Found problems: 85335
2006 MOP Homework, 4
Determine if there exists a strictly increasing sequence of positive integers $a_1$, $a_2$, ... such that $a_n \le n^3$ for every positive integer $n$ and that every positive integer can be written uniquely as the difference of two terms in the sequence.
2021 MIG, 6
Which of the following choices is an even number?
$\textbf{(A) }2 \cdot 0 + 2 - 1\qquad\textbf{(B) }20 + 21\qquad\textbf{(C) }2^0 - 2 + 1\qquad\textbf{(D) }2 - 0 \cdot 2 + 1\qquad\textbf{(E) }2 \cdot 0 + 2 + 1$
2013 IMO, 1
Assume that $k$ and $n$ are two positive integers. Prove that there exist positive integers $m_1 , \dots , m_k$ such that \[1+\frac{2^k-1}{n}=\left(1+\frac1{m_1}\right)\cdots \left(1+\frac1{m_k}\right).\]
[i]Proposed by Japan[/i]
JOM 2015 Shortlist, G7
Let $ABC$ be an acute triangle. Let $H_A,H_B,H_C$ be points on $BC,AC,AB$ respectively such that $AH_A\perp BC, BH_B\perp AC, CH_C\perp AB$. Let the circumcircles $AH_BH_C,BH_AH_C,CH_AH_B$ be $\omega_A,\omega_B,\omega_C$ with circumcenters $O_A,O_B,O_C$ respectively and define $O_AB\cap \omega_B=P_{AB}\neq B$. Define $P_{AC},P_{BA},P_{BC},P_{CA},P_{CB}$ similarly. Define circles $\omega_{AB},\omega_{AC}$ to be $O_AP_{AB}H_C,O_AP_{AC}H_B$ respectively. Define circles $\omega_{BA},\omega_{BC},\omega_{CA},\omega_{CB}$ similarly.
Prove that there are $6$ pairs of tangent circles in the $6$ circles of the form $\omega_{xy}$.
1986 ITAMO, 3
Two numbers are randomly selected from interval $I = [0, 1]$. Given $\alpha \in I$, what is the probability that the smaller of the two numbers does not exceed $\alpha$?
Is the answer $(100 \alpha)$%, it just seems too easy. :|
2006 MOP Homework, 7
Let $S$ denote the set of rational numbers in the interval $(0,1)$. Determine, with proof, if there exists a subset $T$ of $S$ such that every element in $S$ can be uniquely written as the sum of finitely many distinct elements in $T$.
2001 All-Russian Olympiad, 4
Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.
2020 Novosibirsk Oral Olympiad in Geometry, 5
Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.
2013 National Chemistry Olympiad, 57
Methanol can be gently oxidized with hot copper metal. What is(are) the product(s) of this oxidation?
$ \textbf{(A) }\text{Acetic acid}\qquad\textbf{(B) }\text{Carbon dioxide + Water}\qquad\textbf{(C) }\text{Ethanol} \qquad\textbf{(D) }\text{Methanal} \qquad$
2017 Ecuador Juniors, 3
Given an isosceles triangle $ABC$ with $AB = AC$. Let $O$ be the circumcenter of $ABC$, $D$ the midpoint of $AB$ and $E$ the centroid of $ACD$. Prove that $CD \perp EO$.
2011 Dutch Mathematical Olympiad, 3
In a tournament among six teams, every team plays against each other team exactly once. When a team wins, it receives $3$ points and the losing team receives $0$ points. If the game is a draw, the two teams receive $1$ point each.
Can the final scores of the six teams be six consecutive numbers $a,a +1,...,a + 5$?
If so, determine all values of $a$ for which this is possible.
2024 Romania National Olympiad, 2
Let $(\mathbb{K},+, \cdot)$ be a division ring in which $x^2y=yx^2,$ for all $x,y \in \mathbb{K}.$ Prove that $(\mathbb{K},+, \cdot)$ is commutative.
2016 ITAMO, 2
A mathematical contest had $3$ problems, each of which was given a score between $0$ and $7$ ($0$ and $7$ included). It is known that, for any two contestants, there exists at most one problem in which they have obtained the same score (for example, there are no two contestants whose ordered scores are $7,1,2$ and $7,1,5$, but there might be two contestants whose ordered scores are $7,1,2$ and $7,2,1$). Find the maximum number of contestants.
1956 Moscow Mathematical Olympiad, 336
$64$ non-negative numbers whose sum equals $1956$ are arranged in a square table, eight numbers in each row and each column. The sum of the numbers on the two longest diagonals is equal to $112$. The numbers situated symmetrically with respect to any of the longest diagonals are equal.
(a) Prove that the sum of numbers in any column is less than $1035$.
(b) Prove that the sum of numbers in any row is less than $518$.
2015 Romania National Olympiad, 1
Find all real numbers $x, y,z,t \in [0, \infty)$ so that
$$x + y + z \le t, \,\,\, x^2 + y^2 + z^2 \ge t \,\,\, and \,\,\,x^3 + y^3 + z^3 \le t.$$
2014 HMIC, 2
$2014$ triangles have non-overlapping interiors contained in a circle of radius $1$. What is the largest possible value of the sum of their areas?
Kvant 2019, M2573
Two ants are moving along the edges of a convex polyhedron. The route of every ant ends in its starting point, so that one ant does not pass through the same point twice along its way. On every face $F$ of the polyhedron are written the number of edges of $F$ belonging to the route of the first ant and the number of edges of $F$ belonging to the route of the second ant. Is there a polyhedron and a pair of routes described as above, such that only one face contains a pair of distinct numbers?
[i]Proposed by Nikolai Beluhov[/i]
2014 Bosnia Herzegovina Team Selection Test, 3
Find all nonnegative integer numbers such that $7^x- 2 \cdot 5^y = -1$
2006 Irish Math Olympiad, 2
$ABC$ is a triangle with points $D$, $E$ on $BC$ with $D$ nearer $B$; $F$, $G$ on $AC$, with $F$ nearer $C$; $H$, $K$ on $AB$, with $H$ nearer $A$. Suppose that $AH=AG=1$, $BK=BD=2$, $CE=CF=4$, $\angle B=60^\circ$ and that $D$, $E$, $F$, $G$, $H$ and $K$ all lie on a circle. Find the radius of the incircle of triangle $ABC$.
1934 Eotvos Mathematical Competition, 2
Which polygon inscribed in a given circle has the property that the sum of the squares of the lengths of its sides is maximum?
1999 India Regional Mathematical Olympiad, 5
If $a,b,c$ are sides of a triangle, prove that \[ \frac{a}{c+a-b} + \frac{b}{a+b-c} + \frac{c}{b+c-a} \geq 3. \]
2020 Australian Maths Olympiad, 1
Determine all pairs $(a,b)$ of non-negative integers such that
$$ \frac{a+b}{2}-\sqrt{ab}=1.$$
1979 Austrian-Polish Competition, 1
On sides $AB$ and $BC$ of a square $ABCD$ the respective points $E$ and $F$ have been chosen so that $BE = BF$. Let $BN$ be the altitude in triangle $BCE$. Prove that $\angle DNF = 90$.
1959 Poland - Second Round, 1
What necessary and sufficient condition should the coefficients $ a $, $ b $, $ c $, $ d $ satisfy so that the equation
$$ax^3 + bx^2 + cx + d = 0$$
has two opposite roots?
2016 Danube Mathematical Olympiad, 3
Let $ABC$ be a triangle with $AB < AC,$ $I$ its incenter, and $M$ the midpoint of the side $BC$. If $IA=IM,$ determine the smallest possible value of the angle $AIM$.