Found problems: 85335
Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?
Let $f$ be a function defined on the positive integers with $f(n) \ge 0$ and $f(n) \le f(n+1)$ for all $n$. Prove that if
\[\sum_{n = 1}^{\infty} \frac{f(n)}{n^2}\]
diverges, there exists a sequence $a_1, a_2, \dots$ such that the sequence $\tfrac{a_n}{n}$ hits every natural number, while
\[a_{n+m} \le a_n + a_m + f(n+m)\]
holds for every pair $n$, $m$.
The coordinates of the triangle's vertices in the Cartesian system $XOY$ are integers. Prove that the diameter of the circle circumscribed by this triangle is not greater than the product of the lengths of the triangle's sides.
We are given two concentric circles, Construct a square whose two vertices lie on one circle and the other two on the other circle.
Find the real roots of the equation $$(5-x)^4+ (x-2)^ 4 = 17$$ and the real roots of a more general equation $$(a - x) ^4+ (x - b)^4 = c$$
Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$.
Prove that $M_1$ lies on the segment $BH_1$.
A quadrilateral $ABCD$ is circumscribed about a circle. Lines $AC$ and $DC$ meet at point $E$ and lines $DA$ and $BC$ meet at $F$, where $B$ is between $A$ and $E$ and between $C$ and $F$. Let $I_1, I_2$ and $I_3$ be the incenters of triangles $AFB, BEC$ and $ABC$, respectively. The line $I_1I_3$ intersects $EA$ at $K$ and $ED$ at $L$, whereas the line $I_2I_3$ intersects $FC$ at $M$ and $FD$ at $N$. Prove that $EK = EL$ if and only if $FM = FN$
In the beginnig, all nine squares of $3\times 3$ chessboard contain $0$. At each step, we choose two squares sharing a common edge, then we add $1$ to them or $-1$ to them. Show that it is not possible to make all squares $2$, after a finite number of steps.
Let $n \in \mathbb{N}, n \ge 3.$
$a)$ Prove that there exist $z_1,z_2,…,z_n \in \mathbb{C}$ such that $$\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}.$$
$b)$ Which are the values of $n$ for which there exist the complex numbers $z_1,z_2,…,z_n,$ of the same modulus, such that $$\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}?$$
Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12 AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$.
Let $T=\text{TNFTPP}$. How many positive integers are within $T$ of exactly $\lfloor \sqrt T\rfloor$ perfect squares? (Note: $0^2=0$ is considered a perfect square.)
In Sikinia we only pay with coins that have a value of either $11$ or $12$ Kulotnik. In a burglary in one of Sikinia's banks, $11$ bandits cracked the safe and could get away with $5940$ Kulotnik. They tried to split up the money equally - so that everyone gets the same amount - but it just doesn't worked. After a while their leader claimed that it actually isn't possible.
Prove that they didn't get any coin with the value $12$ Kulotnik.
Ali has $6$ types of $2\times2$ squares with cells colored in white or black, and has presented them to Mohammad as forbidden tiles.
$a)$ Prove that Mohammad can color the cells of the infinite table (from each $4$ sides.) in black or white such that there's no forbidden tiles in the table.
$b)$ Prove that Ali can present $7$ forbidden tiles such that Mohammad cannot achieve his goal.
Let $p$ be an odd prime and $n$ a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length $p^{n}$. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by $p^{n+1}$.
[i]Proposed by Alexander Ivanov, Bulgaria[/i]
A parallelepiped is cut by a plane along a 6-gon. Supposed this 6-gon can be put into a certain rectangle $ \pi$ (which means one can put the rectangle $ \pi$ on the parallelepiped's plane such that the 6-gon is completely covered by the rectangle). Show that one also can put one of the parallelepiped' faces into the rectangle $ \pi.$
A best-of-9 series is to be played between two teams; that is, the first team to win 5 games is the winner. The Mathletes have a chance of $\tfrac{2}{3}$ of winning any given game. What is the probability that exactly 7 games will need to be played to determine a winner?
[b]p1.[/b] What is $\sqrt[2015]{2^01^5}$?
[b]p2.[/b] What is the ratio of the area of square $ABCD$ to the area of square $ACEF$?
[b]p3.[/b] $2015$ in binary is $11111011111$, which is a palindrome. What is the last year which also had this property?
[b]p4.[/b] What is the next number in the following geometric series: $1020100$, $10303010$, $104060401$?
[b]p5.[/b] A circle has radius $A$ and area $r$. If $A = r^2\pi$, then what is the diameter, $C$, of the circle?
[b]p6.[/b] If
$$O + N + E = 1$$
$$T + H + R + E + E = 3$$
$$N + I + N + E = 9$$
$$T + E + N = 10$$
$$T + H + I + R + T + E + E + N = 13$$
Then what is the value of $O$?
[b]p7.[/b] By shifting the initial digit, which is $6$, of the positive integer $N$ to the end (for example, $65$ becomes $56$), we obtain a number equal to $\frac{N}{4}$ . What is the smallest such $N$?
[b]p8.[/b] What is $\sqrt[3]{\frac{2015!(2013!)+2014!(2012!)}{2013!(2012!)}}$ ?
[b]p9.[/b] How many permutations of the digits of $1234$ are divisible by $11$?
[b]p10.[/b] If you choose $4$ cards from a normal $52$ card deck (with replacement), what is the probability that you will get exactly one of each suit (there are $4$ suits)?
[b]p11.[/b] If $LMT$ is an equilateral triangle, and $MATH$ is a square, such that point $A$ is in the triangle, then what is $HL/AL$?
[b]p12.[/b] If
$$\begin{tabular}{cccccccc}
& & & & & L & H & S\\
+ & & & & H & I & G & H \\
+ & & S & C & H & O & O & L \\
\hline
= & & S & O & C & O & O & L \\
\end{tabular}$$ and $\{M, A, T,H, S, L,O, G, I,C\} = \{0, 1, 2, 3,4, 5, 6, 7, 8, 9\} $, then what is the ordered pair $(M + A +T + H, [T + e + A +M])$ where $e$ is $2.718...$and $[n]$ is the greatest integer less than or equal to $n$ ?
[b]p13.[/b] There are $5$ marbles in a bag. One is red, one is blue, one is green, one is yellow, and the last is white. There are $4$ people who take turns reaching into the bag and drawing out a marble without replacement. If the marble they draw out is green, they get to draw another marble out of the bag. What is the probability that the $3$rd person to draw a marble gets the white marble?
[b]p14.[/b] Let a "palindromic product" be a product of numbers which is written the same when written back to front, including the multiplication signs. For example, $234 * 545 * 432$, $2 * 2 *2 *2$, and $14 * 41$ are palindromic products whereas $2 *14 * 4 * 12$, $567 * 567$, and $2* 2 * 3* 3 *2$ are not. 2015 can be written as a "palindromic product" in two ways, namely $13 * 5 * 31$ and $31 * 5 * 13$. How many ways can you write $2016$ as a palindromic product without using 1 as a factor?
[b]p15.[/b] Let a sequence be defined as $S_n = S_{n-1} + 2S_{n-2}$, and $S_1 = 3$ and $S_2 = 4$. What is $\sum_{n=1}^{\infty}\frac{S_n}{3^n}$ ?
[b]p16.[/b] Put the numbers $0-9$ in some order so that every $2$-digit substring creates a number which is either a multiple of $7$, or a power of $2$.
[b]p17.[/b] Evaluate
$\dfrac{8+ \dfrac{8+ \dfrac{8+...}{3+...}}{3+ \dfrac{8+...}{3+...}}}{3+\dfrac{8+ \dfrac{8+...}{3+...}}{
3+ \dfrac{8+...}{3+...}}}$, assuming that it is a positive real number.
[b]p18.[/b] $4$ non-overlapping triangles, each of area $A$, are placed in a unit circle. What is the maximum value of $A$?
[b]p19.[/b] What is the sum of the reciprocals of all the (positive integer) factors of $120$ (including $1$ and $120$ itself).
[b]p20.[/b] How many ways can you choose $3$ distinct elements of $\{1, 2, 3,...,4000\}$ to make an increasing arithmetic series?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $G$ be a finite set of real $n \times n$ matrices $\left\{M_{i}\right\}, 1 \leq i \leq r,$ which form a group under matrix multiplication. Suppose that $\textstyle\sum_{i=1}^{r} \operatorname{tr}\left(M_{i}\right)=0,$ where $\operatorname{tr}(A)$ denotes the trace of the matrix $A .$ Prove that $\textstyle\sum_{i=1}^{r} M_{i}$ is the $n \times n$ zero matrix.
Let $a,b, c$ be real numbers with $a+b+c = 2018$.
Suppose $x, y$, and $z$ are the distinct positive real numbers which are satisfied $a = x^2 - yz - 2018, b = y^2 - zx - 2018$ , and $c = z^2 - xy - 2018$.
Compute the value of the following expression $P = \frac{\sqrt{a^3 + b^3 + c^3 - 3abc}}{x^3 + y^3 + z^3 - 3xyz}$
The incircle of the triangle $\triangle{ABC}$ has center at $O$ and it is tangent to the sides $BC$, $AC$ and $AB$ at the points $X$, $Y$ and $Z$, respectively. The lines $BO$ and $CO$ intersect the line $YZ$ at the points $P$ and $Q$, respectively.
Show that if the segments $XP$ and $XQ$ has the same length, then the triangle $\triangle ABC$ is isosceles.
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible comittees that can be formed subject to these requirements.
Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.
In a convex quadrilateral $ABCD$ sides $AB$ and $DC$ are both divided into $m$ equal parts by points $A, S_1 , S_2 , \ldots , S_{m-1} ,B$ and $D,T_1, T_2, \ldots , T_{m-1},C,$ respectively (in this order).
Similarly, sides $BC$ and $AD$ are divided into $n$ equal parts by points $B,U_1,U_2, \ldots, U_{n-1},C$ and $A,V_1,V_2, \ldots,V_{n-1}, D$. Prove that for $1 \leq i \leq m-1$ each of the segments $S_i T_i$ is divided by the segments $U_j V_j$ ($1\leq j \leq n-1$) into $n$ equal parts
Find the largest constant $K\in \mathbb{R}$ with the following property:
if $a_1,a_2,a_3,a_4>0$ are numbers satisfying $a_i^2 + a_j^2 + a_k^2 \geq 2
(a_ia_j + a_ja_k + a_ka_i)$, for every $1\leq i<j<k\leq 4$, then \[a_1^2+a_2^2+a_3^2+a_4^2 \geq K
(a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4).\]