Found problems: 85335
Each face of a regular tetrahedron can be colored one of red, purple, blue, or orange. How many distinct ways can we color the faces of the tetrahedron? Colorings are considered distinct if they cannot reach one another by rotation.
It is given that for some $n \in \mathbb{N}$ there exists a natural number $a$, such that $a^{n-1} \equiv 1 \pmod{n}$ and that for any prime divisor $p$ of $n-1$ we have $a^{\frac{n-1}{p}} \not \equiv 1 \pmod{n}$.
Prove that $n$ is a prime.
Prove that for each positive integer n,the equation
$x^{2}+15y^{2}=4^{n}$
has at least $n$ integer solution $(x,y)$
Let $AB$ be a given segment and $M$ be its midpoint. We consider the set of right-angled triangles $ABC$ with hypotenuses $AB$. Denote by $D$ the foot of the altitude from $C$. Let $K$ and $L$ be feet of perpendiculars from $D$ to the legs $BC$ and $AC$, respectively. Determine the largest possible area of the quadrilateral $MKCL$.
Czech Republic
On a cube, 27 points are marked in the following manner: one point in each corner, one point on the middle of each edge, one point on the middle of each face, and one in the middle the cube. The number of lines containing three out of these points is
A. 33
B. 42
C. 49
D. 72
E. 81
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1,k_2,\ldots,k_n$ for which \[k_1^2+k_2^2+\ldots+k_n^2=2002?\]
$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2).$
[asy]
unitsize(4mm); real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33}; for(real i:boxes){ draw(box((i,0),(i+1,3))); } draw((8,1.5)--(12,1.5),Arrow()); defaultpen(fontsize(20pt)); label(",",(20,0)); label(",",(29,0)); label(",...",(35.5,0));
[/asy]
Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?
$\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)$
Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if
\[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\]
Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$.
[i]Proposed by Andrey Badzyan, Russia[/i]
In rectangular coordinate system, the area which is surrounded by the figure of $f(x)=a\sin ax+\cos ax(a>0)$ on a complete period and the figure of $g(x)=\sqrt{a^2+1}$ is________.
$P(x)$ is a polynomial in $x$ with non-negative integer coefficients. If $P(1)=5$ and $P(P(1))=177$, what is the sum of all possible values of $P(10)$?
In triangle $ABC$ points $M$ and $N$ are the midpoints of sides $AC$ and $AB$, respectively and $D$ is the projection of $A$ into $BC$. Point $O$ is the circumcenter of $ABC$ and circumcircles of $BOC$, $DMN$ intersect at points $R, T$. Lines $DT$, $DR$ intersect line $MN$ at $E$ and $F$, respectively. Lines $CT$, $BR$ intersect at $K$. A point $P$ lies on $KD$ such that $PK$ is the angle bisector of $\angle BPC$. Prove that the circumcircles of $ART$ and $PEF$ are tangent.
[i]Proposed by Mehran Talaei - Iran[/i]
Find all triplets of positive integers $(a, b, c)$ that verify $\left(\frac{1}{a}+1\right)\left(\frac{1}{b}+1\right)\left(\frac{1}{c}+1\right)=2$.
Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7\}$.
Determine
\[\left\lfloor\prod_{n=2}^{2022}\frac{2n+2}{2n+1}\right\rfloor,\]
given that the answer is relatively prime to $2022$.
In a competition, there were $2n+1$ teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team ${ A,B,C }$ if $A$ won against $B$, $B$ won against $C$ , $C$ won against $A$.
(a) Find the minimum of cyclical 3-subset (depending on $n$);
(b) Find the maximum of cyclical 3-subset (depending on $n$).
Circle is divided into $n$ arcs by $n$ marked points on the circle. After that circle rotate an angle $ 2\pi k/n $ (for some positive integer $ k $), marked points moved to $n$ [i] new points [/i], dividing the circle into $ n $ [i] new arcs[/i]. Prove that there is a new arc that lies entirely in the one of the old arсs.
(It is believed that the endpoints of arcs belong to it.)
[i]I. Mitrophanov[/i]
Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \] are pairwise disjoint.
A convex polygon of $n$ sides is considered. All its diagonals are drawn and we suppose that any three of them can only intersect on a vertex and that there is no pair of parallel diagonals. Under these conditions, we wish to compute
a) The total number of intersection points of these diagonals, excluding the vertices.
b) How many points, of these intersections, lie inside the polygon and how many lie outside.
Let $1 < a_1 < a_2 < \cdots < a_N$ be natural numbers. It is known that for any $1\leqslant i\leqslant N$ the product of all these numbers except $a_i$ increased by one, is a multiple of $a_i$. Prove that $a_1\leqslant N$.
Triangle $\vartriangle ABC$ is isosceles with $AB = AC$ and $\angle ABC = 2\angle BAC$. Compute $\frac{AB}{BC}$ .
Alexey and Bogdan play a game with two piles of stones. In the beginning, one of the piles contains $2021$ stones, and the second is empty. In one move, each of the guys has to pick up an even number of stones (more than zero) from an arbitrary pile, then transfer half of the stones taken to another pile, and the other half - to remove from the game. Loses the one who cannot make a move. Who will win this game if both strive to win, and Bogdan begins?
(Oleksii Masalitin)
A quadrilateral whose perimeter is equal to $P$ is inscribed in a circle of radius $R$ and is circumscribed around a circle of radius $r$. Check whether the inequality $P\le \frac{r+\sqrt{r^2+4R^2}}{2}$ holds.
Try to find the corresponding inequalities for the $n$-gon ($n \ge 5$) inscribed in a circle of radius $R$ and circumscribed around a circle of radius $r$.
Let $A,B,C,D$ be four distinct points on a circle such that the lines $(AC)$ and $(BD)$ intersect at $E$, the lines $(AD)$ and $(BC)$ intersect at $F$ and such that $(AB)$ and $(CD)$ are not parallel.
Prove that $C,D,E,F$ are on the same circle if, and only if, $(EF)\bot(AB)$.
Let $a_1=0$, $a_2=1$, and $a_{n+2}=a_{n+1}+a_n$ for all positive integers $n$. Show that there exists an increasing infinite arithmetic progression of integers, which has no number in common in the sequence $\{a_n\}_{n \ge 0}$.
In triangle $ABC$ the medians $AM$ and $CN$ to sides $BC$ and $AB$, respectively, intersect in point $O$. $P$ is the midpoint of side $AC$, and $MP$ intersects $CN$ in $Q$. If the area of triangle $OMQ$ is $n$, then the area of triangle $ABC$ is:
$\text{(A)}\ 16n\qquad
\text{(B)}\ 18n\qquad
\text{(C)}\ 21n\qquad
\text{(D)}\ 24n\qquad
\text{(E)}\ 27n$