Found problems: 85335
Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits.
Find the area of the region enclosed by the graph of $|x-60|+|y|=|x/4|.$
Let $f : Z_+ \to Z_+$ such that:
a) $f(n + 1) > f(n)$ for all $n \in Z_+$
b) $f(n + f(m)) = f(n) + m + 1$ for all $n,m \in Z_+$
Find $f(2006)$.
An acute triangle $ ABC$ is given. Points $ A_1$ and $ A_2$ are taken on the side $ BC$ (with $ A_2$ between $ A_1$ and $ C$), $ B_1$ and $ B_2$ on the side $ AC$ (with $ B_2$ between $ B_1$ and $ A$), and $ C_1$ and $ C_2$ on the side $ AB$ (with $ C_2$ between $ C_1$ and $ B$) so that
\[ \angle AA_1A_2 \equal{} \angle AA_2A_1 \equal{} \angle BB_1B_2 \equal{} \angle BB_2B_1 \equal{} \angle CC_1C_2 \equal{} \angle CC_2C_1.\]
The lines $ AA_1,BB_1,$ and $ CC_1$ bound a triangle, and the lines $ AA_2,BB_2,$ and $ CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.
A set $A$ of squares is given on a chessboard which is infinite in all directions. On each square of this chessboard which does not belong to $A$ there is a king. On a command all kings may be moved in such a way that each king either remains on its square or is moved to an adjacent square, which may have been occupied by another king before the command. Each square may be occupied by at most one king. Does there exist such a number $k$ and such a way of moving the kings that after $k$ moves the kings will occupy all squares of the chessboard? Consider the following cases:
(a) $A$ is the set of all squares, both of whose coordinates are multiples of $100$. (There is a horizontal line numbered by the integers from $-\infty$ to $+\infty$, and a similar vertical line. Each square of the chessboard may be denoted by two numbers, its coordinates with respect to these axes.)
(b) $A$ is the set of all squares which are covered by $100$ fixed arbitrary queens (i.e. each square covered by at least one queen).
Remark:
If $A$ consists of just one square, then $k = 1$ and the required way is the following:
all kings to the left of the square of $A$ make one move to the right.
For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is fibonatic when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not fibonatic integers.
In an $n$-by-$m$ grid, $1$ row and $1$ column are colored blue, the rest of the cells are white. If precisely $\frac{1}{2010}$ of the cells in the grid are blue, how many values are possible for the ordered pair $(n,m)$
In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$
Robert tiles a $420 \times 420$ square grid completely with $1 \times 2$ blocks, then notices that the two diagonals of the grid pass through a total of $n$ blocks. Find the sum of all possible values of $n$.
As shown in figure , it is known that $ABCD$ is parallelogram, $A,B,C$ lie on circle $\odot O_1$, $AD$ and $BD$ intersect $\odot O$ at points $E$ and $F$ respectively, $C,D,F$ lie on circle $\odot O_2$, $AD$ intersects $\odot O_2$ at point $G$. If the radii of circles $\odot O_1$, $\odot O_2$ are $R_1, R_2$ respectively, prove that $\frac{EG}{AD}=\frac{R_2^2}{R_1^2}$.
[img]https://cdn.artofproblemsolving.com/attachments/d/f/1d9925a77d4f3fe068bd24364fb396eaa9a27a.png[/img]
Find the largest value of the expression $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$ , where $p,R, r$ is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle.
Given positive integers $x$ and $y$ such that $xy^2$ is a perfect cube, prove that $x^2y$ is also a perfect cube.
A rectangle with sides $ n$ and $p$ is divided into $np$ unit squares. Initially there are m unitary squares painted black and the remaining painted white. The following processoccurs repeatedly: if a unit square painted white has at minus two sides in common with squares painted black then Its color also turns black. Find the smallest integer $m$ that satisfies the property: there exists an initial position of $m$ black unit squares such that the entire $ n \times p$ rectangle is painted black when repeat the process a finite number of times.
Two teams take part in a competition. There are 7 members (numbered 1 to 7). Two member 1 start a competition first. The failer is sifted out, and the winner start a new competition with member 2 in the other team. ... When all members of a team are out, the competition ends. The number of possible situations is________.
Consider the function $ f(\theta) \equal{} \int_0^1 |\sqrt {1 \minus{} x^2} \minus{} \sin \theta|dx$ in the interval of $ 0\leq \theta \leq \frac {\pi}{2}$.
(1) Find the maximum and minimum values of $ f(\theta)$.
(2) Evaluate $ \int_0^{\frac {\pi}{2}} f(\theta)\ d\theta$.
Chords $BC$ and $DE$ of circle $\omega$ meet at point $A$. The line through $D$ parallel to $BC$ meets $\omega$ again at $F$, and $FA$ meets $\omega$ again at $T$. Let $M = ET \cap BC$ and let $N$ be the reflection of $A$ over $M$. Show that $(DEN)$ passes through the midpoint of $BC$.
Let \[S=\sqrt{1+\dfrac1{1^2}+\dfrac1{2^2}}+\sqrt{1+\dfrac1{2^2}+\dfrac1{3^2}}+\cdots+\sqrt{1+\dfrac1{2007^2}+\dfrac1{2008^2}}.\] Compute $\lfloor S^2\rfloor$.
Let $f:\mathbb{N} \rightarrow \mathbb{N} \setminus \left \{ 1 \right \}$ be a function which satisfies both the inequality $f(a+f(a)) \leq 2a+3$ and the equation
$$f(f(a)+b) = f(a+f(b)),$$
for any two $a,b \in \mathbb{N}$. Let $g:\mathbb{N} \rightarrow \mathbb{N}$ be defined with: $g(a)$ is the largest prime divisor of $f(a)$. Prove that there exist integers $a>b>2024$ such that $b|a$ and $g(a) = g(b)$.
$O$ is the circumcenter of acute $\Delta ABC$, $H$ is the Orthocenter. $AD \bot BC$, $EF$ is the perpendicular bisector of $AO$,$D,E$ on the $BC$. Prove that the circumcircle of $\Delta ADE$ through the midpoint of $OH$.
Let $q_{0},q_{1},...$ be a sequence of integers such that
a) for any $m>n$ we have $m-n\mid q_{m}-q_{n}$, and
b) $|q_{n}|\leq n^{10}, \ \forall n\geq 0$.
Prove there exists a polynomial $Q$ such that $q_{n}=Q(n), \ \forall n\geq 0$.
Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized.
Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$
[i]2010 Saitama University entrance exam/Mathematics[/i]
Last Edited
A finite sequence of natural numbers $a_1, a_2, \dots, a_n$ is given. A sub-sequence $a_{k+1}, a_{k+2}, \dots, a_l$ will be called a [i]repetition[/i] if there exists a natural number $p\leq \frac{l-k}2$ such that $a_i=a_{i+p}$ for $k+1\leq i\leq l-p$, but $a_i\neq a_{i+p}$ for $i=k$ (if $k>0$) and $i=l-p+1$ (if $l<n$).
Show that the sequence contains less than $n$ repetitions.
How many positive integers $n \le \text{lcm}(1,2, \ldots, 100)$ have the property that $n$ gives different remainders when divided by each of $2,3, \ldots, 100$?
Triangle $ ABC$ has $ AB\equal{}13$ and $ AC\equal{}15$, and the altitude to $ \overline{BC}$ has length $ 12$. What is the sum of the two possible values of $ BC$?
$ \textbf{(A)}\ 15\qquad
\textbf{(B)}\ 16\qquad
\textbf{(C)}\ 17\qquad
\textbf{(D)}\ 18\qquad
\textbf{(E)}\ 19$
Quadrangle $ ABCD$ is inscribed in a circle with radius 1 in such a way that the diagonal $ AC$ is a diameter of the circle, while the other diagonal $ BD$ is as long as $ AB$. The diagonals intersect at $ P$. It is known that the length of $ PC$ is $ 2/5$. How long is the side $ CD$?