Found problems: 85335
2014 AMC 8, 17
George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only $2$ miles per hour. At how many miles per hour must George run the last $\frac{1}{2}$ mile in order to arrive just as school begins today?
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad \textbf{(E) }12$
2023/2024 Tournament of Towns, 4
4. There are several (at least two) positive integers written along the circle. For any two neighboring integers one is either twice as big as the other or five times as big as the other. Can the sum of all these integers equal 2023 ?
Sergey Dvoryaninov
2017 EGMO, 2
Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties:
$(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$
$(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$
[i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]
2015 Bosnia Herzegovina Team Selection Test, 5
Let $N$ be a positive integer. It is given set of weights which satisfies following conditions:
$i)$ Every weight from set has some weight from $1,2,...,N$;
$ii)$ For every $i\in {1,2,...,N}$ in given set there exists weight $i$;
$iii)$ Sum of all weights from given set is even positive integer.
Prove that set can be partitioned into two disjoint sets which have equal weight
1997 Brazil Team Selection Test, Problem 2
We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.
2003 Junior Balkan Team Selection Tests - Romania, 1
Consider a rhombus $ABCD$ with center $O$. A point $P$ is given inside the rhombus, but not situated on the diagonals. Let $M,N,Q,R$ be the projections of $P$ on the sides $(AB), (BC), (CD), (DA)$, respectively. The perpendicular bisectors of the segments $MN$ and $RQ$ meet at $S$ and the perpendicular bisectors of the segments $NQ$ and $MR$ meet at $T$. Prove that $P, S, T$ and $O$ are the vertices of a rectangle.
2003 France Team Selection Test, 3
$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.
2008 ITest, 23
Find the number of positive integers $n$ that are solutions to the simultaneous system of inequalities \begin{align*}4n-18&<2008,\\7n+17&>2008.\end{align*}
2021 Malaysia IMONST 1, 17
Determine the sum of all positive integers $n$ that satisfy the following condition:
when $6n + 1$ is written in base $10$, all its digits are equal.
2009 Saint Petersburg Mathematical Olympiad, 5
Call a set of some cells in infinite chess field as board. Set of rooks on the board call as awesome if no one rook can beat another, but every empty cell is under rook attack. There are awesome set with $2008$ rooks and with $2010$ rooks. Prove, that there are awesome set with $2009$ rooks.
1992 China Team Selection Test, 2
A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.
1974 Bundeswettbewerb Mathematik, 4
Peter and Paul gamble as follows. For each natural number, successively, they determine its largest odd divisor and compute its remainder when divided by $4$. If this remainder is $1$, then Peter gives Paul a coin; otherwise, Paul
gives Peter a coin. After some time they stop playing and balance the accounts. Prove that Paul wins.
2022 Kyiv City MO Round 2, Problem 1
Positive reals $x, y, z$ satisfy $$\frac{xy+1}{x+1} = \frac{yz+1}{y+1} = \frac{zx+1}{z+1}$$
Do they all have to be equal?
[i](Proposed by Oleksii Masalitin)[/i]
2007 AMC 8, 7
The average age of $5$ people in a room is $30$ years. An $18$-year-old person leaves
the room. What is the average age of the four remaining people?
$\textbf{(A)}\ 25 \qquad
\textbf{(B)}\ 26 \qquad
\textbf{(C)}\ 29 \qquad
\textbf{(D)}\ 33 \qquad
\textbf{(E)}\ 36$
2006 QEDMO 3rd, 2
Let $ a$, $ b$, $ c$ and $ n$ be positive integers such that $ a^n$ is divisible by $ b$, such that $ b^n$ is divisible by $ c$, and such that $ c^n$ is divisible by $ a$.
Prove that $ \left(a \plus{} b \plus{} c\right)^{n^2 \plus{} n \plus{} 1}$ is divisible by $ abc$.
An even broader [i]generalization[/i], though not part of the QEDMO problem and not quite number theory either:
If $ u$ and $ n$ are positive integers, and $ a_1$, $ a_2$, ..., $ a_u$ are integers such that $ a_i^n$ is divisible by $ a_{i \plus{} 1}$ for every $ i$ such that $ 1\leq i\leq u$ (we set $ a_{u \plus{} 1} \equal{} a_1$ here), then show that $ \left(a_1 \plus{} a_2 \plus{} ... \plus{} a_u\right)^{n^{u \minus{} 1} \plus{} n^{u \minus{} 2} \plus{} ... \plus{} n \plus{} 1}$ is divisible by $ a_1a_2...a_u$.
1987 Tournament Of Towns, (140) 5
A certain number of cubes are painted in six colours, each cube having six faces of different colours (the colours in different cubes may be arranged differently) . The cubes are placed on a table so as to form a rectangle. We are allowed to take out any column of cubes, rotate it (as a whole) along its long axis and replace it in the rectangle. A similar operation with rows is also allowed. Can we always make the rectangle monochromatic (i.e. such that the
top faces of all the cubes are the same colour) by means of such operations?
( D. Fomin , Leningrad)
1951 AMC 12/AHSME, 35
If $ a^x \equal{} c^q \equal{} b$ and $ c^y \equal{} a^z \equal{} d$, then
$ \textbf{(A)}\ xy \equal{} qz \qquad\textbf{(B)}\ \frac {x}{y} \equal{} \frac {q}{z} \qquad\textbf{(C)}\ x \plus{} y \equal{} q \plus{} z \qquad\textbf{(D)}\ x \minus{} y \equal{} q \minus{} z$
$ \textbf{(E)}\ x^y \equal{} q^z$
1997 All-Russian Olympiad, 4
A polygon can be divided into 100 rectangles, but not into 99. Prove that it cannot be divided into 100 triangles.
[i]A. Shapovalov[/i]
Russian TST 2019, P1
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
1989 Tournament Of Towns, (205) 3
What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$?
(M . I. Gusarov)
2025 China National Olympiad, 6
Let $a_1, a_2, \ldots, a_n$ be real numbers such that $\sum_{i=1}^n a_i = n$, $\sum_{i = 1}^n a_i^2 = 2n$, $\sum_{i=1}^n a_i^3 = 3n$.
(i) Find the largest constant $C$, such that for all $n \geqslant 4$, \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C. \]
(ii) Prove that there exists a positive constant $C_2$, such that \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C + C_2 n^{-\frac 32}, \]where $C$ is the constant determined in (i).
2013 District Olympiad, 2
Find all real numbers $x$ for which the number $$a =\frac{2x + 1}{x^2 + 2x + 3}$$ is an integer.
2022 Bulgarian Spring Math Competition, Problem 11.1
Solve the equation
\[(x+1)\log^2_{3}x+4x\log_{3}x-16=0\]
1984 All Soviet Union Mathematical Olympiad, 393
Given three circles $c_1,c_2,c_3$ with $r_1,r_2,r_3$ radiuses, $r_1 > r_2, r_1 > r_3$. Each lies outside of two others. The A point -- an intersection of the outer common tangents to $c_1$ and $c_2$ -- is outside $c_3$. The $B$ point -- an intersection of the outer common tangents to $c_1$ and $c_3$ -- is outside $c_2$. Two pairs of tangents -- from $A$ to $c_3$ and from $B$ to $c_2$ -- are drawn. Prove that the quadrangle, they make, is circumscribed around some circle and find its radius.
2022 JHMT HS, 6
For positive real numbers $a$ and $b,$ let $f(a,b)$ denote the real number $x$ such that area of the (non-degenerate) triangle with side lengths $a,b,$ and $x$ is maximized. Find
\[ \sum_{n=2}^{100}f\left(\sqrt{\tbinom{n}{2}},\sqrt{\tbinom{n+1}{2}}\right). \]