Found problems: 85335
1996 Czech and Slovak Match, 2
Let ⋆ be a binary operation on a nonempty set $M$. That is, every pair $(a,b) \in M$ is assigned an element $a$ ⋆$ b$ in $M$. Suppose that ⋆ has the additional property that $(a $ ⋆ $b) $ ⋆$ b= a$ and $a$ ⋆ $(a$ ⋆$ b)= b$ for all $a,b \in M$.
(a) Show that $a$ ⋆ $b = b$ ⋆ $a$ for all $a,b \in M$.
(b) On which finite sets $M$ does such a binary operation exist?
1987 IMO Longlists, 8
Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.
[hide="Note"]Note. Here (and generally in MathLinks) natural numbers supposed to be positive.[/hide]
2006 Swedish Mathematical Competition, 6
Determine all positive integers $a,b,c$ satisfying $a^{(b^c)}=(b^a)^c$
2012 BMT Spring, round 5
[b]p1.[/b] Let $n$ be the number so that $1 - 2 + 3 - 4 + ... - (n - 1) + n = 2012$. What is $4^{2012}$ (mod $n$)?
[b]p2. [/b]Consider three unit squares placed side by side. Label the top left vertex $P$ and the bottom four vertices $A,B,C,D$ respectively. Find $\angle PBA + \angle PCA + \angle PDA$.
[b]p3.[/b] Given $f(x) = \frac{3}{x-1}$ , then express $\frac{9(x^2-2x+1)}{x^2-8x+16}$ entirely in terms of $f(x)$. In other words, $x$ should not be in
your answer, only $f(x)$.
[b]p4.[/b] Right triangle with right angle $B$ and integer side lengths has $BD$ as the altitude. $E$ and $F$ are the incenters of triangles $ADB$ and $BDC$ respectively. Line $EF$ is extended and intersects $BC$ at $G$, and $AB$ at $H$. If $AB = 15$ and $BC = 8$, find the area of triangle $BGH$.
[b]p5.[/b] Let $a_1, a_2, ..., a_n$ be a sequence of real numbers. Call a $k$-inversion $(0 < k\le n)$ of a sequence to be indices $i_1, i_2, .. , i_k$ such that $i_1 < i_2 < .. < i_k$ but $a_{i1} > a_{i2} > ...> a_{ik}$ . Calculate the expected number of $6$-inversions in a random permutation of the set $\{1, 2, ... , 10\}$.
[b]p6.[/b] Chell is given a strip of squares labeled $1, .. , 6$ all placed side to side. For each $k \in {1, ..., 6}$, she then chooses one square at random in $\{1, ..., k\}$ and places a Weighted Storage Cube there. After she has placed all $6$ cubes, she computes her score as follows: For each square, she takes the number of cubes in the pile and then takes the square (i.e. if there were 3 cubes in a square, her score for that square would be $9$). Her overall score is the sum of the scores of each square. What is the expected value of her score?
PS. You had better use hide for answers.
2012 ISI Entrance Examination, 6
[b]i)[/b] Let $0<a<b$.Prove that amongst all triangles having base $a$ and perimeter $a+b$ the triangle having two sides(other than the base) equal to $\frac {b}{2}$ has the maximum area.
[b]ii)[/b]Using $i)$ or otherwise, prove that amongst all quadrilateral having give perimeter the square has the maximum area.
2000 National High School Mathematics League, 9
If $a+\log_2 3,a+\log_4 3,a+\log_8 3$ are a geometric series, then the common ratio is________.
I Soros Olympiad 1994-95 (Rus + Ukr), 11.8
Let's write down a segment of a series of integers from $0$ to $1995$. Among the numbers written out, two have been crossed out. Let's consider the longest arithmetic progression contained among the remaining $1994$ numbers. Let $K$ be the length of the progression. Which two numbers must be crossed out so that the value of $K$ is the smallest?
2004 Abels Math Contest (Norwegian MO), 1a
If $m$ is a positive integer, prove that $2^m$ cannot be written as a sum of two or more consecutive natural numbers.
2020-21 IOQM India, 3
If $\sum_{k=1}^{N} \frac{2k+1}{(k^2+k)^2}= 0.9999$ then determine the value of $N$.
2000 Stanford Mathematics Tournament, 5
Find the interior angle between two sides of a regular octagon (degrees).
2006 Princeton University Math Competition, 9
Consider all line segments of length $4$ with one end-point on the line $y = x$ and the other end-point on the line $y = 2x$. Find the equation of the locus of the midpoints of these line segments.
1959 Miklós Schweitzer, 9
[b]9.[/b] Let $f(z)= z^n +a_1 z^{n-1}+\dots + a_n$ be a polynomial over the field of the complex numbers and let $E_f$ denote the closed (not necessarily connected) domain of complex numbers $z$ for which $\mid f(z) \mid \leq 1$. Show that there exists a point $z_0 \in E_f$ such that $\mid f'(z_0) \mid \geq n$. [b](F. 5)[/b]
1911 Eotvos Mathematical Competition, 3
Prove that $3^n + 1$ is not divisible by $2^n$ for any integer $n > 1$.
2008 Grigore Moisil Intercounty, 3
Let $ f[0,\infty )\longrightarrow\mathbb{R} $ be a convex and differentiable function with $ f(0)=0. $
[b]a)[/b] Prove that $ \int_0^x f(t)dt\le \frac{x^2}{2}f'(x) , $ for any nonnegative $ x. $
[b]b)[/b] Determine $ f $ if the above inequality is actually an equality.
[i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]
2019 Caucasus Mathematical Olympiad, 7
On sides $BC$, $CA$, $AB$ of a triangle $ABC$ points $K$, $L$, $M$ are chosen, respectively, and a point $P$ is inside $ABC$ is chosen so that $PL\parallel BC$, $PM\parallel CA$, $PK\parallel AB$. Determine if it is possible that each of three trapezoids $AMPL$, $BKPM$, $CLPK$ has an inscribed circle.
1996 Estonia Team Selection Test, 2
Let $a,b,c$ be the sides of a triangle, $\alpha ,\beta ,\gamma$ the corresponding angles and $r$ the inradius. Prove that $$a\cdot sin\alpha+b\cdot sin\beta+c\cdot sin\gamma\geq 9r$$
1998 National High School Mathematics League, 12
In $\triangle ABC$, $\angle C=90^{\circ},\angle B=30^{\circ}, AC=2$. $M$ is the midpoint of $AB$. Fold up $\triangle ACM$ along $CM$, satisfying that $|AB|=2\sqrt2$. The volume of triangular pyramid $A-BCM$ is________.
2002 Iran MO (3rd Round), 2
$f: \mathbb R\longrightarrow\mathbb R^{+}$ is a non-decreasing function. Prove that there is a point $a\in\mathbb R$ that \[f(a+\frac1{f(a)})<2f(a)\]
2023 Malaysian Squad Selection Test, 6
Given a cyclic quadrilateral $ABCD$ with circumcenter $O$, let the circle $(AOD)$ intersect the segments $AB$, $AC$, $DB$, $DC$ at $P$, $Q$, $R$, $S$ respectively. Suppose $X$ is the reflection of $D$ about $PQ$ and $Y$ is the reflection of $A$ about $RS$.
Prove that the circles $(AOD)$, $(BPX)$, $(CSY)$ meet at a common point.
[i]Proposed by Leia Mayssa & Ivan Chan Kai Chin[/i]
2005 Today's Calculation Of Integral, 50
Let $a,b$ be real numbers such that $a<b$.
Evaluate
\[\lim_{b\rightarrow a} \frac{\displaystyle\int_a^b \ln |1+(x-a)(b-x)|dx}{(b-a)^3}\].
1969 Canada National Olympiad, 4
Let $ABC$ be an equilateral triangle, and $P$ be an arbitrary point within the triangle. Perpendiculars $PD,PE,PF$ are drawn to the three sides of the triangle. Show that, no matter where $P$ is chosen, \[ \frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}. \]
2010 Mexico National Olympiad, 1
Let $n$ be a positive integer. In an $n\times4$ table, each row is equal to
\[\begin{tabular}{| c | c | c | c |}
\hline
2 & 0 & 1 & 0 \\
\hline
\end{tabular}\]
A [i]change[/i] is taking three consecutive boxes in the same row with different digits in them and changing the digits in these boxes as follows:
\[0\to1\text{, }1\to2\text{, }2\to0\text{.}\]
For example, a row $\begin{tabular}{| c | c | c | c |}\hline 2 & 0 & 1 & 0 \\ \hline\end{tabular}$ can be changed to the row $\begin{tabular}{| c | c | c | c |}\hline 0 & 1 & 2 & 0 \\ \hline\end{tabular}$ but not to $\begin{tabular}{| c | c | c | c |}\hline 2 & 1 & 2 & 1 \\ \hline\end{tabular}$ because $0$, $1$, and $0$ are not distinct.
Changes can be applied as often as wanted, even to items already changed. Show that for $n<12$, it is not possible to perform a finite number of changes so that the sum of the elements in each column is equal.
1989 IMO Longlists, 93
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.
2020 CMIMC Team, 9
Over all natural numbers $n$ with 16 (not necessarily distinct) prime divisors, one of them maximizes the value of $s(n)/n$, where $s(n)$ denotes the sum of the divisors of $n$. What is the value of $d(d(n))$, where $d(n)$ is the the number of divisors of $n$?
2019 South East Mathematical Olympiad, 8
For positive integer $x>1$, define set $S_x$ as $$S_x=\{p^\alpha|p \textup{ is one of the prime divisor of }x,\alpha \in \mathbb{N},p^\alpha|x,\alpha \equiv v_p(x)(\textup{mod} 2)\},$$
where $v_p(n)$ is the power of prime divisor $p$ in positive integer $n.$ Let $f(x)$ be the sum of all the elements of $S_x$ when $x>1,$ and $f(1)=1.$
Let $m$ be a given positive integer, and the sequence $a_1,a_2,\cdots,a_n,\cdots$ satisfy that for any positive integer $n>m,$ $a_{n+1}=\max\{ f(a_n),f(a_{n-1}+1),\cdots,f(a_{n-m}+m)\}.$ Prove that
(1)there exists constant $A,B(0<A<1),$ such that when positive integer $x$ has at least two different prime divisors, $f(x)<Ax+B$ holds;
(2)there exists positive integer $Q$, such that for any positive integer $n,a_n<Q.$