Found problems: 85335
In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is 15 $\text{cm}^2$ and the area of face $ABD$ is 12 $\text{cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\text{cm}^3$.
Faith has four different integer length segments. It turned out that any three of them can form a triangle. What is the smallest total length of this set of segments?
I have a four-digit palindrome $\underline{a} \ \underline{b} \ \underline{b} \ \underline{a}$ that is divisible by $b$ and is also divisible by the two-digit number $\underline{b} \ \underline{b}.$ Find the number of palindromes satisfying both of these properties.
The tangent lines to the circle $\omega$ at points $B$ and $D$ intersect at point $P$. The line passing through $P$ cuts out from circle chord $AC$. Through an arbitrary point on the segment $AC$ a straight line parallel to $BD$ is drawn. Prove that it divides the lengths of polygonal $ABC$ and $ADC$ in the same ratio.
[hide=last sentence was in Russian: ]Докажите, что она делит длины ломаных ABC и ADC в одинаковых отношениях. [/hide]
$k>2$ is an integer. $a_0,a_1,...$ is an integer sequence such that $a_0=0$, $a_{n+1}=ka_n-a_{n-1}$. Prove that for any positive integer $m$, $(2m)!|a_1a_2...a_{3m}$.
A regular hexagon of side $5$ is cut into unit equilateral triangles by lines parallel to the sides of the hexagon. We call the vertices of these triangles knots. If more than half of all knots are marked, show that there exist five marked knots that lie on a circle.
Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$.
Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^o$. Find the measure of $\angle ABC$.
Let $f_n$ be a sequence defined by $f_0=2020$ and
\[f_{n+1} = \frac{f_n + 2020}{2020f_n + 1}\]
for all $n \geq 0$. Determine $f_{2020}$.
[b]p1.[/b] Find all real solutions of the system
$$x^5 + y^5 = 1$$
$$x^6 + y^6 = 1$$
[b]p2.[/b] The centers of three circles of the radius $R$ are located in the vertexes of equilateral triangle. The length of the sides of the triangle is $a$ and $\frac{a}{2}< R < a$. Find the distances between the intersection points of the
circles, which are outside of the triangle.
[b]p3.[/b] Prove that no positive integer power of $2$ ends with four equal digits.
[b]p4.[/b] A circle is divided in $10$ sectors. $90$ coins are located in these sectors, $9$ coins in each sector. At every move you can move a coin from a sector to one of two neighbor sectors. (Two sectors are called neighbor if they are adjoined along a segment.) Is it possible to move all coins into one sector in exactly$ 2004$ moves?
[b]p5.[/b] Inside a convex polygon several points are arbitrary chosen. Is it possible to divide the polygon into smaller convex polygons such that every one contains exactly one given point? Justify your answer.
[b]p6.[/b] A troll tried to spoil a white and red $8\times 8$ chessboard. The area of every square of the chessboard is one square foot. He randomly painted $1.5\%$ of the area of every square with black ink. A grasshopper jumped on the spoiled chessboard. The length of the jump of the grasshopper is exactly one foot and at every jump only one point of the chessboard is touched. Is it possible for the grasshopper to visit every square of the chessboard without touching any black point? Justify your answer.
PS. You should use hide for answers.
Does there exist a functional equation[sup]1[/sup] that has a solution and the range of any of its solutions is the set of integers?
[sup]1[/sup][size=75]A [i]functional equation[/i] has the form $\mbox{\footnotesize \(E = 0\)}$, where $\mbox{\footnotesize \(E\)}$ is a function form. The set of function forms is the smallest set $\mbox{\footnotesize \(\mathcal{F}\)}$ which contains the variables $\mbox{\footnotesize \(x_1, x_2, \dots\)}$, the real numbers $\mbox{\footnotesize \(r \in \mathbb{R}\)}$, and for which $\mbox{\footnotesize \(E, E_1, E_2 \in \mathcal{F}\)}$ implies $\mbox{\footnotesize \(E_1+E_2 \in \mathcal{F}\)}$, $\mbox{\footnotesize \(E_1 \cdot E_2 \in \mathcal{F}\)}$, and $\mbox{\footnotesize \(f(E) \in \mathcal{F}\)}$, where $\mbox{\footnotesize \(f\)}$ is a fixed function symbol. The solution of the functional equation $\mbox{\footnotesize \(E = 0\)}$ is a function $\mbox{\footnotesize \(f: \mathbb{R} \to \mathbb{R}\)}$ such that $\mbox{\footnotesize \(E = 0\)}$ holds for all values of the variables. E.g. $\mbox{\footnotesize \(f\big(x_1 + f(\sqrt{2} \cdot x_2 \cdot x_2)\big) + (-\pi) + (-1) \cdot x_1 \cdot x_1 \cdot x_2 = 0\)}$ is a functional equation.[/size]
a，b，c>0，abc=1，prove that(a/b)+(b/c)+(c/a)≥a+b+c.
Let $n>2$ be an integer. Anna, Edda and Magni play a game on a hexagonal board tiled with regular hexagons, with $n$ tiles on each side. The figure shows a board with 5 tiles on each side. The central tile is marked.
[asy]unitsize(.25cm);
real s3=1.73205081;
pair[] points={(-4,4*s3),(-2,4*s3),(0,4*s3),(2,4*s3),(4,4*s3),(-5,3*s3), (-3,3*s3), (-1,3*s3), (1,3*s3), (3,3*s3), (5,3*s3), (-6,2*s3),(-4,2*s3), (-2,2*s3), (0,2*s3), (2,2*s3), (4,2*s3),(6,2*s3),(-7,s3), (-5,s3), (-3,s3), (-1,s3), (1,s3), (3,s3), (5,s3),(7,s3),(-8,0), (-6,0), (-4,0), (-2,0), (0,0), (2,0), (4,0), (6,0), (8,0),(-7,-s3),(-5,-s3), (-3,-s3), (-1,-s3), (1,-s3), (3,-s3), (5,-s3), (7,-s3), (-6,-2*s3), (-4,-2*s3), (-2,-2*s3), (0,-2*s3), (2,-2*s3), (4,-2*s3), (6,-2*s3), (-5,-3*s3), (-3,-3*s3), (-1,-3*s3), (1,-3*s3), (3,-3*s3), (5,-3*s3), (-4,-4*s3), (-2,-4*s3), (0,-4*s3), (2,-4*s3), (4,-4*s3)};
void draw_hexagon(pair p)
{
draw(shift(p)*scale(2/s3)*(dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--dir(30)));
}
{for (int i=0;i<61;++i){draw_hexagon(points[i]);}}
label((0,0), "\Large $*$");
[/asy]
The game begins with a stone on a tile in one corner of the board. Edda and Magni are on the same team, playing against Anna, and they win if the stone is on the central tile at the end of any player's turn. Anna, Edda and Magni take turns moving the stone: Anna begins, then Edda, then Magni, then Anna, and so on.
The rules for each player's turn are:
[list]
[*] Anna has to move the stone to an adjacent tile, in any direction.
[*] Edda has to move the stone straight by two tiles in any of the $6$ possible directions.
[*] Magni has a choice of passing his turn, or moving the stone straight by three tiles in any of the $6$ possible directions.
[/list]
Find all $n$ for which Edda and Magni have a winning strategy.
Find all positive integers $n \ge 3$ such that there exists a positive constant $M_n$ satisfying the following inequality for any $n$ positive reals $a_1, a_2,\dots\>,a_n$: \[\displaystyle \frac{a_1+a_2+\cdots\>+a_n}{\sqrt[n]{a_1a_2\cdots\>a_n}} \le M_n \biggl( \frac{a_2}{a_1} + \frac{a_3}{a_2} +\cdots\>+ \frac{a_n}{a_{n-1}} + \frac {a_1}{a_n} \biggr).\] Moreover, find the minimum value of $M_n$ for such $n$.
The difficulty is finding $M_n$...
Paul owes Paula $35$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$
In a football tournament, $n$ teams play one round ($n \vdots 2$). In each round should play $n / 2$ pairs of teams that have not yet played. Schedule of each round takes place before its holding. For which smallest natural $k$ such that the following situation is possible:
after $k$ tours, making a schedule of $k + 1$ rounds already is not possible,
i.e. these $n$ teams cannot be divided into $n / 2$ pairs, in each of which there are teams that have not played in the previous $k$ rounds.
PS. The 3 vertical dots notation in the first row, I do not know what it means.
Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$
[i]Proposed by Milan Haiman[/i]
Let $ABC$ be a triangle, and points $D,E$ are on $BA,CA$ respectively such that $DB=BC=CE$. Let $O,I$ be the circumcenter, incenter of $\triangle ABC$. Prove that the circumradius of $\triangle ADE$ is equal to $OI$.
Let $x,y$ be positive integers such that $3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect square.
Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations.
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $n = 2^{2015} - 1$. For any integer $1 \le x < n$, let \[f_n(x) = \sum\limits_p s_p(n-x) + s_p(x) - s_p(n),\] where $s_q(k)$ denotes the sum of the digits of $k$ when written in base $q$ and the summation is over all primes $p$. Let $N$ be the number of values of $x$ such that $4 | f_n(x)$. What is the remainder when $N$ is divided by $1000?$
$C_0:x^2+y^2=1,C_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}(a>b>0)$. Find all $(a,b)$ such that for any point $P$ on $C_1$, we can find a parallelogram with an apex $P$, and it is externally tangent to $C_0$, inscribed to $C_1$.
In triangle $ABC$, points $D, E$, and $F$ are marked on sides $AC, BC$, and $AB$ respectively, so that $AD = AB$, $EC = DC$, $BF = BE$. After that, they erased everything except points $E, F$ and $D$. Reconstruct the triangle $ABC$ (no study required).
Line $PQ$ is tangent to the incircle of triangle $ABC$ in such a way that the points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively. On the sides $AB$ and $AC$ are selected points $M$ and $N$, respectively, so that $AM = BP$ and $AN = CQ$. Prove that all lines constructed in this manner $MN$ pass through one point
Let $n$ be a positive integer. A [i]Japanese triangle[/i] consists of $1 + 2 + \dots + n$ circles arranged in an equilateral triangular shape such that for each $i = 1$, $2$, $\dots$, $n$, the $i^{th}$ row contains exactly $i$ circles, exactly one of which is coloured red. A [i]ninja path[/i] in a Japanese triangle is a sequence of $n$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with $n = 6$, along with a ninja path in that triangle containing two red circles.
[asy]
// credit to vEnhance for the diagram (which was better than my original asy):
size(4cm);
pair X = dir(240); pair Y = dir(0);
path c = scale(0.5)*unitcircle;
int[] t = {0,0,2,2,3,0};
for (int i=0; i<=5; ++i) {
for (int j=0; j<=i; ++j) {
filldraw(shift(i*X+j*Y)*c, (t[i]==j) ? lightred : white);
draw(shift(i*X+j*Y)*c);
}
}
draw((0,0)--(X+Y)--(2*X+Y)--(3*X+2*Y)--(4*X+2*Y)--(5*X+2*Y),linewidth(1.5));
path q = (3,-3sqrt(3))--(-3,-3sqrt(3));
draw(q,Arrows(TeXHead, 1));
label("$n = 6$", q, S);
label("$n = 6$", q, S);
[/asy]
In terms of $n$, find the greatest $k$ such that in each Japanese triangle there is a ninja path containing at least $k$ red circles.
A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or
[*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter.
[i]Proposed by Aron Thomas[/i]