Found problems: 85335
Prove that $m^7- m$ is divisible by $42$ for any positive integer $m$.
For every $3$-digit natural number $n$ (leading digit of $n$ is nonzero), we consider the number $n_0$ obtained from $n$ eliminating all possible digits that are zero. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$, for which $n_0$ is a divisor of $n$ different from $n$.
Find $\frac{area(CDF)}{area(CEF)}$ in the figure.
[asy]
/* File unicodetex not found. */
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(5.75cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -2, xmax = 21, ymin = -2, ymax = 16; /* image dimensions */
/* draw figures */
draw((0,0)--(20,0));
draw((13.48,14.62)--(7,0));
draw((0,0)--(15.93,9.12));
draw((13.48,14.62)--(20,0));
draw((13.48,14.62)--(0,0));
label("6",(15.16,12.72),SE*labelscalefactor);
label("10",(18.56,5.1),SE*labelscalefactor);
label("7",(3.26,-0.6),SE*labelscalefactor);
label("13",(13.18,-0.71),SE*labelscalefactor);
label("20",(5.07,8.33),SE*labelscalefactor);
/* dots and labels */
dot((0,0),dotstyle);
label("$B$", (-1.23,-1.48), NE * labelscalefactor);
dot((20,0),dotstyle);
label("$C$", (19.71,-1.59), NE * labelscalefactor);
dot((7,0),dotstyle);
label("$D$", (6.77,-1.64), NE * labelscalefactor);
dot((13.48,14.62),dotstyle);
label("$A$", (12.36,14.91), NE * labelscalefactor);
dot((15.93,9.12),dotstyle);
label("$E$", (16.42,9.21), NE * labelscalefactor);
dot((9.38,5.37),dotstyle);
label("$F$", (9.68,4.5), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
Assume the earth is a perfect sphere with a circumference of $60$ units. A great circle is a circle on a sphere whose center is also the center of the sphere. There are three train tracks on three great circles of the earth. One is along the equator and the other two pass through the poles, intersecting at a $90$ degree angle. If each track has a train of length $L$ traveling at the same speed, what is the maximum value of $L$ such that the trains can travel without crashing?
Consider a $5\times 5$ grid with $25$ cells. What is the least number of cells that should be colored, such that every $2\times 3$ or $3\times 2$ rectangle in the grid has at least two colored cells?
A consecutive pythagorean triple is a pythagorean triple of the form $a^2 + (a+1)^2 = b^2$, $a$ and $b$ positive integers. Given that $a$, $a+1$, and $b$ form the third consecutive pythagorean triple, find $a$.
Let $a$ and $b$ be two integers which are coprime and let $n$ be one variable integer. Determine probability that number of solutions $(x,y)$, where $x$ and $y$ are nonnegative integers, of equation $ax+by=n$ is $\left\lfloor \frac{n}{ab} \right\rfloor + 1$
Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$.
Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.
Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.
Let $T = (a,b,c)$ be a triangle with sides $a,b$ and $c$ and area $\triangle$. Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a, b' = h_b, c' = h_c$) and denote its area by $\triangle '$. Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes of $T'$, and denote its area by $\triangle ''$. Given that $\triangle ' = 30$ and $\triangle '' = 20$, find $\triangle$.
Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. A point $C$ is constructed such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Prove that $CX = CY$.
(Oleksii Masalitin)
Let $x$ be the positive integer satisfying $5^2 +28^2 +39^2 = 24^2 +35^2 + x^2$. Find $x$.
Determine the smallest non-negative integer $n$ such that
\[\sqrt{(6n+11)(6n+14)(20n+19)}\in\mathbb Q.\]
[i]Mihai Bunget[/i]
Five points are arbitrarily put inside a given equilateral triangle $ABC$ whose area is equal to $1$. Show that we can draw three equilateral triangles within triangle $ABC$ such that the following conditions are all satisfied:
i) the five points are covered by the three equilateral triangles;
ii) any side of the three equilateral triangles is parallel to a certain side of the triangle $ABC$;
iii) the sum of the areas of the three equilateral triangles is not larger than $0.64$.
A chess tournament took place between $2n+1$ players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly $k$ games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than $n-\sqrt{2k}$ and no more than $n+\sqrt{2k}$ games.
A circle $k$ of radius $r$ is inscribed in $\vartriangle ABC$, tangent to the circle $k$, which are parallel respectively to the sides $AB, BC$ and $CA$ intersect the other sides of $\vartriangle ABC$ at points $M, N; P, Q$ and $L, T$ ($P, T \in AB$, $L, N \in BC$ and $M, Q\in AC$). Denote by $r_1,r_2,r_3$ the radii of inscribed circles in triangles $MNC, PQA$ and $LTB$. Prove that $r_1+r_2+r_3=r$.
Three friends Archie, Billie, and Charlie play a game. At the beginning of the game, each of them has a pile of $2024$ pebbles. Archie makes the first move, Billie makes the second, Charlie makes the third and they continue to make moves in the same order. In each move, the player making the move must choose a positive integer $n$ greater than any previously chosen number by any player, take $2n$ pebbles from his pile and distribute them equally to the other two players. If a player cannot make a move, the game ends and that player loses the game.
$\hspace{5px}$ Determine all the players who have a strategy such that, regardless of how the other two players play, they will not lose the game.
[i]Proposed by Ilija Jovčeski, Macedonia[/i]
On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$.
$(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$.
$(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.
Find the number of positive integers $10 \le n \le 99$ with last digit at most $5$ such that the last two digits of $n^n$ are the same as $n.$
Let $ x,y,z$ be nonnegative real numbers such that $ x \plus{} y \plus{} z \equal{} 2$. Prove that $ x^{2}y^{2} \plus{} y^{2}z^{2} \plus{} z^{2}x^{2} \plus{} xyz\leq 1$. When does the equality occur?
Given positive integers $d \ge 3$, $r>2$ and $l$, with $2d \le l <rd$. Every vertice of the graph $G(V,E)$ is assigned to a positive integer in $\{1,2,\cdots,l\}$, such that for any two consecutive vertices in the graph, the integers they are assigned to, respectively, have difference no less than $d$, and no more than $l-d$.
A proper coloring of the graph is a coloring of the vertices, such that any two consecutive vertices are not the same color. It's given that there exist a proper subset $A$ of $V$, such that for $G$'s any proper coloring with $r-1$ colors, and for an arbitrary color $C$, either all numbers in color $C$ appear in $A$, or none of the numbers in color $C$ appear in $A$.
Show that $G$ has a proper coloring within $r-1$ colors.
Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \]
[i]Ion Rasa[/i]
Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes.
\[ 2^m p^2 + 1 = q^5 \]
There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.
In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps:
(a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$.
(b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group.
Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning.
[i]Czech Republic[/i]
The finite sets $A_1$, $A_2$, $\ldots$, $A_n$ are given and we denote by $d(n)$ the number of elements which appear exactly in an odd number of sets chosen from $A_1$, $A_2$, $\ldots$, $A_n$. Prove that for any $k$, $1\leq k\leq n$ the number \[{ d(n) - \sum\limits^n_{i=1} |A_i| + 2\sum\limits_{ i<j} |A_i \cap A_j | - \cdots + (-1)^k2^{k-1} \sum\limits_{i_1 <i_2 <\cdots < i_k} | A_{i_1} \cap A_{i_2} \cap \cdots \cap A_{i_k}}| \] is divisible by $2^k$.
[i]Ioan Tomescu, Dragos Popescu[/i]