Found problems: 85335
Kvant 2021, M2670
There are 100 points on the plane so that any 10 of them are vertices of a convex polygon. Does it follow from this that all these points are the vertices of a convex 100-gon?
[i]From the folklore[/i]
2016 Postal Coaching, 5
Is it possible to define an operation $\star$ on $\mathbb Z$ such that[list=a][*] for any $a, b, c$ in $\mathbb Z, (a \star b) \star c = a \star (b \star c)$ holds;
[*] for any $x, y$ in $\mathbb Z, x \star x \star y = y \star x \star x=y$?[/list]
1998 Tournament Of Towns, 5
Pinocchio claims that he can divide an isoceles triangle into three triangles, any two of which can be put together to form a new isosceles triangle. Is Pinocchio lying?
(A Shapovalov)
2015 Estonia Team Selection Test, 10
Let $n$ be an integer and $a, b$ real numbers such that $n > 1$ and $a > b > 0$. Prove that $$(a^n - b^n) \left ( \frac{1}{b^{n- 1}} - \frac{1}{a^{n -1}}\right) > 4n(n -1)(\sqrt{a} - \sqrt{b})^2$$
2019 Novosibirsk Oral Olympiad in Geometry, 7
Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$
2020 Saint Petersburg Mathematical Olympiad, 5.
The altitudes $BB_1$ and $CC_1$ of the acute triangle $\triangle ABC$ intersect at $H$. The circle centered at $O_b$ passes through points $A,C_1$, and the midpoint of $BH$. The circle centered at $O_c$ passes through $A,B_1$ and the midpoint of $CH$. Prove that $B_1 O_b +C_1O_c > \frac{BC}{4}$
2025 China National Olympiad, 5
Let $p$ be a prime number and $f$ be a bijection from $\left\{0,1,\ldots,p-1\right\}$ to itself. Suppose that for integers $a,b \in \left\{0,1,\ldots,p-1\right\}$, $|f(a) - f(b)|\leqslant 2024$ if $p \mid a^2 - b$. Prove that there exists infinite many $p$ such that there exists such an $f$ and there also exists infinite many $p$ such that there doesn't exist such an $f$.
2015 District Olympiad, 1
On a blackboard there are written the numbers $ 11 $ and $ 13. $ One [i]step[/i] means to sum two written numbers and write it. Show that:
[b]a)[/b] after any number of steps, the number $ 86 $ will not be written.
[b]b)[/b] after some number of steps, the number $ 2015 $ may be written.
2015 IMO Shortlist, N8
For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define
\[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\]
That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity.
Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that
\[\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.\]
[i]Proposed by Rodrigo Sanches Angelo, Brazil[/i]
2017 MIG, 4
Percy buys $3$ apples for $6$ dollars, $4$ pears for $16$ dollars, and $1$ watermelon for $5$ dollars. Assuming the rates stay the same, how much would it cost to buy $10$ apples, $3$ pears, and $2$ watermelons?
$\textbf{(A) } 38\qquad\textbf{(B) } 39\qquad\textbf{(C) } 40\qquad\textbf{(D) } 41\qquad\textbf{(E) } 42$
2016 Postal Coaching, 1
If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x), f(x)g(x), f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3 - 3x^2 + 5$ and $x^2 - 4x$ are written on the blackboard. Can we write a nonzero polynomial of the form $x^n - 1$ after a finite number of steps? Justify your answer.
2013 Purple Comet Problems, 6
Pete's research shows that the number of nuts collected by the squirrels in any park is proportional to the square of the number of squirrels in that park. If Pete notes that four squirrels in a park collect $60$ nuts, how many nuts are collected by $20$ squirrels in a park?
2002 China Team Selection Test, 2
$ m$ and $ n$ are positive integers. In a $ 8 \times 8$ chessboard, $ (m,n)$ denotes the number of grids a Horse can jump in a chessboard ($ m$ horizontal $ n$ vertical or $ n$ horizontal $ m$ vertical ). If a $ (m,n) \textbf{Horse}$ starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a $ \textbf{H Route}$. For example, the $ (1,2) \textbf{Horse}$ has its $ \textbf{H Route}$. Find the smallest positive integer $ t$, such that from any grid of the chessboard, the $ (t,t\plus{}1) \textbf{Horse}$ does not has any $ \textbf{H Route}$.
2019 Azerbaijan Senior NMO, 2
A positive number $a$ is given, such that $a$ could be expressed as difference of two inverses of perfect squares ($a=\frac1{n^2}-\frac1{m^2}$). Is it possible for $2a$ to be expressed as difference of two perfect squares?
2013 Sharygin Geometry Olympiad, 5
Points $E$ and $F$ lie on the sides $AB$ and $AC$ of a triangle $ABC$. Lines $EF$ and $BC$ meet at point $S$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. The line passing through $A$ and parallel to $MN$ meets $BC$ at point $K$. Prove that $\frac{BK}{CK}=\frac{FS}{ES}$ .
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2010 AMC 8, 8
As Emily is riding her bike on a long straight road, she spots Ermenson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour. Ermenson skates at a constant rate of $8$ miles per hour. For how many minutes can Emily see Ermenson?
$ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 16 $
1977 IMO Longlists, 3
In a company of $n$ persons, each person has no more than $d$ acquaintances, and in that company there exists a group of $k$ persons, $k\ge d$, who are not acquainted with each other. Prove that the number of acquainted pairs is not greater than $\left[ \frac{n^2}{4}\right]$.
2024 Argentina Iberoamerican TST, 2
On a $5 \times 5$ board, pieces made up of $4$ squares are placed, as seen in the figure, each covering exactly $4$ squares of the board. The pieces can be rotated or turned over. They can also overlap, but they cannot protrude from the board. Suppose that each square on the board is covered by at most two pieces. Determine the maximum number of squares on the board that can be covered (by one or two pieces).
[asy]
size(3cm);
draw((0,0)--(0,1)--(1,1)--(1,0)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,2)--(2,2)--(2,1)--(3,1)--(3,2)--(2,2));
[/asy]
2024 Harvard-MIT Mathematics Tournament, 7
Positive integers $a, b,$ and $c$ have the property that $a^b, b^c,$ and $c^a$ end in $4, 2,$ and $9,$ respectively. Compute the minimum possible value of $a+b+c.$
2007 South africa National Olympiad, 1
Determine whether $ \frac{1}{\sqrt{2}} \minus{} \frac{1}{\sqrt{6}}$ is less than or greater than $ \frac{3}{10}$.
1993 China Team Selection Test, 2
Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$
2011 National Olympiad First Round, 10
How many interger tuples $(x,y,z)$ are there satisfying $0\leq x,y,z < 2011$, $xy+yz+zx \equiv 0 \pmod{2011}$, and $x+y+z \equiv 0 \pmod{2011}$ ?
$\textbf{(A)}\ 2010 \qquad\textbf{(B)}\ 2011 \qquad\textbf{(C)}\ 2012 \qquad\textbf{(D)}\ 4021 \qquad\textbf{(E)}\ 4023$
2017 Online Math Open Problems, 30
Let $p = 2017$ be a prime. Given a positive integer $n$, let $T$ be the set of all $n\times n$ matrices with entries in $\mathbb{Z}/p\mathbb{Z}$. A function $f:T\rightarrow \mathbb{Z}/p\mathbb{Z}$ is called an $n$-[i]determinant[/i] if for every pair $1\le i, j\le n$ with $i\not= j$, \[f(A) = f(A'),\] where $A'$ is the matrix obtained by adding the $j$th row to the $i$th row.
Let $a_n$ be the number of $n$-determinants. Over all $n\ge 1$, how many distinct remainders of $a_n$ are possible when divided by $\dfrac{(p^p - 1)(p^{p - 1} - 1)}{p - 1}$?
[i]Proposed by Ashwin Sah[/i]
2021 USAMO, 4
A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.)
Given this information, find all possible values for the number of elements of $S$.
2011 Belarus Team Selection Test, 2
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]