Found problems: 85335
2024 Thailand October Camp, 1
In a test, $201$ students are trying to solve $6$ problems.We know that for each of $5$ first problems, there are at least $140$ students, who can solve it. Moreover, there is exactly $60$ students, who can solve $6^{th}$ problem. Show that there exist $2$ students, such that two of them combined are able to solve all $6$ question. (For example, number $1$ do $1,2,3,4$ and number $2$ do $3,5,6$)
1976 Dutch Mathematical Olympiad, 3
In how many ways can the king in the chessboard reach the eighth rank in $7$ moves from its original square on the first row?
2015 Chile TST Ibero, 2
In the country of Muilejistan, there exists a network of roads connecting all its cities. The network has the particular property that for any two cities, there is a unique path without backtracking (i.e., a path where the traveler never returns along the same road).
The longest possible path between two cities is 600 kilometers. For instance, the path from the city of Mlar to the city of Nlar is 600 kilometers. Similarly, the path from the city of Klar to the city of Glar is also 600 kilometers.
1. If Jalim departs from Mlar towards Nlar at noon and Kalim departs from Klar towards Glar also at noon, both traveling at the same speed, prove that they meet at some point on their journey.
2. If the distance in kilometers between any two cities is an integer, prove that the distance from Glar to Mlar is even.
2006 Irish Math Olympiad, 1
The rooms of a building are arranged in a $m\times n$ rectangular grid (as shown below for the $5\times 6$ case). Every room is connected by an open door to each adjacent room, but the only access to or from the building is by a door in the top right room. This door is locked with an elaborate system of $mn$ keys, one of which is located in every room of the building. A person is in the bottom left room and can move from there to any adjacent room. However, as soon as the person leaves a room, all the doors of that room are instantly and automatically locked. Find, with proof, all $m$ and $n$ for which it is possible for the person to collect all the keys and escape the building.
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label("starting position",(15.3,3));
[/asy]
2022 Saudi Arabia BMO + EGMO TST, p3
We consider all partitions of a positive integer n into a sum of (nonnegative integer) exponents of $2$ (i.e. $1$, $2$, $4$, $8$ , $ . . .$ ). A number in the sum is allowed to repeat an arbitrary number of times (e.g. $7 = 2 + 2 + 1 + 1 + 1$) and two partitions differing only in the order of summands are considered to be equal (e.g. $8 = 4 + 2 + 1 + 1$ and $8 = 1 + 2 + 1 + 4$ are regarded to be the same partition). Let $E(n)$ be the number of partitions in which an even number of exponents appear an odd number of times and $O(n)$ the number of partitions in which an odd number of exponents appear an odd number of times. For example, for $n = 5$ partitions counted in $E(n)$ are $5 = 4 + 1$ and $5 = 2 + 1 + 1 + 1$, whereas partitions counted in O(n) are $5 = 2 + 2 + 1$ and $5 = 1 + 1 + 1 + 1 + 1$, hence $E(5) = O(5) = 2$. Find $E(n) - O(n)$ as a function of $n$.
2022 Iran-Taiwan Friendly Math Competition, 2
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that:
$\bullet$ $f(x)<2$ for all $x\in (0,1)$;
$\bullet$ for all real numbers $x,y$ we have:
$$max\{f(x+y),f(x-y)\}=f(x)+f(y)$$
Proposed by Navid Safaei
2016 Tuymaada Olympiad, 5
The ratio of prime numbers $p$ and $q$ does not exceed 2 ($p\ne q$).
Prove that there are two consecutive positive integers such that
the largest prime divisor of one of them is $p$ and that of the other is $q$.
2018 HMNT, 2
Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain?
1954 Putnam, B4
Given the focus $F$ and the directrix $D$ of a parabola $P$ and a line $L$, describe a euclidean construction for the point or points of intersection of $P$ and $L.$ Be sure to identify the case for which there are no points of intersection.
2020 Taiwan TST Round 1, 6
There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.
In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps:
(a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$.
(b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group.
Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning.
[i]Czech Republic[/i]
2021 Winter Stars of Mathematics, 3
Determine all integers $n>1$ whose positive divisors add up to a power of $3.$
[i]Andrei Bâra[/i]
1976 IMO Longlists, 17
Show that there exists a convex polyhedron with all its vertices on the surface of a sphere and with all its faces congruent isosceles triangles whose ratio of sides are $\sqrt{3} :\sqrt{3} :2$.
2013 Bosnia Herzegovina Team Selection Test, 2
The sequence $a_n$ is defined by $a_0=a_1=1$ and $a_{n+1}=14a_n-a_{n-1}-4$,for all positive integers $n$.
Prove that all terms of this sequence are perfect squares.
STEMS 2024 Math Cat B, P3
Let $r$, $s$ be real numbers, find maximum $t$ so that if $a_1, a_2, \ldots$ is a sequence of positive real numbers satisfying
\[ a_1^r + a_2^r + \cdots + a_n^r \le 2023 \cdot n^t \]
for all $n \ge 2023$ then the sum
\[ b_n = \frac 1{a_1^s} + \cdots + \frac 1{a_n^s} \]
is unbounded, i.e for all positive reals $M$ there is an $n$ such that $b_n > M$.
1998 Romania National Olympiad, 4
Let $ABCD$ be an arbitrary tetrahedron. The bisectors of the angles $\angle BDC$, $\angle CDA$ and $\angle ADB$ intersect $BC$, $CA$ and $AB$, in the points $M$, $N$, $P$, respectively.
a) Show that the planes $(ADM)$, $(BDN)$ and $(CDP)$ have a common line $d$.
b) Let the points $A' \in (AD)$, $B' \in (BD)$ and $C' \in (CD)$ be such that $(AA') = (BB') = (CC')$ ; show that if $G$ and $G'$ are the centroids of $ABC$ and $A'B'C'$ then the lines $GG'$ and $d$ are either parallel or identical.
2009 Philippine MO, 2
[b](a)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n + 1 = x^2$.
[b](b)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n = x^2 + 1$.
1992 IMO Longlists, 28
Two circles $ \Omega_{1}$ and $ \Omega_{2}$ are externally tangent to each other at a point $ I$, and both of these circles are tangent to a third circle $ \Omega$ which encloses the two circles $ \Omega_{1}$ and $ \Omega_{2}$.
The common tangent to the two circles $ \Omega_{1}$ and $ \Omega_{2}$ at the point $ I$ meets the circle $ \Omega$ at a point $ A$. One common tangent to the circles $ \Omega_{1}$ and $ \Omega_{2}$ which doesn't pass through $ I$ meets the circle $ \Omega$ at the points $ B$ and $ C$ such that the points $ A$ and $ I$ lie on the same side of the line $ BC$.
Prove that the point $ I$ is the incenter of triangle $ ABC$.
[i]Alternative formulation.[/i] Two circles touch externally at a point $ I$. The two circles lie inside a large circle and both touch it. The chord $ BC$ of the large circle touches both smaller circles (not at $ I$). The common tangent to the two smaller circles at the point $ I$ meets the large circle at a point $ A$, where the points $ A$ and $ I$ are on the same side of the chord $ BC$. Show that the point $ I$ is the incenter of triangle $ ABC$.
1995 China National Olympiad, 2
Let $a_1,a_2,\cdots ,a_{10}$ be pairwise distinct natural numbers with their sum equal to 1995. Find the minimal value of $a_1a_2+a_2a_3+\cdots +a_9a_{10}+a_{10}a_1$.
1969 IMO Longlists, 42
$(MON 3)$ Let $A_k (1 \le k \le h)$ be $n-$element sets such that each two of them have a nonempty intersection. Let $A$ be the union of all the sets $A_k,$ and let $B$ be a subset of $A$ such that for each $k (1\le k \le h)$ the intersection of $A_k$ and $B$ consists of exactly two different elements $a_k$ and $b_k$. Find all subsets $X$ of the set $A$ with $r$ elements satisfying the condition that for at least one index $k,$ both elements $a_k$ and $b_k$ belong to $X$.
2009 Sharygin Geometry Olympiad, 4
Let $ P$ and $ Q$ be the common points of two circles. The ray with origin $ Q$ reflects from the first circle in points $ A_1$, $ A_2$,$ \ldots$ according to the rule ''the angle of incidence is equal to the angle of reflection''. Another ray with origin $ Q$ reflects from the second circle in the points $ B_1$, $ B_2$,$ \ldots$ in the same manner. Points $ A_1$, $ B_1$ and $ P$ occurred to be collinear. Prove that all lines $ A_iB_i$ pass through P.
2001 China Team Selection Test, 1
For a given natural number $n > 3$, the real numbers $x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}$ satisfy the conditions $0
< x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}$. Find the minimum possible value of
\[\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j +
1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k
x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l
x_{l + 1}})}\] and find all $(n + 2)$-tuplets of real numbers $(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2})$ which gives this value.
2008 Harvard-MIT Mathematics Tournament, 3
A $ 3\times3\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $ 3\times3\times1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $ 45^\circ$ angle with the horizontal plane.
PEN G Problems, 10
Show that $\frac{1}{\pi} \arccos \left( \frac{1}{\sqrt{2003}} \right)$ is irrational.
1994 Tournament Of Towns, (422) 3
Find five positive integers such that the greatest common divisor of each pair is equal to the difference between them.
(SI Tokarev)
2021 Cono Sur Olympiad, 1
We say that a positive integer is guarani if the sum of the number with its reverse is a number that only has odd digits. For example, $249$ and $30$ are guarani, since $249 + 942 = 1191$ and $30 + 03 = 33$.
a) How many $2021$-digit numbers are guarani?
b) How many $2023$-digit numbers are guarani?