Found problems: 85335
Which of the following does not divide the number of ordered pairs $(x,y)$ of integers satisfying the equation $x^3 - 13y^3 = 1453$?
$
\textbf{a)}\ 2
\qquad\textbf{b)}\ 3
\qquad\textbf{c)}\ 5
\qquad\textbf{d)}\ 7
\qquad\textbf{e)}\ \text{None of above}
$
Solve the equation $\cos 12x = 5\sin 3x+9\ tan ^2x+\ cot ^2x$
Let $ABC$ be an acute-angled triangle. $M ,N$ are any two points on the sides $AB , AC$ respectively. The circles with the diameters $BN$ and $CM$ intersect at points $P$ and $Q$. Show that the points $P, Q$ and the orthocenter of the triangle $ABC$ lie on a straight line.
Each vertex of the 3-dimensional Euclidean space either is coloured red or blue. Prove that within those squares being possible in this space with edge length 1 there is at least one square either with three red vertices or four blue vertices !
Let $x,y,z$ be positive reals such that $xyz=1$. Show that
$$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$
When does equality happen?
Let $n\ge3$ be an integer. Each row in an $(n-2)\times n$ array consists of the numbers 1,2,...,$n$ in some order, and the numbers in each column are all different. Prove that this array can be expanded into an $n\times n$ array such that each row and each column consists of the numbers 1,2,...,$n$.
In a chess festival that is held in a school with $2017$ students, each pair of students played at most one match versus each other. In the end, it is seen that for any pair of students which have played a match versus each other, at least one of them has played at most $22$ matches. What is the maximum possible number of matches in this event?
Georg and his mother love pizza. They buy a pizza shaped as an equilateral triangle. Georg demands to be allowed to divide the pizza by a straight cut and then make the first choice. The mother accepts this reluctantly, but she wants to choose a point of the pizza through which the cut must pass. Determine the largest fraction of the pizza which the mother is certain to get by this procedure.
Let $S$ be the set of all ordered pairs of integers $(m,n)$ satisfying $m>0$ and $n<0.$ Let $<$ be a partial ordering on $S$ defined by the statement $(m,n)<(m',n')$ if and only if $m\le m'$ and $n\le n'.$ An example is $(5,-10)<(8,-2).$ Now let $O$ be a completely ordered subset of $S,$ in other words if $(a,b)\in O$ and $(c,d) \in O,$ then $(a,b)<(c,d)$ or $(c,d)<(a,b).$ Also let $O'$ denote the collection of all such completely ordered sets.
(a) Determine whether and arbitrary $O\in O'$ is finite.
(b) Determine whether the carnality $|O|$ of $O$ is bounded for $O\in O'.$
(c) Determine whether $|O|$ can be countable infinite for any $O\in O'.$
Let us denote the midpoint of $AB$ with $O$. The point $C$, different from $A$ and $B$ is on the circle $\Omega$ with center $O$ and radius $OA$ and the point $D$ is the foot of the perpendicular from $C$ to $AB$. The circle with center $C$ and radius $CD$ and $\omega$ intersect at $M$, $N$. Prove that $MN$ cuts $CD$ in two equal segments.
Two half-lines $a$ and $b$, with the common endpoint $O$, make an acute angle $\alpha$. Let $A$ on $a$ and $B$ on $b$ be points such that $OA=OB$, and let $b$ be the line through $A$ parallel to $b$. Let $\beta$ be the circle with centre $B$ and radius $BO$. We construct a sequence of half-lines $c_1,c_2,c_3,\ldots $, all lying inside the angle $\alpha$, in the following manner:
(i) $c_i$ is given arbitrarily;
(ii) for every natural number $k$, the circle $\beta$ intercepts on $c_k$ a segment that is of the same length as the segment cut on $b'$ by $a$ and $c_{k+1}$.
Prove that the angle determined by the lines $c_k$ and $b$ has a limit as $k$ tends to infinity and find that limit.
Elisa has $2023$ treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of the unlocked chests of her choice, and afterwards, a fairy acts according to the following rules:
[list=disc]
[*]if more than one chests are unlocked, it locks one of them, or
[*]if there is only one unlocked chest, it unlocks all the chests.
[/list]
Given that this process goes on forever, prove that there is a constant $C$ with the following property: Elisa can ensure that the difference between the numbers of gems in any two chests never exceeds $C$, regardless of how the fairy chooses the chests to unlock.
The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$.
[i]Proposed by Croatia[/i]
Let $N$ be the number of subsets $B$ of the set $\{1,2,\dots,2018\}$ such that the sum of the elements of $B$ is congruent to $2018$ modulo $2048$. Find the remainder when $N$ is divided by $1000$.
Three different points were randomly selected from the vertices of the regular $2n$-gon. Let $ p_n $ be the probability of the event that the triangle with vertices at the selected points is acute-angled. Calculate $ \lim_{n\to \infty} p_n $.
Attention. We assume that all choices of three different points are equally likely.
Let $R$ be a commutative ring with $1$ such that the number of elements of $R$ is equal to $p^3$ where $p$ is a prime number. Prove that if the number of elements of $\text{zd}(R)$ be in the form of $p^n$ ($n \in \mathbb{N^*}$) where $\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}$, then $R$ has exactly one maximal ideal.
A scout troop buys $ 1000$ candy bars at a price of five for $ \$2$. They sell all the candy bars at a price of two for $ \$1$. What was their profit, in dollars?
$ \textbf{(A)}\ 100 \qquad
\textbf{(B)}\ 200 \qquad
\textbf{(C)}\ 300 \qquad
\textbf{(D)}\ 400 \qquad
\textbf{(E)}\ 500$
In the scalene acute triangle $ABC$, $O$ is the circumcenter. $AD, BE, CF$ are three altitudes. And $H$ is the orthocenter. Let $G$ be the reflection point of $O$ through $BC$. Draw the diameter $EK$ in $\odot (GHE)$, and the diameter $FL$ in $\odot (GHF)$.
a) If $AK, AL$ and $DE, DF$ intersect at $U, V$ respectively, prove that $UV\parallel EF$.
b) Suppose $S$ is the intersection of the two tangents of the circumscribed circle of $\triangle ABC$ at $B$ and $C$. $T$ is the intersection of $DS$ and $HG$. And $M,N$ are the projection of $H$ on $TE,TF$ respectively. Prove that $M,N,E,F$ are concyclic.
Let $\triangle ABC$ be an acute triangle. Point $Z$ is on $A$ altitude and points $X$ and $Y$ are on the $B$ and $C$ altitudes out of the triangle respectively, such that:
$\angle AYB=\angle BZC=\angle CXA=90$
Prove that $X$,$Y$ and $Z$ are collinear, if and only if the length of the tangent drawn from $A$ to the nine point circle of $\triangle ABC$ is equal with the sum of the lengths of the tangents drawn from $B$ and $C$ to the nine point circle of $\triangle ABC$.
There are $ n$ voters and $ m$ candidates. Every voter makes a certain arrangement list of all candidates (there is one person in every place $ 1,2,...m$) and votes for the first $ k$ people in his/her list. The candidates with most votes are selected and say them winners. A poll profile is all of this $ n$ lists.
If $ a$ is a candidate, $ R$ and $ R'$ are two poll profiles. $ R'$ is $ a\minus{}good$ for $ R$ if and only if for every voter; the people which in a worse position than $ a$ in $ R$ is also in a worse position than $ a$ in $ R'$. We say positive integer $ k$ is monotone if and only if for every $ R$ poll profile and every winner $ a$ for $ R$ poll profile is also a winner for all $ a\minus{}good$ $ R'$ poll profiles. Prove that $ k$ is monotone if and only if $ k>\frac{m(n\minus{}1)}{n}$.
A village has a circular wall around it, and the wall has four gates pointing north, south, east and west. A tree stands outside the village, $16 \, \mathrm{m}$ north of the north gate, and it can be [i]just[/i] seen appearing on the horizon from a point $48 \, \mathrm{m}$ east of the south gate. What is the diamter in meters, of the wall that surrounds the village?
Line $\ell$ is perpendicular to one of the medians of the triangle. The median perpendiculars to the sides of this triangle intersect the line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the other two.
Let $M$ be the midpoint of side $AC$ of the triangle $ABC$. Let $P$ be a point on the side $BC$. If $O$ is the point of intersection of $AP$ and $BM$ and $BO = BP$, determine the ratio $\frac{OM}{PC}$ .
The first three terms of a geometric sequence are the integers $a,\,720,$ and $b,$ where $a<720<b.$ What is the sum of the digits of the least possible value of $b?$
$\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21$
In triangle $ABC$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O$, and that \[\frac{AO}{OA'}+\frac{BO}{OB'}+\frac{CO}{OC'}=92,\] find \[\frac{AO}{OA'}\cdot \frac{BO}{OB'}\cdot \frac{CO}{OC'}.\]