Found problems: 85335
Prove that there is only one triangle whose sides are consecutive natural numbers and one of the angles is twice the other angle.
What is the value of the product$$\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?$$
$\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{4}{3}\qquad\textbf{(C) }\frac{7}{2}\qquad\textbf{(D) }7\qquad\textbf{(E) }8$
Let $a, b$, and $n$ be integers such that $a + b$ is divisible by $n$ and $a^2 + b^2$ is divisible by $n^2$. Prove that $a^m + b^m$ is divisible by $n^m$ for all positive integers $m$.
Let $A, B, C$, and $D$ be points on a circle, in that order, such that $\overline{AD}$ is a diameter of the circle. Let $E$ be the intersection of $\overleftrightarrow{AB}$ and $\overleftrightarrow{DC}$, let $F$ be the intersection of $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$, and let $G$ be the intersection of $\overleftrightarrow{EF}$ and $\overleftrightarrow{AD}$. If $AD = 8$, $AE = 9$, and $DE = 7$, compute $EG$.
$n$ lines are given in the plane so that no three of them concur and no two are parallel.
Show that there is a non-self-intersecting path consisting of $n$ straight segments so that each of the given lines contains exactly one of the segments of the path.
A point in the plane is called [b]integral[/b] if both its $x$ and $y$ coordinates are integers. We are given a triangle whose vertices are integral. Its sides do not contain any other integral points. Inside the triangle, there are exactly 4 integral points. Must those 4 points lie on one line?
Three integers are written on a blackboard. At every step one of them is erased and the sum of the other two decreased by $1$ is written instead. Is it possible to obtain the numbers $17,75,91$ if the three initial numbers were:
$\textbf{(a)}~2,2,2$;
$\textbf{(b)}~3,3,3$?
A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors).
Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.)
[hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]
An ant is currently on a vertex of the top face on a 6-sided die. The ant wants to travel to the opposite vertex of the die (the vertex that is farthest from the start), and the ant can travel along edges of the die to other vertices that are on the top face of the die.
Every second, the ant picks a valid edge to move along, and the die randomly flips to an adjacent face. If the ant is on any of the bottom vertices after the flip, it is crushed and dies. What is the probability that the ant makes it to its target? (If the ant makes it to the target and the die rolls to crush it, it achieved its dreams before dying, so this counts.)
[i]Proposed by Lohith Tummala[/i]
Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are the centers of the circles?
On a $2004\times 2004$ chessboard we place $2004$ white knights$^1$ in the upper row, and $2004$ black ones in the lowest row. After a finite number of regular chess moves$^2$ , we get the opposite situation where the black ones are on the top and the white ones on the bottom lines.
In a [i]turn [/i] we make a move with each of the pieces of a color. If you know that each square except those on which the knights originally lie, must not be used more than once in this process, and that after each turn no $2$ knights of the same color can be attacking each other$^3$ , determine the number of ways in which this can be accomplished.
$^1$ also known as horses
$^2$ the knight can be moved either one square horizontally and two vertically or two squares horizontally and one vertically, in any direction on both horizontal and vertical lines
$^3$ a knight is attacking another knight, if in one chess move, the first one can be placed on the second one’s place
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.
Among all triples $(a,b,c)$ of natural numbers satisfying
\[(a+14\sqrt{3})(b-14c\sqrt{3})=2024,\]
determine the one with the maximal value of $a$.
Find the sum of all integers $n$ satisfying the following inequality:
\[\frac{1}{4}<\sin\frac{\pi}{n}<\frac{1}{3}.\]
Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that
$$n!+f(m)!|f(n)!+f(m!)$$
for all $m,n\in\mathbb{N}$
[i]Proposed by Valmir Krasniqi and Dorlir Ahmeti, Albania[/i]
Suppose that $a^{\log_{b}c}+b^{\log_{c}a}=m$.
Find the value of $c^{\log_{b}a}+a^{\log_{c}b}$ .
Diagonals of an isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ are perpendicular. Let $DE$ be the perpendicular from $D$ to $AB$, and let $CF$ be the perpendicular from $C$ to $DE$. Prove that angle $DBF$ is equal to half of angle $FCD$.
The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number.
[list=a]
[*] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$.
[*] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$.
[/list]
Peter has a deck of $1001$ cards, and with a blue pen he has written the numbers $1,2,\ldots,1001$ on the cards (one number on each card). He replaced cards in a circle so that blue numbers were on the bottom side of the card. Then, for each card $C$, he took $500$ consecutive cards following $C$ (clockwise order), and denoted by $f(C)$ the number of blue numbers written on those $500$ cards that are greater than the blue number written on $C$ itself. After all, he wrote this $f(C)$ number on the top side of the card $C$ with a red pen. Prove that Peter's friend Basil, who sees all the red numbers on these cards, can determine the blue number on each card.
Suppose $P$ is a point inside a convex $2n$-gon, such that $P$ does not lie on any diagonal. Show that $P$ lies inside an even number of triangles whose vertices are vertices of the polygon.
Let $a,b,c,d$ be non-negative reals such that $a+b+c+d=4$. Prove the inequality
\[\frac{a}{a^3+8}+\frac{b}{b^3+8}+\frac{c}{c^3+8}+\frac{d}{d^3+8}\le\frac{4}{9}\]
Let $ABC$ be a triangle. The internal bisector of $\angle B$ meets $AC$ in $P$ and $I$ is the incenter of $ABC$. Prove that if $AP+AB = CB$, then $API$ is an isosceles triangle.
If $x$ is a real number such that $(x - 3)(x - 1)(x + 1)(x + 3) + 16 = 116^2$, what is the largest possible value of $x$?
Let N be a positive integer greater than 2. We number the vertices
of a regular 2n-gon clockwise with the numbers 1, 2, . . . ,N,−N,−N +
1, . . . ,−2,−1. Then we proceed to mark the vertices in the following way.
In the first step we mark the vertex 1. If ni is the vertex marked in the
i-th step, in the i+1-th step we mark the vertex that is |ni| vertices away
from vertex ni, counting clockwise if ni is positive and counter-clockwise
if ni is negative. This procedure is repeated till we reach a vertex that has
already been marked. Let $f(N)$ be the number of non-marked vertices.
(a) If $f(N) = 0$, prove that 2N + 1 is a prime number.
(b) Compute $f(1997)$.
A positive integer is [i]detestable[/i] if the sum of its digits is a multiple of $11$. How many positive integers below $10000$ are detestable?
[i]Proposed by Giacomo Rizzo[/i]