Found problems: 85335
Solid camphor is insoluble in water but is soluble in vegetable oil. The best explanation for this behavior is that camphor is a(n)
${ \textbf{(A)}\ \text{Ionic solid} \qquad\textbf{(B)}\ \text{Metallic solid} \qquad\textbf{(C)}\ \text{Molecular solid} \qquad\textbf{(D)}\ \text{Network solid} } $
Given an infinite sequence of numbers $a_1, a_2,...$, in which there are no two equal members. Segment $a_i, a_{i+1}, ..., a_{i+m-1}$ of this sequence is called a monotone segment of length $m$, if $a_i < a_{i+1} <...<a_{i+m-1}$ or $a_i > a_{i+1} >... > a_{i+m-1}$. It turned out that for each natural $k$ the term $a_k$ is contained in some monotonic segment of length $k + 1$. Prove that there exists a natural $N$ such that the sequence $a_N , a_{N+1} ,...$ monotonic.
Find all triples of primes $(p, q, r)$ such that $p^q=2021+r^3$.
If $x$ is a real number so $3^x=27x$, compute $\log_3 \left(\tfrac{3^{3^x}}{x^{3^3}}\right)$.
Determine a subset $ A\subset \mathbb{N}^*$ having $ 5$ different elements, so that the sum of the squares of its elements equals their product.
Do not simply post the subset, show how you found it.
$15\times 36$-checkerboard is covered with square tiles. There are two kinds of tiles, with side $7$ or $5.$
Tiles are supposed to cover whole squares of the board and be non-overlapping.
What is the maximum number of squares to be covered?
[b]a)[/b] Prove that there exist two functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ having the properties:
$ \text{(i)}\quad f\circ g=g\circ f $
$\text{(ii)}\quad f\circ f=g\circ g $
$ \text{(iii)}\quad f(x)\neq g(x), \quad \forall x\in\mathbb{R} $
[b]b)[/b] Show that if there are two functions $ f_1,g_1:\mathbb{R}\longrightarrow\mathbb{R} $ with the properties $ \text{(i)} $ and $ \text{(iii)} $ from above, then $ \left( f_1\circ f_1\right)(x) \neq \left( g_1\circ g_1 \right)(x) , $ for all real numbers $ x. $
Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively
defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there
exists some $i \in \mathbb{N}$ with $a_i = m^2$.
[i]Proposed by Nikola Velov, North Macedonia[/i]
$N$ people have a series of calls. Each call is between two people, and is started by exactly one of them. Each person starts at most $10$ calls. Two people can call at most once. In any group of $3$ people, there are at least two people who have a call. Find the maximum possible value of $N$.
[i]Proposed by Serena Xu[/i]
Let $c = \frac{2\pi}{11}.$ What is the value of
$$\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?$$
$\textbf{(A)}\ -1 \qquad\textbf{(B)}\ \frac{\sqrt{-11}}{5} \qquad\textbf{(C)}\ \frac{\sqrt{11}}{5} \qquad\textbf{(D)}\
\frac{10}{11} \qquad\textbf{(E)}\ 1$
Let $\triangle{ABC}$ be a scalene triangle with incentre $I$ and circumcircle $\omega$. Lines $AI, BI, CI$ intersect $\omega$ for the second time at points $D, E, F$, respectively. The parallel lines from $I$ to the sides $BC, AC, AB$ intersect $EF, DF, DE$ at points $K, L, M$, respectively. Prove that the points $K, L, M$ are collinear.
[i](Cyprus)[/i]
Find all two-variable polynomials $p(x,y)$ such that for each $a,b,c\in\mathbb R$:
\[p(ab,c^2+1)+p(bc,a^2+1)+p(ca,b^2+1)=0\]
Let $ABCD$ be inscribed in a circle $\omega$.
Let the line parallel to the tangent to $\omega$ at $A$ and passing $D$ meet $\omega$ at $E$.
$F$ is a point on $\omega$ such that lies on the different side of $E$ wrt $CD$.
If $AE \cdot AD \cdot CF = BE \cdot BC \cdot DF$ and $\angle CFD = 2\angle AFB$,
Show that the tangent to $\omega$ at $A, B$ and line $EF$ concur at one point.
($A$ and $E$ lies on the same side of $CD$)
Triangle $ABC$ is right-angled at $C$. $AQ$ is drawn parallel to $BC$ with $Q$ and $B$ on opposite sides of $AC$ so that when $BQ$ is drawn, intersecting $AC$ at $P$, we have $PQ = 2AB$. Prove that $\angle ABC = 3\angle PBC$.
The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, it intersects the graph $y=f(x)$ at precisely one point, which is $\frac{1}{\sqrt{m}}$ units from the origin. Let $a$ be the unique real number for which $f$ takes on its maximum value at $x=a$ (you may assume that such an $a$ exists). Find $\int_{0}^{a}f(x) \, dx$.
$B$ is a point on the line which is tangent to a circle at the point $A$. The line segment $AB$ is rotated about the centre of the circle through some angle to the line segment $A'B'$. Prove that the line $AA'$ passes through the midpoint of $BB'$.
Source: 2018 Canadian Open Math Challenge Part C Problem 4
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Given a positive integer $N$, Matt writes $N$ in decimal on a blackboard, without writing any of the leading 0s. Every minute he takes two consicutive digits, erases them, and replaces them with the last digit of their product. Any leading zeroes created this way are also erased. He repeats this process for as long as he likes. We call the positive integer $M$ [i]obtainable[/i] from $N$ if starting from $N$, there is a finite sequence of moves that Matt can make to produce the number $M$. For example, 10 is obtainible from 251023 via
\[2510\underline{23}\rightarrow\underline{25} 106\rightarrow 1\underline{06}\rightarrow 10\]
$\text{(a)}$ Show that 2018 is obtainablefrom 2567777899.
$\text{(b)}$ Find two positive integers $A$ and $B$ for which there is no positive integer $C$
[color=transparent](B.)[/color] such that both $A$ and $B$ are obtainablefrom $C$
$\text{(c)}$ Let $S$ be any finite set of positive integers, none of which contains the digit 5
[color=transparent](C.)[/color] in its decimal representation. Prove that there exists a positive integer $N$
[color=transparent](C.)[/color] for which all elements of $S$ are obtainable from $N$.
We say the point $i$ in the permutation $\sigma$ [b]ongoing[/b] if for every $j<i$ we have $\sigma (j)<\sigma (i)$.
[b]a)[/b] prove that the number of permutations of the set $\{1,....,n\}$ with exactly $r$ ongoing points is $s(n,r)$.
[b]b)[/b] prove that the number of $n$-letter words with letters $\{a_1,....,a_k\},a_1<.....<a_k$. with exactly $r$ ongoing points is $\sum_{m}\dbinom{k}{m} S(n,m) s(m,r)$.
$1019$ stones are placed into two non-empty boxes. Each second Alex chooses a box with an even amount of stones and shifts half of these stones into another box.
Prove that for each $k$, $1\le k\le1018$, at some moment there will be a box with exactly $k$ stones.
[i](O. Izhboldin)[/i]
Find all positive integers $n$ that are equal to the sum of digits of $n^2$.
Let $P$ be the intersection point of the diagonals of the quadrangle $ABCD$, $M$ the intersection point of the lines connecting the midpoints of its opposite sides, $O$ the intersection point of the perpendicular bisectors of the diagonals, $H$ the intersection point of the lines connecting the orthocenters of the triangles $APD$ and $BCP$, $APB$ and $CPD$. Prove that $M$ is the midpoint of $OH$.
Consider the set $ S$ of all numbers of the form $ a(n)\equal{}n^2\plus{}n\plus{}1, n \in \mathbb{N}.$ Show that the product $ a(n)a(n\plus{}1)$ is in $ S$ for all $ n \in \mathbb{N}$ and give an example of two elements $ s,t$ of $ S$ such that $ s,t \notin S$.
Five players $(A,B,C,D,E)$ take part in a bridge tournament. Every two players must play (as partners) against every other two players. Any two given players can be partners not more than once per a day. What is the least number of days needed for this tournament?
Let $A$ be a finite subset of prime numbers and $a> 1$ be a positive integer. Show that the number of positive integers $m$ for which all prime divisors of $a^m-1$ are in $A$ is finite.
Three concentric circles centered at $O$ have radii of $1$, $2$, and $3$. Points $B$ and $C$ lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle $BOC$, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of $\angle{BOC}$ in degrees?
[asy]
size(100);
import graph;
draw(circle((0,0),3));
real radius = 3;
real angleStart = -54; // starting angle of the sector
real angleEnd = 54; // ending angle of the sector
label("$O$",(0,0),W);
pair O = (0, 0);
filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, lightgray);
filldraw(circle((0,0),2),lightgray);
filldraw(circle((0,0),1),white);
draw((1.763,2.427)--(0,0)--(1.763,-2.427));
label("$B$",(1.763,2.427),NE);
label("$C$",(1.763,-2.427),SE);
[/asy]
$\textbf{(A)}\ 108 \qquad \textbf{(B)}\ 120 \qquad \textbf{(C)}\ 135 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 150$