Found problems: 85335
2021 Azerbaijan IMO TST, 1
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
1988 Tournament Of Towns, (183) 6
Consider a sequence of words , consisting of the letters $A$ and $B$ .
The first word in the sequence is "$A$" . The k-th word i s obtained from the $(k-1)$-th by means of the following transformation : each $A$ is substituted by $AAB$ , and each $B$ is substituted by $A$. It is easily seen that every word is an initial part of the next word. The initial parts of these words coincide to give a sequence of letters $AABAABAAA BAABAAB...$
(a) In which place of this sequence is the $1000$-th letter $A$?
(b ) Prove that this sequence is not periodic.
(V . Galperin , Moscows)
1999 AMC 12/AHSME, 26
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $ 1$. The polygons meet at a point $ A$ in such a way that the sum of the three interior angles at $ A$ is $ 360^\circ$. Thus the three polygons form a new polygon with $ A$ as an interior point. What is the largest possible perimeter that this polygon can have?
$ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 21\qquad \textbf{(E)}\ 24$
2018 Centroamerican and Caribbean Math Olympiad, 2
Let $\Delta ABC$ be a triangle inscribed in the circumference $\omega$ of center $O$. Let $T$ be the symmetric of $C$ respect to $O$ and $T'$ be the reflection of $T$ respect to line $AB$. Line $BT'$ intersects $\omega$ again at $R$. The perpendicular to $CT$ through $O$ intersects line $AC$ at $L$. Let $N$ be the intersection of lines $TR$ and $AC$. Prove that $\overline{CN}=2\overline{AL}$.
2018 Slovenia Team Selection Test, 1
Let $n$ be a positive integer. On the table, we have $n^2$ ornaments in $n$ different colours, not necessarily $n$ of each colour. Prove that we can hang the ornaments on $n$ Christmas trees in such a way that there are exactly $n$ ornaments on each tree and the ornaments on every tree are of at most $2$ different colours.
2014-2015 SDML (High School), 2
What is the maximum value of the function $$\frac{1}{\left|x+1\right|+\left|x+2\right|+\left|x-3\right|}?$$
$\text{(A) }\frac{1}{3}\qquad\text{(B) }\frac{1}{4}\qquad\text{(C) }\frac{1}{5}\qquad\text{(D) }\frac{1}{6}\qquad\text{(E) }\frac{1}{7}$
2024 Macedonian Mathematical Olympiad, Problem 4
In two wooden boxes, there are $1994$ and $2024$ marbles, respectively. Spiro and Cvetko play the following game: alternately, each player takes a turn and removes some marbles from one of the boxes, so that the number of removed marbles in that turn is a divisor of the current number of marbles in the other box. The winner of the game is the one after whose turn both boxes are empty. Spiro takes the first turn. Which of the players has a winning strategy?
2002 Cono Sur Olympiad, 2
Given a triangle $ABC$, with right $\angle A$, we know: the point $T$ of tangency of the circumference inscribed in $ABC$ with the hypotenuse $BC$, the point $D$ of intersection of the angle bisector of $\angle B$ with side AC and the point E of intersection of the angle bisector of $\angle C$ with side $AB$ . Describe a construction with ruler and compass for points $A$, $B$, and $C$. Justify.
2006 May Olympiad, 2
Several prime numbers (some repeated) are written on the board. Mauro added the numbers on the board and Fernando multiplied the numbers on the board. The result obtained by Fernando is equal to $40$ times the result obtained by Mauro. Determine what the numbers on the board can be. Give all chances.
2012 Indonesia TST, 2
A TV station holds a math talent competition, where each participant will be scored by 8 people. The scores are F (failed), G (good), or E (exceptional). The competition is participated by three people, A, B, and C. In the competition, A and B get the same score from exactly 4 people. C states that he has differing scores with A from at least 4 people, and also differing scores with B from at least 4 people. Assuming C tells the truth, how many scoring schemes can occur?
1995 Austrian-Polish Competition, 3
Let $P(x) = x^4 + x^3 + x^2 + x + 1$. Show that there exist two non-constant polynomials $Q(y)$ and $R(y)$ with integer coefficients such that for all $Q(y) \cdot R(y)= P(5y^2)$ for all $y$ .
2015 Tournament of Towns, 2
A $10 \times 10$ square on a grid is split by $80$ unit grid segments into $20$ polygons of equal area (no one of these segments belongs to the boundary of the square). Prove that all polygons are congruent.
[i]($6$ points)[/i]
2014 Spain Mathematical Olympiad, 1
Let $(x_n)$ be a sequence of positive integers defined by $x_1=2$ and $x_{n+1}=2x_n^3+x_n$ for all integers $n\ge1$. Determine the largest power of $5$ that divides $x_{2014}^2+1$.
1978 Putnam, A1
Let $A$ be any set of $20$ distinct integers chosen from the arithmetic progression $1, 4, 7,\ldots,100$. Prove that there must be two distinct integers in $A$ whose sum is $104$.
2024 JBMO TST - Turkey, 8
There is $207$ boxes on the table which numbered $1,2, \dots , 207$ respectively. Firstly Aslı puts a red ball in each of the $100$ boxes that she chooses and puts a white ball in each of the remaining ones. After that Zehra, writes a pair $(i,j)$ on the blackboard such that $1\leq i \leq j \leq 207$. Finally, Aslı tells Zehra that for every pair; whether the color of the balls which is inside the box which numbered by these numbers are the same or not. Find the least possible value of $N$ such that Zehra can guarantee finding all colors that has been painted to balls in each of the boxes with writing $N$ pairs on the blackboard.
2021 MOAA, 20
Compute the sum of all integers $x$ for which there exists an integer $y$ such that
\[x^3+xy+y^3=503.\]
[i]Proposed by Nathan Xiong[/i]
2014 AMC 12/AHSME, 25
The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coefficients is it true that $|4x+3y|\leq 1000$?
$\textbf{(A) }38\qquad
\textbf{(B) }40\qquad
\textbf{(C) }42\qquad
\textbf{(D) }44\qquad
\textbf{(E) }46\qquad$
2006 Princeton University Math Competition, 7
Given parallelogram $ABCD$, construct point $F$ so that $CF\perp BC$, as shown. Also $F$ is placed so that $\angle DFC = 120^o$. If $DF = 4$ and $BC =CF = 2$, what is the area of the parallelogram?
[img]https://cdn.artofproblemsolving.com/attachments/7/4/0cdb0752760686acb891580da55f55212098fb.jpg[/img]
2002 Switzerland Team Selection Test, 8
In a group of $n$ people, every weekend someone organizes a party in which he invites all of his acquaintances. Those who meet at a party become acquainted. After each of the $n$ people has organized a party, there still are two people not knowing each other. Show that these two will never get to know each other at such a party.
2014 Cezar Ivănescu, 1
[b]a)[/b] Find the real numbers $ x,y $ such that the set $ \{ x,y \}\cup\left\{ 31/20,29/30,27/40,11/120 \right\} $ contains six elements that can represent an arithmetic progression.
[b]b)[/b] Let be four real numbers in arithmetic progression $ b_1<b_2<b_3<b_4. $ Are there sets $ S $ of $ 6 $ elements that represent an arithmetic progression such that $ \left\{ b_1,b_2,b_3,b_4 \right\}\subset S? $
2019 AMC 12/AHSME, 2
Suppose $a$ is $150\%$ of $b$. What percent of $a$ is $3b$?
$\textbf{(A) } 50 \qquad \textbf{(B) } 66\frac{2}{3} \qquad \textbf{(C) } 150 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 450$
2009 HMNT, 10
Compute \[ \displaystyle\prod_{n=0}^{\infty} \left( 1 - \left( \frac {1}{2} \right)^{3^n} + \left( \frac {1}{4} \right)^{3^n} \right). \]
1940 Moscow Mathematical Olympiad, 057
Draw a circle that has a given radius $R$ and is tangent to a given line and a given circle. How many solutions does this problem have?
2023 AMC 12/AHSME, 5
You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensue that at least one of your guessed squares is covered by the rectangle?
$\textbf{(A)}~3\qquad\textbf{(B)}~5\qquad\textbf{(C)}~4\qquad\textbf{(D)}~8\qquad\textbf{(E)}~6$
2008 IMO Shortlist, 6
For $ n\ge 2$, let $ S_1$, $ S_2$, $ \ldots$, $ S_{2^n}$ be $ 2^n$ subsets of $ A \equal{} \{1, 2, 3, \ldots, 2^{n \plus{} 1}\}$ that satisfy the following property: There do not exist indices $ a$ and $ b$ with $ a < b$ and elements $ x$, $ y$, $ z\in A$ with $ x < y < z$ and $ y$, $ z\in S_a$, and $ x$, $ z\in S_b$. Prove that at least one of the sets $ S_1$, $ S_2$, $ \ldots$, $ S_{2^n}$ contains no more than $ 4n$ elements.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]