Found problems: 85335
2002 Croatia National Olympiad, Problem 3
Points $E$ and $F$ are taken on the diagonals $AB_1$ and $CA_1$ of the lateral faces $ABB_1A_1$ and $CAA_1C_1$ of a triangular prism $ABCA_1B_1C_1$ so that $EF\parallel BC_1$. Find the ratio of the lengths of $EF$ and $BC_1$.
1977 Polish MO Finals, 3
Consider the set $A = \{0, 1, 2, . . . , 2^{2n} - 1\}$. The function $f : A \rightarrow A$ is given by: $f(x_0 + 2x_1 + 2^2x_2 + ... + 2^{2n-1}x_{2n-1})=$$(1 - x_0) + 2x_1 + 2^2(1 - x_2) + 2^3x_3 + ... + 2^{2n-1}x_{2n-1}$
for every $0-1$ sequence $(x_0, x_1, . . . , x_{2n-1})$. Show that if $a_1, a_2, . . . , a_9$ are consecutive terms of an arithmetic progression, then the sequence $f(a_1), f(a_2), . . . , f(a_9)$ is not increasing.
2019 China Team Selection Test, 4
Prove that there exist a subset $A$ of $\{1,2,\cdots,2^n\}$ with $n$ elements, such that for any two different non-empty subset of $A$, the sum of elements of one subset doesn't divide another's.
2020 Durer Math Competition Finals, 5
The hexagon $ABCDEF$ has all angles equal . We know that four consecutive sides of the hexagon have lengths $7, 6, 3$ and $5$ in this order. What is the sum of the lengths of the two remaining sides?
2023 Assara - South Russian Girl's MO, 8
The girl continues the sequence of letters $ASSARA... $, adding one of the three letters $A$, $R$ or $S$. When adding the next letter, the girl makes sure that no two written sevens of consecutive letters coincide. At some point it turned out that it was impossible to add a new letter according to these rules. What letter could be written last?
1998 Belarus Team Selection Test, 3
Let $s,t$ be given nonzero integers, $(x,y)$ be any (ordered) pair of integers. A sequence of moves is performed as follows: per move $(x,y)$ changes to $(x+t, y-s)$. The pair (x,y) is said to be [i]good [/i] if after some (may be, zero) number of moves described a pair of integers arises that are not relatively prime.
a) Determine whether $(s,t)$ is itself a good pair;
bj Prove that for any nonzero $s$ and $t$ there is a pair $(x,y)$ which is not good.
2015 ELMO Problems, 1
Define the sequence $a_1 = 2$ and $a_n = 2^{a_{n-1}} + 2$ for all integers $n \ge 2$. Prove that $a_{n-1}$ divides $a_n$ for all integers $n \ge 2$.
[i]Proposed by Sam Korsky[/i]
2014 Singapore Senior Math Olympiad, 34
Let $x_1,x_2,\dots,x_{100}$ be real numbers such that $|x_1|=63$ and $|x_{n+1}|=|x_n+1|$ for $n=1,2\dots,99$.
Find the largest possible value of $(-x_1-x_2-\cdots-x_{100})$.
1966 IMO Longlists, 10
How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?
2012 BMT Spring, 5
Let $p > 1$ be relatively prime to $10$. Let $n$ be any positive number and$ d$ be the last digit of $n$. Define $f(n) = \lfloor \frac{n}{10} \rfloor + d \cdot m$. Then, we can call $m$ a [i]divisibility multiplier[/i] for $p$, if $f(n)$ is divisible by $p$ if and only if $n$ is divisible by $p$. Find a divisibility multiplier for $2013$.
2010 SEEMOUS, Problem 4
Suppose that $A$ and $B$ are $n\times n$ matrices with integer entries, and $\det B\ne0$. Prove that there exists $m\in\mathbb N$ such that the product $AB^{-1}$ can be represented as
$$AB^{-1}=\sum_{k=1}^mN_k^{-1},$$where $N_k$ are $n\times n$ matrices with integer entries for all $k=1,\ldots,m$, and $N_i\ne N_j$ for $i\ne j$.
2021 Science ON all problems, 4
Consider positive real numbers $x,y,z$. Prove the inequality
$$\frac 1x+\frac 1y+\frac 1z+\frac{9}{x+y+z}\ge 3\left (\left (\frac{1}{2x+y}+\frac{1}{x+2y}\right )+\left (\frac{1}{2y+z}+\frac{1}{y+2z}\right )+\left (\frac{1}{2z+x}+\frac{1}{x+2z}\right )\right ).$$
[i] (Vlad Robu \& Sergiu Novac)[/i]
2005 Romania Team Selection Test, 3
Let $\mathbb{N}=\{1,2,\ldots\}$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that for all $m,n\in \mathbb{N}$ the number $f^2(m)+f(n)$ is a divisor of $(m^2+n)^2$.
2004 District Olympiad, 4
Divide a $ 2\times 4 $ rectangle into $ 8 $ unit squares to obtain a set of $ 15 $ vertices denoted by $ \mathcal{M} . $ Find the points $ A\in\mathcal{M} $ that have the property that the set $ \mathcal{M}\setminus \{ A\} $ can form $ 7 $ pairs $ \left( A_1,B_1\right) ,\left( A_2,B_2\right) ,\ldots ,\left( A_7,B_7\right)\in\mathcal{M}\times\mathcal{M} $ such that
$$ \overrightarrow{A_1B_1} +\overrightarrow{A_2B_2} +\cdots +\overrightarrow{A_7B_7} =\overrightarrow{O} . $$
PEN M Problems, 25
Let $\{a_{n}\}_{n \ge 1}$ be a sequence of positive integers such that \[0 < a_{n+1}-a_{n}\le 2001 \;\; \text{for all}\;\; n \in \mathbb{N}.\] Show that there are infinitely many pairs $(p, q)$ of positive integers such that $p>q$ and $a_{q}\; \vert \; a_{p}$.
2021 India National Olympiad, 3
Betal marks $2021$ points on the plane such that no three are collinear, and draws all possible segments joining these. He then chooses any $1011$ of these segments, and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using [b]only[/b] a straightedge. Can Vikram always complete this challenge?
[i]Note.[/i] A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points.
[i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]
2020 Turkey Team Selection Test, 1
Find all pairs of $(a,b)$ positive integers satisfying the equation:
$$\frac {a^3+b^3}{ab+4}=2020$$
2017 ASDAN Math Tournament, 10
Let $\zeta=e^{2\pi i/36}$. Compute
$$\prod_{\stackrel{a=1}{\gcd(a,36)=1}}^{35}(\zeta^a-2).$$
2024 CCA Math Bonanza, I10
Let $ABC$ be a triangle with side lengths $AB = 7$, $BC = 8$, and $CA = 9$. Let $O$ be the circumcenter of $\triangle ABC$, and let $AO$, $BO$, $CO$ intersect the circumcircle of $\triangle ABC$ again at $D$, $E$, and $F$, respectively. The area of convex hexagon $AFBDCE$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is square-free. Find $m + n$.
[i]Individual #10[/i]
2003 Chile National Olympiad, 5
Prove that there is a natural number $N$ of the form $11...1100...00$ which is divisible by $2003$. (The natural numbers are: $1,2,3,...$)
2005 China Western Mathematical Olympiad, 6
In isosceles right-angled triangle $ABC$, $CA = CB = 1$. $P$ is an arbitrary point on the sides of $ABC$. Find the maximum of $PA \cdot PB \cdot PC$.
2005 Alexandru Myller, 4
Let $K$ be a finite field and $f:K\to K^*$. Prove that there is a reducible polynomial $P\in K[X]$ s.t. $P(x)=f(x),\forall x\in K$.
[i]Marian Andronache[/i]
2024 Princeton University Math Competition, 11
Austen has a regular icosahedron ($20$-sided polyhedron with all triangular faces). He randomly chooses $3$ distinct points among the vertices and constructs the circle through these three points. The expected value of the total number of the icosahedron’s vertices that lie on this circle can be written as $m/n$ for relatively prime positive integers $m$ and $n.$ Find $m + n.$
2017 Korea Winter Program Practice Test, 3
Let $\triangle ABC$ be a triangle with $\angle A \neq 60^\circ$. Let $I_B, I_C$ be the $B, C$-excenters of triangle $ABC$, let $B^\prime$ be the reflection of $B$ with respect to $AC$, and let $C^\prime$ be the reflection of $C$ with respect to $AB$. Let $P$ be the intersection of $I_C B^\prime$ and $I_B C^\prime$. Denote by $P_A, P_B, P_C$ the reflections of the point $P$ with respect to $BC, CA, AB$. Show that the three lines $A P_A, B P_B, C P_C$ meet at a single point.
2013 Danube Mathematical Competition, 3
Show that, for every integer $r \ge 2$, there exists an $r$-chromatic simple graph (no loops, nor multiple edges) which has no cycle of less than $6$ edges