure until we reach the column on the left where we place the numbers from top to bottom $241$, $242$, $ ...$, $260$.
Determines the integers inserted in the boxes located in the same position in both grids.
2002 Dutch Mathematical Olympiad, 4
Five pairs of cartoon characters, Donald and Katrien Duck, Asterix and Obelix, Suske and Wiske, Tom and Jerry, Heer Bommel and Tom Poes, sit around a round table with $10$ chairs. The two members of each pair ensure that they sit next to each other. In how many different ways can the ten seats be occupied?
Two ways are different if they cannot be transferred to each other by a rotation.
2016 Brazil Team Selection Test, 1
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.
[i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]
2004 Croatia National Olympiad, Problem 4
Determine all real numbers $\alpha$ with the property that all numbers in the sequence $\cos\alpha,\cos2\alpha,\cos2^2\alpha,\ldots,\cos2^n\alpha,\ldots$ are negative.
2022 Belarusian National Olympiad, 8.8
Vitya and Masha are playing a game. At first, Vitya thinks of three different integers. In one move Masha can ask one of the following three numbers: the sum of the numbers, the product of the numbers or the sum of pairwise products of the numbers. Masha asks questions and Vitya immediately answers before Masha asks the next question.
a) Prove that Masha can always guess Vitya's numbers.
b) What is the least amount of questions Masha needs to ask to guaranteely guess them?
LMT Team Rounds 2021+, A 24
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Using the four words
“Hi”, “hey”, “hello”, and “haiku”,
How many haikus
Can somebody make?
(Repetition is allowed,
Order does matter.)
[i]Proposed by Jeff Lin[/i]
2021 Ukraine National Mathematical Olympiad, 2
Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$. Find all pairs of natural numbers $(k,d)$ for which there exists a $n$ such that for any polynomial $p \in P^{(n)}$, number $P(k)$ is divisible by the number $d$.
(Oleksii Masalitin)
2019 Adygea Teachers' Geometry Olympiad, 1
Inside the quadrangle, a point is taken and connected with the midpoint of all sides. Areas of the three out of four formed quadrangles are $S_1, S_2, S_3$. Find the area of the fourth quadrangle.
2008 Bulgarian Autumn Math Competition, Problem 12.3
Find all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
\[(f(x)f(y)-1)f(x+y)=2f(x)f(y)-f(x)-f(y)\quad \forall x,y\in \mathbb{R}\]
2016-2017 SDML (Middle School), 5
What is the measure in degrees of the acute angle formed by the hands of a $12$-hour clock at $3:20$ PM?
$\text{(A) }18\qquad\text{(B) }20\qquad\text{(C) }22\qquad\text{(D) }25\qquad\text{(E) }30$
1965 Leningrad Math Olympiad, grade 8
[b]8.1[/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. Draw another straight line so that after that the rectangle was divided into the largest possible number of parts.
[b]8.2[/b] Engineers always tell the truth, but businessmen always lie. F and G are engineers. A declares that, B asserts that, C asserts that, D says that, E insists that, F denies that G is an businessman. C also announces that D is a businessman. If A is a businessman, then how much total businessmen in this company?
[b]8.3 [/b]There is a straight road through the field. A tourist stands on the road at a point ?. It can walk along the road at a speed of 6 km/h and across the field at a speed of 3 km/h. Find the locus of the points where the tourist can get there within an hour's walk.
[b]8.4 / 7.5 [/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ .
[b]8.5.[/b] In some state, every two cities are connected by a road. Each road is only allowed to move in one direction. Prove that there is a city from which you can travel around everything. state, having visited each city exactly once.
[b]8.6[/b] Find all eights of prime numbers such that the sum of the squares of the numbers in the eight is 992 less than their quadruple product. [hide=original wording]Найдите все восьмерки простых чисел такие, что сумма квадратов чисел в восьмерке на 992 меньше, чем их учетверенное произведение.[/hide]
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].
Cono Sur Shortlist - geometry, 2005.G5
Let $O$ be the circumcenter of an acute triangle $ABC$ and $A_1$ a point of the minor arc $BC$ of the circle $ABC$ . Let $A_2$ and $A_3$ be points on sides $AB$ and $AC$ respectively such that $\angle BA_1A_2=\angle OAC$ and $\angle CA_1A_3=\angle OAB$ . Points $B_2, B_3, C_2$ and $C_3$ are similarly constructed, with $B_2$ in $BC, B_3$ in $AB, C_2$ in $AC$ and $C_3$ in $BC$. Prove that lines $A_2A_3, B_2B_3$ and $C_2C_3$ are concurrent.
2005 Tuymaada Olympiad, 3
The organizers of a mathematical congress found that if they accomodate any participant in a room the rest can be accomodated in double rooms so that 2 persons living in each room know each other. Prove that every participant can organize a round table on graph theory for himself and an even number of other people so that each participant of the round table knows both his neigbours.
[i]Proposed by S. Berlov, S. Ivanov[/i]