Found problems: 85335
1994 Chile National Olympiad, 1
A railway line is divided into ten sections by stations $E_1, E_2,..., E_{11}$. The distance between the first and the last station is $56$ km. A trip through two consecutive stations never exceeds $ 12$ km, and a trip through three consecutive stations is at least $17$ Km. Calculate the distance between $E_2$ and $E_7$.
1973 Putnam, A3
Let $n$ be a fixed positive integer and let $b(n)$ be the minimum value of
$$k+\frac{n}{k},$$
where $k$ is allowed to range through all positive integers. Prove that $\lfloor b(n) \rfloor= \lfloor \sqrt{4n+1} \rfloor.$
2005 Taiwan National Olympiad, 1
$P,Q$ are two fixed points on a circle centered at $O$, and $M$ is an interior point of the circle that differs from $O$. $M,P,Q,O$ are concyclic. Prove that the bisector of $\angle PMQ$ is perpendicular to line $OM$.
2023 Miklós Schweitzer, 9
Let $C[-1,1]$ be the space of continuous real functions on the interval $[-1,1]$ with the usual supremum norm, and let $V{}$ be a closed, finite-codimensional subspace of $C[-1,1].$ Prove that there exists a polynomial $p\in V$ with norm at most one, which satisfies $p'(0)>2023.$
2023 Romanian Master of Mathematics, 1
Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$
1980 VTRMC, 7
Let $S$ be the set of all ordered pairs of integers $(m,n)$ satisfying $m>0$ and $n<0.$ Let $<$ be a partial ordering on $S$ defined by the statement $(m,n)<(m',n')$ if and only if $m\le m'$ and $n\le n'.$ An example is $(5,-10)<(8,-2).$ Now let $O$ be a completely ordered subset of $S,$ in other words if $(a,b)\in O$ and $(c,d) \in O,$ then $(a,b)<(c,d)$ or $(c,d)<(a,b).$ Also let $O'$ denote the collection of all such completely ordered sets.
(a) Determine whether and arbitrary $O\in O'$ is finite.
(b) Determine whether the carnality $|O|$ of $O$ is bounded for $O\in O'.$
(c) Determine whether $|O|$ can be countable infinite for any $O\in O'.$
2002 May Olympiad, 2
Let $k$ be a fixed positive integer, $k \le 10$. Given a list of ten numbers, the allowed operation is: choose $k$ numbers from the list, and add $1$ to each of them. Thus, a new list of ten numbers is obtained. If you initially have the list $1,2,3,4,5,6,7,8,9,10$, determine the values of $k$ for which it is possible, through a sequence of allowed operations, to obtain a list that has the ten equal numbers. In each case indicate the sequence.
2014 Paraguay Mathematical Olympiad, 5
Let $ABC$ be a triangle with area $92$ square centimeters. Calculate the area of another triangle whose sides have the same lengths as the medians of triangle $ABC$.
2024 Junior Balkan Team Selection Tests - Moldova, 12
[b]Version 1.[/b] Find all primes $p$ satisfying the following conditions:
(i) $\frac{p+1}{2}$ is a prime number.
(ii) There are at least three distinct positive integers $n$ for which $\frac{p^2+n}{p+n^2}$ is an integer.
[b]Version 2.[/b] Let $p \neq 5$ be a prime number such that $\frac{p+1}{2}$ is also a prime. Suppose there exist positive integers $a <b$ such that $\frac{p^2+a}{p+a^2}$ and $\frac{p^2+b}{p+b^2}$ are integers. Show that $b=(a-1)^2+1$.
LMT Guts Rounds, 2020 F1
Find the remainder when $2020!$ is divided by $2020^2.$
[i]Proposed by Kevin Zhao[/i]
2009 Germany Team Selection Test, 2
In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.
1984 Czech And Slovak Olympiad IIIA, 3
Let the sequence $\{a_n\}$ , $n \ge 0$ satisfy the recurrence relation
$$a_{n + 2} =4a_{n + 1}-3a_n, \ \ (1) $$
Let us define the sequence $\{b_n\}$ , $n \ge 1$ by the relation
$$b_n= \left[ \frac{a_{n+1}}{a_{n-1}} \right]$$
where we put $b_n =1$ for $a_{n-1}=0$. Prove that, starting from a certain term, the sequence also satisfies the recurrence relation (1).
Note: $[x]$ indicates the whole part of the number $x$.
KoMaL A Problems 2022/2023, A.837
Let all the edges of tetrahedron \(A_1A_2A_3A_4\) be tangent to sphere \(S\). Let \(\displaystyle a_i\) denote the length of the tangent from \(A_i\) to \(S\). Prove that
\[\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg). \]
[i]Submitted by Viktor Vígh, Szeged[/i]
2001 Argentina National Olympiad, 2
Let $\vartriangle ABC$ be a triangle such that angle $\angle ABC$ is less than angle $\angle ACB$. The bisector of angle $\angle BAC$ cuts side $BC$ at $D$. Let $E$ be on side $AB$ such that $\angle EDB = 90^o$ and $F$ on side $AC$ such that $\angle BED = \angle DEF$. Prove that $\angle BAD = \angle FDC$.
2015 BMT Spring, 18
A value $x \in [0, 1]$ is selected uniformly at random. A point $(a, b)$ is called [i]friendly [/i] to $x$ if there exists a circle between the lines $y = 0$ and $y = 1$ that contains both $(a, b)$ and $(0, x)$. Find the area of the region of the plane determined by possible locations of friendly points.
2016 ASDAN Math Tournament, 9
Let $P(x)$ be a monic cubic polynomial. The line $y=0$ and $y=m$ intersect $P(x)$ at points $A,C,E$ and $B,D<F$ from left to right for a positive real number $m$. If $AB=\sqrt{7}$, $CD=\sqrt{15}$, and $EF=\sqrt{10}$, what is the value of $m$?
1968 IMO Shortlist, 9
Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality
\[abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.\]
Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.
2002 AIME Problems, 10
While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\frac{m\pi}{n-\pi}$ and $\frac{p\pi}{q+\pi},$ where $m,$ $n,$ $p$ and $q$ are positive integers. Find $m+n+p+q.$
2000 Taiwan National Olympiad, 3
Consider the set $S=\{ 1,2,\ldots ,100\}$ and the family $\mathcal{P}=\{ T\subset S\mid |T|=49\}$. Each $T\in\mathcal{P}$ is labelled by an arbitrary number from $S$. Prove that there exists a subset $M$ of $S$ with $|M|=50$ such that for each $x\in M$, the set $M\backslash\{ x\}$ is not labelled by $x$.
2005 Moldova Team Selection Test, 4
Given functions $f,g:N^*\rightarrow N^*$, $g$ is surjective and $2f(n)^2=n^2+g(n)^2$, $\forall n>0$. Prove that if $|f(n)-n|\le2005\sqrt n$, $\forall n>0$, then $f(n)=n$ for infinitely many $n$.
1999 Ukraine Team Selection Test, 2
Show that there exist integers $j,k,l,m,n$ greater than $100$ such that $j^2 +k^2 +l^2 +m^2 +n^2 = jklmn-12$.
2016 Denmark MO - Mohr Contest, 1
A class consisting of $24$ students has participated in the first round of the Georg Mohr Contest, where one could obtain between $0$ and $20$ points. Three of the students obtained exactly the class’s average. If each of the students that scored below the average had scored $4$ points more, the average would have been $3$ points higher. How many students scored above the class’s average?
1992 Poland - Second Round, 2
Given a natural number $ n \geq 2 $. Let $ a_1, a_2, \ldots , a_n $, $ b_1, b_2, \ldots , b_n $ be real numbers. Prove that the following conditions are equivalent:
- For any real numbers $ x_1 \leq x_2 \leq \ldots \leq x_n $ holds the inequality
$$\sum_{i=1}^n a_i x_i \leq \sum_{i=1}^n b_i x_i.$$
- For every natural number $ k\in \{1,2,\ldots, n-1\} $ holds the inequality
$$
\sum_{i=1}^k a_i \geq \sum_{i=1}^k b_i, \ \ \text{ and } \\ \ \sum_{i=1}^n a_i = \sum_{i=1 }^n b_i.$$
2010 ELMO Shortlist, 7
The game of circulate is played with a deck of $kn$ cards each with a number in $1,2,\ldots,n$ such that there are $k$ cards with each number. First, $n$ piles numbered $1,2,\ldots,n$ of $k$ cards each are dealt out face down. The player then flips over a card from pile $1$, places that card face up at the bottom of the pile, then next flips over a card from the pile whose number matches the number on the card just flipped. The player repeats this until he reaches a pile in which every card has already been flipped and wins if at that point every card has been flipped. Hamster has grown tired of losing every time, so he decides to cheat. He looks at the piles beforehand and rearranges the $k$ cards in each pile as he pleases. When can Hamster perform this procedure such that he will win the game?
[i]Brian Hamrick.[/i]
2011 Saudi Arabia IMO TST, 1
Find all integers $n$, $n \ge 2$, such that the numbers $1!, 2 !,..., (n - 1)!$ give distinct remainders when divided by $n$.