Found problems: 85335
A Sudoku matrix is defined as a $ 9\times9$ array with entries from $ \{1, 2, \ldots , 9\}$ and with the constraint that each row, each column, and each of the nine $ 3 \times 3$ boxes that tile the array contains each digit from $ 1$ to $ 9$ exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has equal probability of being chosen). We know two of the squares in this matrix, as shown. What is the probability that the square marked by ? contains the digit $ 3$?
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We consider the real sequence ($x_n$) defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2 x_{n}$ for $n=0,1,2,...$
We define the sequence ($y_n$) by $y_n=x^2_n+2^{n+2}$ for every nonnegative integer $n$.
Prove that for every $n>0, y_n$ is the square of an odd integer.
Let $f(x)=x+\cfrac{1}{2x+\cfrac{1}{2x+\cfrac{1}{2x+\cdots}}}$. Find $f(99)f^\prime (99)$.
A set of $8$ problems was prepared for an examination. Each student was given $3$ of them. No two students received more than one common problem. What is the largest possible number of students?
We have a hexagon such that all its edges touch a circle. If five of the edges have lengths 1,2,3,4, and 5 as on the figure, how long is the last edge?
[img]http://i250.photobucket.com/albums/gg265/geometry101/HexagonImage.jpg[/img]
A. 1
B. 3
C. 15/8
D. $ \sqrt{15}$
E. Not uniquely determined, more than one possibility
Given a subset of $\{1,2,...,n\}$, we define its [i]alternating sum [/i] in the following way: we order the elements of the subset in descending order and, starting with the largest, we alternately add and subtract the successive numbers. For example, the [i]alternating sum[/i] of the set $\{1,3,4,6,8\}$ is $8-6+4-3+1 = 4$. Determines the sum of the alternating sums of all subsets of $\{1,2,...,10\}$ with an odd number of elements.
Let $K,L, M$ be the midpoints of $BC,CA,AB$ repectively on a given triangle $ABC$. Let $\Gamma$ be a circle passing through $B$ and tangent to the circumcircle of $KLM$, say at $X$. Suppose that $LX$ and $BC$ meet at $\Gamma$ . Show that $CX$ is perpendicular to $AB$.
A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$?
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label("$1$",(-3.187,1.5513),S);[/asy]$ \textbf{(A)}\ 2\sqrt {5} \minus{} \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} \minus{} \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 \plus{} 2\sqrt {3}}{2}$
Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.
A positive integers has exactly $81$ divisors, which are located in a $9 \times 9$ table such that for any two numbers in the same row or column one of them is divisible by the other one.
Find the maximum possible number of distinct prime divisors of $n$
[b]1.[/b] Let $a_{v} $ and $b_{v} $, ${v= 1,2,\dots,n} $, be real numbers such that
$a_{1}\geq a_{2} \geq a_{3}\geq\dots\geq a_{n}> 0 $ and
$b_{1}\geq a_{1}, b_{1}b_{2}\geq a_{1}a_{2},\dots,b_{1}b_{2}\dots b_{n}\geq a_{1}a_{2}\dots a_{n} $
Show that $b_{1}+b_{2}+\dots+b_{n}\geq a_{1}+a_{2}+\dots+a_{n} $ [b](S. 4)[/b]
Two players play alternately. The first player is given a pair of positive integers $(x_1, y_1)$. Each player must replace the pair $(x_n, y_n)$ that he is given by a pair of non-negative integers $(x_{n+1}, y_{n+1})$ such that $x_{n+1} = min(x_n, y_n)$ and $y_{n+1} = max(x_n, y_n)- k\cdot x_{n+1}$ for some positive integer $k$. The first player to pass on a pair with $y_{n+1} = 0$ wins. Find for which values of $x_1/y_1$ the first player has a winning strategy.
A game is played with two players and an initial stack of $n$ pennies $(n \geq 3)$. The players take turns choosing one of the stacks of pennies on the table and splitting it into two stacks. The winner is the player who makes a move that causes all stacks to be of height $1$ or $2.$ For which starting values of n does the player who goes first win, assuming best play by both players?
A plane is drawn through the height of a regular tetrahedron, which intersects the planes of the lateral faces along $ 3 $ lines that form angles $ \alpha $, $ \beta $, $ \gamma $ with the plane of the tetrahedron's base. Prove that
$$ tg^2 \alpha + tg^2 \beta + tg^2 \gamma =12.$$
$I$ is the incenter of $ABC$. $D$ is the intersection of $AI$ and the circumcircle of $ABC$, not $A$. And $P$ is a midpoint of $BI$. If $CI=2AI$, show that $AB=PD$.
Solve in $ \mathbb{R} $ the equation $ 4^{x^2-x}=\log_2 x+\sqrt{x-1} +14. $
[i]Marin Tolosi[/i]
For each positive integer $n$, let $f_n$ be a real-valued symmetric function of $n$ real variables. Suppose that for all $n$ and all real numbers $x_1,\ldots,x_n, x_{n+1},y$ it is true that
$\;(1)\; f_{n}(x_1 +y ,\ldots, x_n +y) = f_{n}(x_1 ,\ldots, x_n) +y,$
$\;(2)\;f_{n}(-x_1 ,\ldots, -x_n) =-f_{n}(x_1 ,\ldots, x_n),$
$\;(3)\; f_{n+1}(f_{n}(x_1,\ldots, x_n),\ldots, f_{n}(x_1,\ldots, x_n), x_{n+1}) =f_{n+1}(x_1 ,\ldots, x_{n}).$
Prove that $f_{n}(x_{1},\ldots, x_n) =\frac{x_{1}+\cdots +x_{n}}{n}.$
At a party, five of Ryan’s friends arrive, each hanging their coats on the coat rack. When they leave, Ryan hands out coats in a random order to his friends. What is the probability that at least half of them receive the right coat? (Half of them is $3$ or more)
[i]2015 CCA Math Bonanza Team Round #7[/i]
Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots.
(Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$).
[i]Proposed by Malay Mahajan[/i]
Let $N$ be the fifth largest number that can be created by combining $2021$ $1$'s using addition, multiplication, and exponentiation, in any order (parentheses are allowed). If $f(x)=\log_2(x)$, and $k$ is the least positive integer such that $f^k(N)$ is not a power of $2$, what is the value of $f^k(N)$?
(Note: $f^k(N)=f(f(\cdots(f(N))))$, where $f$ is applied $k$ times.)
[i]Proposed by Adam Bertelli[/i]
In the isosceles triangle $ABC$, with $AB=AC$, $D$ is a point on the extension of $CA$ such that $DB$ is perpendicular to $BC$, $E$ is a point on the extension of $BC$ such that $CE=2BC$, and $F$ is a point on $ED$ such that $FC$ is parallel to $AB$. Prove that $FA$ is parallel to $BC$.
Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex).
Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.
The sequence $a_1, a_2, \dots, a_n$ consists of the numbers $1, 2, \dots, n$ in some order. For which positive integers $n$ is it possible that the $n+1$ numbers $0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n$ all have di fferent remainders when divided by $n + 1$?
Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.
Let $P(x)$ be a polynomial in one variable with integer coefficients. Prove that the number of pairs $(m,n)$ of positive integers such that $2^n + P(n) = m!$, is finite.