Found problems: 85335
The numbers from 1 to 1996 are written down ------ 12345678910111213.... How many zeros are written?
A. 489
B. 699
C. 796
D. 996
E. None of these
Suppose that $X$ and $Y$ are two metric spaces and $f:X \longrightarrow Y$ is a continious function. Also for every compact set $K \subseteq Y$, it's pre-image $f^{pre}(K)$ is a compact set in $X$. Prove that $f$ is a closed function, i.e for every close set $C\subseteq X$, it's image $f(C)$ is a closed subset of $Y$.
Find the maximum number of pairwise disjoint sets of the form \[S_{a,b}=\{n^{2}+an+b \; \vert \; n \in \mathbb{Z}\},\] with $a,b \in \mathbb{Z}$.
In the following base-$10$ equation, each of the letter represents a unique digit: $AM\cdot PM=ZZZ$. Find the sum of $A+M+P+Z$.
$\text{(A) }15\qquad\text{(B) }17\qquad\text{(C) }19\qquad\text{(D) }20\qquad\text{(E) }21$
We need not restrict our number system radix to be an integer. Consider the phinary numeral system in which the radix is the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$ and the digits $0$ and $1$ are used. Compute $1010100_{\phi}-.010101_{\phi}$
Let $k$ be a circle with diameter $AB$. Let $C$ be a point on the straight line $AB$, so that $B$ between $A$ and $C$ lies. Let $T$ be a point on $k$ such that $CT$ is a tangent to $k$. Let $l$ be the parallel to $CT$ through $A$ and $D$ the intersection of $l$ and the perpendicular to $AB$ through $T$. Show that the line $DB$ bisects segment $CT$.
What is the volume of tetrahedron $ABCD$ with edge lengths $AB=2, AC=3, AD=4, BC=\sqrt{13}, BD=2\sqrt{5},$ and $CD=5$?
$\textbf{(A) }3 \qquad \textbf{(B) }2\sqrt{3} \qquad \textbf{(C) }4 \qquad \textbf{(D) }3\sqrt{3} \qquad \textbf{(E) }6$
Which number is bigger : the number of positive integers not exceeding 1000000 that can be represented by the form $2x^{2}-3y^{2}$ with integral $x$ and $y$ or that of positive integers not exceeding 1000000 that can be represented by the form $10xy-x^{2}-y^{2}$ with integral $x$ and $y?$
[i]Proposed by A. Golovanov[/i]
How many five-digit multiples of 3 end with the digit 6 ?
A positive integer $n$ is [i]nice[/i] if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. How many numers in the set $\{2010, 2011, 2012,\ldots,2019\}$ are nice?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$ with $AC \perp BD$. Let $P=AC \cap BD$ and $W,X,Y,Z$ be the projections of $P$ on the lines $AB, BC, CD, DA$ respectively. Let $E,F,G,H$ be the mid-points of sides $AB, BC, CD, DA$ respectively.
(a) Prove that $E,F,G,H,W,X,Y,Z$ are concyclic.
(b) If $R$ is the radius of $\Omega$ and $d$ is the distance between its centre and $P$, then find the radius of the circle in (a) in terms of $R$ and $d$.
On the control board of a nuclear station, there are $n$ electric switches ($n > 0$), all in one row. Each switch has two possible positions: up and down. The switches are connected to each other in such a way that, whenever a switch moves down from its upper position, its right neighbour (if it exists) automatically changes position. At the beginning, all switches are down. The operator of the board first changes the position of the leftmost switch once, then the position of the second leftmost switch twice etc., until eventually he changes the position of the rightmost switch n times. How many switches are up after all these operations?
Determine all real solutions $x,y$ to the system of equations
$$\begin{cases} x^2 + y^2 = 1 \\ x^6 + y^6 = \dfrac{7}{16} \end{cases}$$
Define set $S_n=\{1,2,\cdots,n\}$. $X$ is a subset of $S_n$. We call sum of all numbers in $X$ [i]capacity[/i] ([i]capacity[/i] of empty set is $0$). If [i]capacity[/i] of $X$ is odd/even, then we call it [i]odd/even subset[/i].
[b](a)[/b] Prove that the number of [i]odd subsets[/i] and [i]even subsets[/i] of $S_n$ are the same.
[b](b)[/b] Prove that the sum of [i]capacity[/i] of all [i]odd subsets[/i] and [i]even subsets[/i] are the same when $n\geq3$.
[b](c)[/b] Calculate the sum of [i]capacity[/i] of all [i]odd subsets[/i] when $n\geq3$.
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.
We write the decimal expressions of $\sqrt{2}$ and $\sqrt{3}$ as \[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\] where each $a_i$ or $b_i$ is a digit between 0 and 9. Prove that there exist at least 1000 values of $i$ between $1$ and $10^{1000}$ such that $a_i\neq b_i$.
Find all triplets of positive rational numbers $(m,n,p)$ such that the numbers $m+\frac 1{np}$, $n+\frac 1{pm}$, $p+\frac 1{mn}$ are integers.
[i]Valentin Vornicu, Romania[/i]
$10$ children each have a lunchbox which they store in a basket before entering their classroom. However, being messy children, their lunchboxes get mixed up. When leaving the classroom each student picks up a lunchbox at random. Define a [i]cyclic triple[/i] of students $(A, B, C)$ to be three distinct students such that $A$ has $B$’s lunchbox, $B$ has $C$’s lunchbox, and $C$ has $A$’s lunchbox. Two cyclic triples are considered the same if they contain the same three students (even if in a different order). Determine the expected value of the number of cyclic triples.
[i]2015 CCA Math Bonanza Individual Round #14[/i]
Each of the $n$ students in a class sent a card to each of his $m$ colleagues. Prove that if $2m + 1 > n$, then at least two students sent cards to each other.
Let $ f_m: \mathbb R \to \mathbb R$, $ f_m(x)\equal{}(m^2\plus{}m\plus{}1)x^2\minus{}2(m^2\plus{}1)x\plus{}m^2\minus{}m\plus{}1,$ where $ m \in \mathbb R$.
1) Find the fixed common point of all this parabolas.
2) Find $ m$ such that the distance from that fixed point to $ Oy$ is minimal.
Bisector $AL$ is drawn in an acute triangle $ABC$. On the line $LA$ beyond the point $A$, the point K is chosen with $AK = AL$. Circumcirles of triangles $BLK$ and $CLK$ intersect segments $AC$ and $AB$ at points $P$ and $Q$
respectively. Prove that lines $PQ$ and $BC$ are parallel.
Evaluate $$\prod_{k=1}^{14} cos \big(\frac{k\pi}{15}\big)$$
There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points?
Given a polynomial $P$ of degree $2016$ with real coefficients and a quadratic polynomial $Q$ with real coefficients. Is it possible that the roots of the polynomial $P (Q(x))$ are exactly all these numbers:
$$-2015, -2014, . . . , -2, -1, 1, 2, . . . , 2016, 2017?$$
Find all functions \( f:\mathbb{Z} \to \mathbb{Z} \) such that
\[
f(x + f(y) - 2y) + f(f(y)) = f(x)
\]
for all integers \( x \) and \( y \).