Found problems: 85335
2004 Iran MO (3rd Round), 14
We define $ f: \mathbb{N} \rightarrow \mathbb{N}$, $ f(n) \equal{} \sum_{k \equal{} 1}^{n}(k,n)$.
a) Show that if $ \gcd(m,n)\equal{}1$ then we have $ f(mn)\equal{}f(m)\cdot f(n)$;
b) Show that $ \sum_{d|n}f(d) \equal{} nd(n)$.
1989 Cono Sur Olympiad, 2
Find the sum\[1+11+111+\cdots+\underbrace{111\ldots111}_{n\text{ digits}}.\]
2017 Purple Comet Problems, 5
A store had $376$ chocolate bars. Min bought some of the bars, and Max bought $41$ more of the bars than Min bought. After that, the store still had three times as many chocolate bars as Min bought. Find the number of chocolate bars that Min bought.
2002 Tournament Of Towns, 3
In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
1987 IMO Shortlist, 19
Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$ \beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma $ we can construct a triangle and that its area is not greater than
\[A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).\]
[i]Proposed by Soviet Union[/i]
III Soros Olympiad 1996 - 97 (Russia), 10.2
It is known that the equation $x^3 + px^2 + q = 0$ where $q$ is non-zero, has three different integer roots, the absolute values of two of which are prime numbers. Find the roots of this equation.
2021 Romanian Master of Mathematics, 4
Consider an integer \(n \ge 2\) and write the numbers \(1, 2, \ldots, n\) down on a board. A move consists in erasing any two numbers \(a\) and \(b\), then writing down the numbers \(a+b\) and \(\vert a-b \vert\) on the board, and then removing repetitions (e.g., if the board contained the numbers \(2, 5, 7, 8\), then one could choose the numbers \(a = 5\) and \(b = 7\), obtaining the board with numbers \(2, 8, 12\)). For all integers \(n \ge 2\), determine whether it is possible to be left with exactly two numbers on the board after a finite number of moves.
[i]Proposed by China[/i]
2004 Estonia National Olympiad, 1
Find all pairs of integers $(a, b)$ such that $a^2 + ab + b^2 = 1$
2021 Irish Math Olympiad, 8
A point $C$ lies on a line segment $AB$ between $A$ and $B$ and circles are drawn having $AC$ and $CB$ as diameters. A common tangent to both circles touches the circle with $AC$ as diameter at $P \ne C$ and the circle with $CB$ as diameter at $Q \ne C$.
Prove that $AP, BQ$ and the common tangent to both circles at $C$ all meet at a single point which lies on the circumference of the circle with $AB$ as diameter.
2004 Paraguay Mathematical Olympiad, 3
In an equilateral triangle $ABC$, whose side is $4$, the line perpendicular to $AB$ is drawn through the point $ A$, the line perpendicular to $BC$ through point $ B$ and the line perpendicular to $CA$ through point $C$. These three lines determine another triangle. Calculate the perimeter of this triangle
2022 Harvard-MIT Mathematics Tournament, 1
Let $ABC$ be a triangle with $\angle A = 60^o$. Line $\ell$ intersects segments $AB$ and $AC$ and splits triangle $ABC$ into an equilateral triangle and a quadrilateral. Let $X$ and $Y$ be on $\ell$ such that lines $BX$ and $CY$ are perpendicular to ℓ. Given that $AB = 20$ and $AC = 22$, compute $XY$ .
1969 Putnam, B3
The terms of a sequence $(T_n)$ satisfy $T_n T_{n+1} =n$ for all positive integers $n$ and
$$\lim_{n\to \infty} \frac{ T_{n} }{ T_{n+1}}=1.$$
Show that $ \pi T_{1}^{2}=2.$
1988 IMO Longlists, 42
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.
2011 Kurschak Competition, 2
Let $n$ be a positive integer. Denote by $a(n)$ the ways of expression $n=x_1+x_2+\dots$ where $x_1\leqslant x_2 \leqslant\dots$ are positive integers and $x_i+1$ is a power of $2$ for each $i$. Denote by $b(n)$ the ways of expression $n=y_1+y_2+\dots$ where $y_i$ is a positive integer and $2y_i\leqslant y_{i+1}$ for each $i$.
Prove that $a(n)=b(n)$.
2018 Tajikistan Team Selection Test, 9
Problem 9. The numbers 1,2,…,〖97〗^2 are written in the cells of a 97×97 board. In the center of each cell, there is a tower with the height equal to the number of that cell. Is it possible to see the top of any tower from the top of any other tower? (one point A can see the other point B, iff there is no other point on the segment AB).
1962 Miklós Schweitzer, 6
Let $ E$ be a bounded subset of the real line, and let $ \Omega$ be a system of (non degenerate) closed intervals such that for
each $ x \in E$ there exists an $ I \in \Omega$ with left endpoint $ x$. Show that for every $ \varepsilon > 0$ there exists a finite number of pairwise non overlapping intervals belonging to $ \Omega$ that cover $ E$ with the exception of a subset of outer measure less than $ \varepsilon$. [J. Czipszer]
2020 Canadian Mathematical Olympiad Qualification, 6
In convex pentagon $ABCDE, AC$ is parallel to $DE, AB$ is perpendicular to $AE$, and $BC$ is perpendicular to $CD$. If $H$ is the orthocentre of triangle $ABC$ and $M$ is the midpoint of segment $DE$, prove that $AD, CE$ and $HM$ are concurrent.
2000 AMC 12/AHSME, 6
Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
$ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$
1991 Arnold's Trivium, 90
Calculate the sum of matrix commutators $[A, [B, C]] + [B, [C, A]] + [C, [A, B]]$, where $[A, B] = AB-BA$
1982 IMO Longlists, 38
Numbers $u_{n,k} \ (1\leq k \leq n)$ are defined as follows
\[u_{1,1}=1, \quad u_{n,k}=\binom{n}{k} - \sum_{d \mid n, d \mid k, d>1} u_{n/d, k/d}.\]
(the empty sum is defined to be equal to zero). Prove that $n \mid u_{n,k}$ for every natural number $n$ and for every $k \ (1 \leq k \leq n).$
2011 VJIMC, Problem 2
Let $(a_n)^\infty_{n=1}$ be an unbounded and strictly increasing sequence of positive reals such that the arithmetic mean of any four consecutive terms $a_n,a_{n+1},a_{n+2},a_{n+3}$ belongs to the same sequence. Prove that the sequence $\frac{a_{n+1}}{a_n}$ converges and find all possible values of its limit.
2017 ELMO Shortlist, 2
Let $ABC$ be a scalene triangle with $\angle A = 60^{\circ}$. Let $E$ and $F$ be the feet of the angle bisectors of $\angle ABC$ and $\angle ACB$, respectively, and let $I$ be the incenter of $\triangle ABC$. Let $P,Q$ be distinct points such that $\triangle PEF$ and $\triangle QEF$ are equilateral. If $O$ is the circumcenter of of $\triangle APQ$, show that $\overline{OI}\perp \overline{BC}$.
[i]Proposed by Vincent Huang
1997 AMC 12/AHSME, 4
If $ a$ is $ 50\%$ larger than $ c$, and $ b$ is $ 25\%$ larger than $ c$,then $ a$ is what percent larger than $ b$?
$ \textbf{(A)}\ 20\%\qquad \textbf{(B)}\ 25\%\qquad \textbf{(C)}\ 50\%\qquad \textbf{(D)}\ 100\%\qquad \textbf{(E)}\ 200\%$
2015 Kosovo Team Selection Test, 1
a)Prove that for every n,natural number exist natural numbers a and b such that
$(1-\sqrt{2})^n=a-b\sqrt{2}$ and $a^2-2b^2=(-1)^n$
b)Using first equation prove that for every n exist m such that
$(\sqrt{2}-1)^n=\sqrt{m}-\sqrt{m-1}$
1997 May Olympiad, 1
On a square board with $9$ squares (three by three), nine elements of the set $S=\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ must be placed, different from each other, so that each one is in a box and the following conditions are met:
$\bullet$ The sums of the numbers in the second and third rows are, respectively, double and triple the sum of the numbers in the first row.
$\bullet$ The sum of the numbers in the second and third columns are, respectively, double and triple the sum of the numbers in the first column.
Show all the possible ways to place elements of $S$ on the board, fulfilling the indicated conditions.