Found problems: 85335
2002 Mongolian Mathematical Olympiad, Problem 2
For a natural number $p$, one can move between two points with integer coordinates if the distance between them equals $p$. Find all prime numbers $p$ for which it is possible to reach the point $(2002,38)$ starting from the origin $(0,0)$.
2015 IMO Shortlist, G7
Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.
2009 Hanoi Open Mathematics Competitions, 2
Show that there is a natural number $n$ such that the number $a = n!$ ends exactly in $2009$ zeros.
Durer Math Competition CD Finals - geometry, 2018.C+2
Given an $ABC$ triangle. Let $D$ be an extension of section $AB$ beyond $A$ such that that $AD = BC$ and $E$ is the extension of the section $BC$ beyond $B$ such that $BE = AC$. Prove that the circumcircle of triangle $DEB$ passes through the center of the inscribed circle of triangle $ABC$.
2021 Poland - Second Round, 6
Let $p\ge 5$ be a prime number. Consider the function given by the formula $$f (x_1,..., x_p) = x_1 + 2x_2 +... + px_p.$$
Let $A_k$ denote the set of all these permutations $(a_1,..., a_p)$ of the set $\{1,..., p\}$, for integer number $f (a_1,..., a_p) - k$ is divisible by $p$ and $a_i \ne i$ for all $i \in \{1,..., p\}$. Prove that the sets $A_1$ and $A_4$ have the same number of elements.
2018 Kyiv Mathematical Festival, 4
Do there exist positive integers $a$ and $b$ such that each of the numbers $2^a+3^b,$ $3^a+5^b$ and $5^a+2^b$ is divisible by 29?
1996 AMC 8, 15
The remainder when the product $1492\cdot 1776\cdot 1812\cdot 1996$ is divided by $5$ is
$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4$
2008 Harvard-MIT Mathematics Tournament, 10
([b]8[/b]) Evaluate the integral $ \int_0^1\ln x \ln(1\minus{}x)\ dx$.
LMT Team Rounds 2010-20, 2020.S28
A particular country has seven distinct cities, conveniently named $C_1,C_2,\dots,C_7.$ Between each pair of cities, a direction is chosen, and a one-way road is constructed in that direction connecting the two cities. After the construction is complete, it is found that any city is reachable from any other city, that is, for distinct $1 \leq i, j \leq 7,$ there is a path of one-way roads leading from $C_i$ to $C_j.$ Compute the number of ways the roads could have been configured. Pictured on the following page are the possible configurations possible in a country with three cities, if every city is reachable from every other city.
[Insert Diagram]
[i]Proposed by Ezra Erives[/i]
2019 ASDAN Math Tournament, 3
Consider an equilateral triangle $\vartriangle ABC$ with side length $1$. Let $D$ and $E$ lie on segments $AB$ and $AC$ respectively such that $\angle ADE = 30^o$ and $DE$ is tangent to the incircle of $\vartriangle ABC$. Compute the perimeter of $\vartriangle ADE$.
1999 Slovenia National Olympiad, Problem 3
Let $O$ be the circumcenter of a triangle $ABC$, $P$ be the midpoint of $AO$, and $Q$ be the midpoint of $BC$. If $\angle ABC=4\angle OPQ$ and $\angle ACB=6\angle OPQ$, compute $\angle OPQ$.
2008 Junior Balkan Team Selection Tests - Romania, 2
Let $ a,b,c$ be positive reals with $ ab \plus{} bc \plus{} ca \equal{} 3$. Prove that:
\[ \frac {1}{1 \plus{} a^2(b \plus{} c)} \plus{} \frac {1}{1 \plus{} b^2(a \plus{} c)} \plus{} \frac {1}{1 \plus{} c^2(b \plus{} a)}\le \frac {1}{abc}.
\]
1966 IMO Longlists, 7
For which arrangements of two infinite circular cylinders does their intersection lie in a plane?
2011 China Team Selection Test, 2
Let $n$ be a positive integer and let $\alpha_n $ be the number of $1$'s within binary representation of $n$.
Show that for all positive integers $r$,
\[2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.\]
OIFMAT II 2012, 2
Find all functions $ f: N \rightarrow N $ such that:
$\bullet$ $ f (m) = 1 \iff m = 1 $;
$\bullet$ If $ d = \gcd (m, n) $, then $ f (mn) = \frac {f (m) f (n)} {f (d)} $; and
$\bullet$ $ \forall m \in N $, we have $ f ^ {2012} (m) = m $.
Clarification: $f^n (a) = f (f^{n-1} (a))$
2012 Balkan MO Shortlist, A6
Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.
2012 Argentina National Olympiad Level 2, 1
For each natural number $x$, let $S(x)$ be the sum of its digits. Find the smallest natural number $n$ such that $9S(n) = 16S(2n)$.
1947 Putnam, A1
If $(a_n)$ is a sequence of real numbers such that for $n \geq 1$
$$(2-a_n )a_{n+1} =1,$$
prove that $\lim_{n\to \infty} a_n =1.$
I Soros Olympiad 1994-95 (Rus + Ukr), 9.5
Kolya and Vasya each have $8$ cards with numbers from $1$ to $8$ (each has all the numbers from $1$ to $8$). Kolya put $4$ cards on the table, and Vasya put a card with a larger number on each of them. Now Vasya puts his remaining $4$ cards on the table.
a) Can Kolya always put his own card with a larger number on each of Vasya’s cards?
b) Can Kolya always put on each of Vasya’s cards his own card with a number no less than on Vasya’s card?
2018 Singapore Senior Math Olympiad, 5
Starting with any $n$-tuple $R_0$, $n\ge 1$, of symbols from $A,B,C$, we define a sequence $R_0, R_1, R_2,\ldots,$ according to the following rule: If $R_j= (x_1,x_2,\ldots,x_n)$, then $R_{j+1}= (y_1,y_2,\ldots,y_n)$, where $y_i=x_i$ if $x_i=x_{i+1}$ (taking $x_{n+1}=x_1$) and $y_i$ is the symbol other than $x_i, x_{i+1}$ if $x_i\neq x_{i+1}$. Find all positive integers $n>1$ for which there exists some integer $m>0$ such that $R_m=R_0$.
2022 Dutch IMO TST, 2
Let $n > 1$ be an integer. There are $n$ boxes in a row, and there are $n + 1$ identical stones. A [i]distribution [/i] is a way to distribute the stones over the boxes, in which every stone is in exactly one of the boxes. We say that two of such distributions are a [i]stone’s throw away[/i] from each other if we can obtain one distribution from the other by moving exactly one stone from one box to another. The [i]cosiness [/i] of a distribution $a$ is defined as the number of distributions that are a stone’s throw away from $a$. Determine the average cosiness of all possible distributions.
2016 Croatia Team Selection Test, Problem 4
Find all pairs $(p,q)$ of prime numbers such that
$$ p(p^2 - p - 1) = q(2q + 3) .$$
2011 NIMO Problems, 10
The edges and diagonals of convex pentagon $ABCDE$ are all colored either red or blue. How many ways are there to color the segments such that there is exactly one monochromatic triangle with vertices among $A$, $B$, $C$, $D$, $E$; that is, triangles, whose edges are all the same color?
[i]Proposed by Eugene Chen
[/i]
1973 IMO Shortlist, 1
Let a tetrahedron $ABCD$ be inscribed in a sphere $S$. Find the locus of points $P$ inside the sphere $S$ for which the equality
\[\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4\]
holds, where $A_1,B_1, C_1$, and $D_1$ are the intersection points of $S$ with the lines $AP,BP,CP$, and $DP$, respectively.
1992 Tournament Of Towns, (323) 4
A circle is divided into $7$ arcs. The sum of the angles subtending any two neighbouring arcs is no more than $103^o$. Find the maximal number $A$ such that any of the $7$ arcs is subtended by no less than $A^o$. Prove that this value $A$ is really maximal.
(A. Tolpygo, Kiev)