Found problems: 85335
2024 Canadian Junior Mathematical Olympiad, 4
Jane writes down $2024$ natural numbers around the perimeter of a circle. She wants the $2024$ products of adjacent pairs of numbers to be exactly the set $\{ 1!, 2!, \ldots, 2024! \}.$ Can she accomplish this?
2009 China National Olympiad, 2
Let $ P$ be a convex $ n$ polygon each of which sides and diagnoals is colored with one of $ n$ distinct colors. For which $ n$ does: there exists a coloring method such that for any three of $ n$ colors, we can always find one triangle whose vertices is of $ P$' and whose sides is colored by the three colors respectively.
2019 All-Russian Olympiad, 8
Let $P(x)$ be a non-constant polynomial with integer coefficients and let $n$ be a positive integer. The sequence $a_0,a_1,\ldots$ is defined as follows: $a_0=n$ and $a_k=P(a_{k-1})$ for all positive integers $k.$ Assume that for every positive integer $b$ the sequence contains a $b$th power of an integer greater than $1.$ Show that $P(x)$ is linear.
2015 Sharygin Geometry Olympiad, P11
Let $H$ be the orthocenter of an acute-angled triangle A$BC$. The perpendicular bisector to segment $BH$ meets $BA$ and $BC$ at points $A_0, C_0$ respectively. Prove that the perimeter of triangle $A_0OC_0$ ($O$ is the circumcenter of triangle $ABC$) is equal to $AC$.
2007 Cono Sur Olympiad, 2
Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers.
1997 Romania National Olympiad, 2
Let $A$ be a square matrix of odd order (at least $3$) whose entries are odd integers. Prove that if $A$ is invertible, then it is not possible for all the minors of the entries of a row to have equal absolute values.
2017 Estonia Team Selection Test, 7
Let $n$ be a positive integer. In how many ways can an $n \times n$ table be filled with integers from $0$ to $5$ such that
a) the sum of each row is divisible by $2$ and the sum of each column is divisible by $3$
b) the sum of each row is divisible by $2$, the sum of each column is divisible by $3$ and the sum of each of the two diagonals is divisible by $6$?
2000 Mongolian Mathematical Olympiad, Problem 2
Circles $\omega_1,\omega_2,\omega_3$ with centers $O_1,O_2,O_3$, respectively, are externally tangent to each other. The circle $\omega_1$ touches $\omega_2$ at $P_1$ and $\omega_3$ at $P_2$. For any point $A$ on $\omega_1$, $A_1$ denotes the point symmetric to $A$ with respect to $O_1$. Show that the intersection points of $AP_2$ with $\omega_3$, $A_1P_3$ with $\omega_2$, and $AP_3$ with $A_1P_2$ lie on a line.
2022 Stanford Mathematics Tournament, 3
Every day you go to the music practice rooms at a random time between $12\text{AM}$ and $8\text{AM}$ and practice for $3$ hours, while your friend goes at a random time from $5\text{AM}$ to $11\text{AM}$ and practices for $1$ hour (the block of practice time need not be contained in he given time range for the arrival). What is the probability that you and your meet on at least $2$ days in a given span of $5$ days?
1998 Slovenia National Olympiad, Problem 4
Alf was attending an eight-year elementary school on Melmac. At the end of each school year, he showed the certificate to his father. If he was promoted, his father gave him the number of cats equal to Alf’s age times the number of the grade he passed. During elementary education, Alf failed one grade and had to repeat it. When he finished elementary education he found out that the total number of cats he had received was divisible by $1998$. Which grade did Alf fail?
2010 Purple Comet Problems, 14
There are positive integers $b$ and $c$ such that the polynomial $2x^2 + bx + c$ has two real roots which differ by $30.$ Find the least possible value of $b + c.$
2023 Thailand TST, 3
Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple:
\begin{align*}
\mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\
\mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022}))
\end{align*}
and then write this tuple on the blackboard.
It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?
May Olympiad L2 - geometry, 2011.3
In a right triangle rectangle $ABC$ such that $AB = AC$, $M$ is the midpoint of $BC$. Let $P$ be a point on the perpendicular bisector of $AC$, lying in the semi-plane determined by $BC$ that does not contain $A$. Lines $CP$ and $AM$ intersect at $Q$. Calculate the angles that form the lines $AP$ and $BQ$.
2013 Benelux, 3
Let $\triangle ABC$ be a triangle with circumcircle $\Gamma$, and let $I$ be the center of the incircle of $\triangle ABC$. The lines $AI$, $BI$ and $CI$ intersect $\Gamma$ in $D \ne A$, $E \ne B$ and $F \ne C$. The tangent lines to $\Gamma$ in $F$, $D$ and $E$ intersect the lines $AI$, $BI$ and $CI$ in $R$, $S$ and $T$, respectively. Prove that
\[\vert AR\vert \cdot \vert BS\vert \cdot \vert CT\vert = \vert ID\vert \cdot \vert IE\vert \cdot \vert IF\vert.\]
2004 National Chemistry Olympiad, 50
How many valence electrons are in the pyrophosphate ion, $\ce{P2O7}^{4-}?$
$ \textbf{(A) } 48\qquad\textbf{(B) } 52\qquad\textbf{(C) } 54\qquad\textbf{(D) } 56\qquad $
1998 Dutch Mathematical Olympiad, 2
Let $TABCD$ be a pyramid with top vertex $T$, such that its base $ABCD$ is a square of side length 4. It is given that, among the triangles $TAB$, $TBC$, $TCD$ and $TDA$, one can find an isosceles triangle and a right-angled triangle. Find all possible values for the volume of the pyramid.
EMCC Guts Rounds, 2022
[u]Round 5[/u]
[b]p13.[/b] Find the number of six-digit positive integers that satisfy all of the following conditions:
(i) Each digit does not exceed $3$.
(ii) The number $1$ cannot appear in two consecutive digits.
(iii) The number $2$ cannot appear in two consecutive digits.
[b]p14.[/b] Find the sum of all distinct prime factors of $103040301$.
[b]p15.[/b] Let $ABCA'B'C'$ be a triangular prism with height $3$ where bases $ABC$ and $A'B'C'$ are equilateral triangles with side length $\sqrt6$. Points $P$ and $Q$ lie inside the prism so that $ABCP$ and $A'B'C'Q$ are regular tetrahedra. The volume of the intersection of these two tetrahedra can be expressed in the form $\frac{\sqrt{m}}{n}$ , where $m$ and $n$ are positive integers and $m$ is not divisible by the square of any prime. Find $m + n$.
[u]Round 6[/u]
[b]p16.[/b] Let $a_0, a_1, ...$ be an infinite sequence such that $a^2_n -a_{n-1}a_{n+1} = a_n -a_{n-1}$ for all positive integers $n$. Given that $a_0 = 1$ and $a_1 = 4$, compute the smallest positive integer $k$ such that $a_k$ is an integer multiple of $220$.
[b]p17.[/b] Vincent the Bug is on an infinitely long number line. Every minute, he jumps either $2$ units to the right with probability $\frac23$ or $3$ units to the right with probability $\frac13$ . The probability that Vincent never lands exactly $15$ units from where he started can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$?
[b]p18.[/b] Battler and Beatrice are playing the “Octopus Game.” There are $2022$ boxes lined up in a row, and inside one of the boxes is an octopus. Beatrice knows the location of the octopus, but Battler does not. Each turn, Battler guesses one of the boxes, and Beatrice reveals whether or not the octopus is contained in that box at that time. Between turns, the octopus teleports to an adjacent box and secretly communicates to Beatrice where it teleported to. Find the least positive integer $B$ such that Battler has a strategy to guarantee that he chooses the box containing the octopus in at most $B$ guesses.
[u]Round 7[/u]
[b]p19.[/b] Given that $f(x) = x^2-2$ the number $f(f(f(f(f(f(f(2.5)))))))$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Find the greatest positive integer $n$ such that $2^n$ divides $ab+a+b-1$.
[b]p20.[/b] In triangle $ABC$, the shortest distance between a point on the $A$-excircle $\omega$ and a point on the $B$-excircle $\Omega$ is $2$. Given that $AB = 5$, the sum of the circumferences of $\omega$ and $\Omega$ can be written in the form $\frac{m}{n}\pi$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? (Note: The $A$-excircle is defined to be the circle outside triangle $ABC$ that is tangent to the rays $\overrightarrow{AB}$ and $\overrightarrow{AC}$ and to the side $ BC$. The $B$-excircle is defined similarly for vertex $B$.)
[b]p21.[/b] Let $a_0, a_1, ...$ be an infinite sequence such that $a_0 = 1$, $a_1 = 1$, and there exists two fixed integer constants $x$ and $y$ for which $a_{n+2}$ is the remainder when $xa_{n+1}+ya_n$ is divided by $15$ for all nonnegative integers $n$. Let $t$ be the least positive integer such that $a_t = 1$ and $a_{t+1} = 1$ if such an integer exists, and let $t = 0$ if such an integer does not exist. Find the maximal value of t over all possible ordered pairs $(x, y)$.
[u]Round 8[/u]
[b]p22.[/b] A mystic square is a $3$ by $3$ grid of distinct positive integers such that the least common multiples of the numbers in each row and column are the same. Let M be the least possible maximal element in a mystic square and let $N$ be the number of mystic squares with $M$ as their maximal element. Find $M + N$.
[b]p23.[/b] In triangle $ABC$, $AB = 27$, $BC = 23$, and $CA = 34$. Let $X$ and $Y$ be points on sides $ AB$ and $AC$, respectively, such that $BX = 16$ and $CY = 7$. Given that $O$ is the circumcenter of $BXY$ , the value of $CO^2$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
[b]p24.[/b] Alan rolls ten standard fair six-sided dice, and multiplies together the ten numbers he obtains. Given that the probability that Alan’s result is a perfect square is $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers, compute $a$.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949416p26408251]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2016 Mathematical Talent Reward Programme, SAQ: P 6
Consider the set $A=\{1,2,3,4,5,6,7,8,9\}$.A partition $\Pi $ of $A$ is collection of disjoint sets whose union is $A$. For example, $\Pi_1=\{\{1,2\},\{3,4,5\},\{6,7,8,9\}\}$ and $\Pi _2 =\{\{1\},\{2,5\},\{3,7\},\{4,5,6,7,8,9\}\}$ can be considered as partitions of $A$. For, each $\Pi$ partition ,we consider the function $\pi$ defined on the elements of$A$. $\pi (x)$ denotes the cardinality of the subset in $\Pi$ which contains $x$. For, example in case of $\Pi_1$ , $\pi_1(1)=\pi_1(2)=2$, $\pi_1(3)=\pi_1(4)=\pi_1 (5)=3$, and $\pi_1(6)=\pi_1(7)=\pi_1(8)=\pi_1(9)=4$. For $\Pi_2$ we have $\pi_2(1)=1$, $\pi_2(2)=\pi_2(5)=2$, $\pi_2(3)=\pi_2(7)=2$ and $\pi_2(4)=\pi_2(6)=\pi_2(8)=\pi_2(9)=4$ Given any two partitions $\Pi$ and $\Pi '$, show that there are two numbers $x$ and $y$ in $A$, such that $\pi (x)= \pi '(x)$ and $ \pi (y)= \pi'(y)$.[[b]Hint:[/b] Consider the case where there is a block of size greater than or equal to 4 in a partition and the alternative case]
2025 Israel National Olympiad (Gillis), P7
For a positive integer $n$, let $A_n$ be the set of quadruplets $(a,b,c,d)$ of integers, satisfying the following properties simultaneously:
[list]
[*] $0\le a\le c\le n,$
[*] $0\le b\le d\le n,$
[*] $c+d>n,$ and
[*] $bc=ad+1.$
[/list]
Moreover, define
$$\alpha_n=\sum_{(a,b,c,d)\in A_n}\frac{1}{ab+cd}.$$
Find all real numbers $t$ such that $\alpha_n>t$ for every positive integer $n$.
2021 Regional Competition For Advanced Students, 4
Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$.
(Walther Janous)
2010 Purple Comet Problems, 1
Let $x$ satisfy $(6x + 7) + (8x + 9) = (10 + 11x) + (12 + 13x).$ There are relatively prime positive integers so that $x = -\tfrac{m}{n}$. Find $m + n.$
1996 Romania National Olympiad, 3
Let $A$ be a commutative ring with $0 \neq 1$ such that for any $x \in A \setminus \{0\}$ there exist positive integers $m,n$ such that $(x^m+1)^n=x.$ Prove that any endomorphism of $A$ is an automorphism.
2014 Contests, 2
Let $a,b\in\mathbb{R}_+$ such that $a+b=1$. Find the minimum value of the following expression:
\[E(a,b)=3\sqrt{1+2a^2}+2\sqrt{40+9b^2}.\]
2013 Sharygin Geometry Olympiad, 10
The incircle of triangle $ABC$ touches the side $AB$ at point $C'$; the incircle of triangle $ACC'$ touches the sides $AB$ and $AC$ at points $C_1, B_1$; the incircle of triangle $BCC'$ touches the sides $AB$ and $BC$ at points $C_2$, $A_2$. Prove that the lines $B_1C_1$, $A_2C_2$, and $CC'$ concur.
2004 AIME Problems, 3
A convex polyhedron $P$ has 26 vertices, 60 edges, and 36 faces, 24 of which are triangular, and 12 of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does $P$ have?