Found problems: 85335
A real function has been assigned to every cell of an $n \times n$ table. Prove that a function can be assigned to each row and each column of this table such that the function assigned to each cell is equivalent to the combination of functions assigned to the row and the column containing it.
Let $(a,b)$ be a pair of natural numbers. Henning and Paul play the following game. At the beginning there are two piles of $a$ and $b$ coins respectively. We say that $(a,b)$ is the [i]starting position [/i]of the game. Henning and Paul play with the following rules:
$\bullet$ They take turns alternatively where Henning begins.
$\bullet$ In every step each player either takes a positive integer number of coins from one of the two piles or takes same natural number of coins from both piles.
$\bullet$ The player how take the last coin wins.
Let $A$ be the set of all positive integers like $a$ for which there exists a positive integer $b<a$ such that Paul has a wining strategy for the starting position $(a,b)$. Order the elements of $A$ to construct a sequence $a_1<a_2<a_3<\dots$
$(a)$ Prove that $A$ has infinity many elements.
$(b)$ Prove that the sequence defined by $m_k:=a_{k+1}-a_{k}$ will never become periodic. (This means the sequence $m_{k_0+k}$ will not be periodic for any choice of $k_0$)
Determine all functions $f: R\to R$, such that $f (2x + f (y)) = x + y + f (x)$ for all $x, y \in R$.
(Gerhard Kirchner)
Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly.
Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.
The king assembled 300 wizards and gave them the following challenge. For this challenge, 25 colors can be used, and they are known to the wizards. Each of the wizards receives a hat of one of those 25 colors. If for each color the number of used hats would be written down then all these number would be different, and the wizards know this. Each wizard sees what hat was given to each other wizard but does not see his own hat. Simultaneously each wizard reports the color of his own hat. Is it possible for the wizards to coordinate their actions beforehand so that at least 150 of them would report correctly?
Real numbers $a, b, c$ are not equal $0$ and are solution of the system:
$\begin{cases} a^2 + a = b^2 \\ b^2 + b = c^2 \\ c^2 +c = a^2 \end{cases}$
Prove that $(a - b)(b - c)(c - a) = 1$.
Let $H$ be the orthocenter of an acute-angled triangle $ABC$; $A_1, B_1, C_1$ be the touching points of the incircle with $BC, CA, AB$ respectively; $E_A, E_B, E_C$ be the midpoints of $AH, BH, CH$ respectively. The circle centered at $E_A$ and passing through $A$ meets for the second time the bisector of angle $A$ at $A_2$; points $B_2, C_2$ are defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.
In rectangle $ABCD$, points $E$ and $F$ are on sides $\overline{BC}$ and $\overline{AD}$, respectively. Two congruent semicircles are drawn with centers $E$ and $F$ such that they both lie entirely on or inside the rectangle, the semicircle with center $E$ passes through $C$, and the semicircle with center $F$ passes through $A$. Given that $AB=8$, $CE=5$, and the semicircles are tangent, find the length $BC$.
[i]Proposed by Ada Tsui[/i]
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(n))+f(n)=2n+2001 \text{ or }2n+2002.\]
Kiko and Ñoño play with a rod of length $2n$ where $n \le 3$ is an integer. Kiko cuts the rod in $ k \le 2n$ pieces of integer lengths. Then Ñoño has to arrange these pieces so that they form a hexagon of equal opposite sides and equal angles. The pieces can not be split and they all have to be used. If Ñoño achieves his goal, he wins, in any other case, Kiko wins. Determine which victory can be secured based on $k$.
If $a,b,c$ be the lengths of the sides of a triangle. Let $R$ denote its circumradius. Prove that
\[ R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}\]
When does equality hold?
Evaluate
$\frac{\log_{10}20^2 \cdot \log_{20}30^2 \cdot \log_{30}40^2 \cdot \cdot \cdot \log_{990}1000^2}{\log_{10}11^2 \cdot \log_{11}12^2 \cdot \log_{12}13^2 \cdot \cdot \cdot \log_{99}100^2}$
.
Let $ABCD$ be a rectangle, and let $\omega$ be a circular arc passing through the points $A$ and $C$.
Let $\omega_{1}$ be the circle tangent to the lines $CD$ and $DA$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$.
Similiarly let $\omega_{2}$ be the circle tangent to the lines $AB$ and $BC$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$.
Denote by $r_{1}$ and $r_{2}$ the radii of the circles $\omega_{1}$ and $\omega_{2}$, respectively, and by $r$ the inradius of triangle $ABC$.
[b](a)[/b] Prove that $r_{1}+r_{2}=2r$.
[b](b)[/b] Prove that one of the two common internal tangents of the two circles $\omega_{1}$ and $\omega_{2}$ is parallel to the line $AC$ and has the length $\left|AB-AC\right|$.
The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is
$ \textbf{(A)}\ 20 \qquad
\textbf{(B)}\ 18 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 8$
Given is an integer sequence $\{a_n\}_{n \ge 0}$ such that $a_{0}=2$, $a_{1}=3$ and, for all positive integers $n \ge 1$, $a_{n+1}=2a_{n-1}$ or $a_{n+1}= 3a_{n} - 2a_{n-1}$. Does there exist a positive integer $k$ such that $1600 < a_{k} < 2000$?
Nelson is having his friend drop his unique bouncy ball from a $12$ foot building, and Nelson will only catch the ball at the peak of its trajectory between bounces. On any given bounce, there is an $80\%$ chance that the next peak occurs at $\frac13$ the height of the previous peak and a $20\%$ chance that the next peak occurs at $3$ times the height of the previous peak (where the first peak is at $12$ feet). If Nelson can only reach $4$ feet into the air and will catch the ball as soon as possible, the probability that Nelson catches the ball after exactly $13$ bounces is $2^a \times 3^b \times 5^c \times 7^d \times 11^e$ for integers $a, b, c, d$, and $e$. Find $|a| + |b| + |c| + |d| + |e|$.
Prove that there are infinitely many natural numbers, the arithmetic mean and geometric mean of the divisors which are both integers.
Complex numbers $a,b,w,x,y,z,p$ satisfy
\begin{align*}
\frac{(x-w)\lvert a-w \rvert}{(a-w)\lvert x-w \rvert}&=\text{(cyclic variants)};\\
\frac{(z-w)\lvert b-w \rvert}{(b-w)\lvert z-w \rvert}&=\text{(cyclic variants)};\\
p &= \frac{\sum_{\text{cyc}} \frac w{\lvert p-w \rvert}}{\sum_{\text{cyc}}\frac1{\lvert p-w \rvert}};
\end{align*}
where cyclic sums, equations, etc. are wrt $w,x,y,z$.
Prove that there exists a real $k$ such that
\[\sum_{\text{cyc}} \frac{(x-w)(a-w)}{\lvert x-w\rvert (p-w)}
=k\sum_{\text{cyc}} \frac{(z-w)(b-w)}{\lvert z-w\rvert(p-w)}.\]
[i]Neal Yan[/i]
Let $S$ be the sum of all possible values of $a \cdot c$ such that $$a^3+3ab^2-72ab+432a=4c^3$$ if $a$, $b$, and $c$ are positive integers, $a+b > 11$, $a > b-13$, and $c \le 1000$. Find the sum of all distinct prime factors of $S$.
[i]Proposed by Kevin Zhao[/i]
How many ordered triples of positive integers $(a,b,c)$ are there such that $(2a+b)(2b+a)=2^c$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the preceding}
$
The sequence $a_{n,k} \ , k = 1, 2, 3,\ldots, 2^n \ , n = 0, 1, 2,\ldots,$ is defined by the following recurrence formula:
\[a_1 = 2,\qquad a_{n,k} = 2a_{n-1,k}^3, \qquad , a_{n,k+2^{n-1}} =\frac 12 a_{n-1,k}^3\]\[\text{for} \quad k = 1, 2, 3,\ldots, 2^{n-1} \ , n = 0, 1, 2,\ldots\]
Prove that the numbers $a_{n,k}$ are all different.
1-Let $A$ and $B$ be two diametrically opposite points on a circle with radius $1$. Points $P_1,P_2,...,P_n$ are arbitrarily chosen on the circle. Let a and b be the geometric means of the distances of $P_1,P_2,...,P_n$ from $A$ and $B$, respectively. Show that at least one of the numbers $a$ and $b$ does not exceed $\sqrt{2}$
Robert has five beads in his hand, with the letters $C, M, I, M,$ and $C,$ and he wants to make a circular bracelet spelling "$CMIMC.$" However, the power went out, so Robert can no longer see the beads in his hand. Thus, he puts the five beads on the bracelet randomly, hoping that the bracelet, when possibly rotated or flipped, spells out "$CMIMC.$" What is the probability that this happens? (Robert doesn’t care whether some letters appear upside down or backwards.)
Let $a$ and $b$ be relatively prime positive integers such that \[\dfrac ab=\dfrac1{2^1}+\dfrac2{3^2}+\dfrac3{2^3}+\dfrac4{3^4}+\dfrac5{2^5}+\dfrac6{3^6}+\cdots,\] where the numerators always increase by $1$, and the denominators alternate between powers of $2$ and $3$, with exponents also increasing by $1$ for each subsequent term. Compute $a+b$.
How many integers $ N$ less than 1000 can be written as the sum of $ j$ consecutive positive odd integers from exactly 5 values of $ j\ge 1$?