Found problems: 85335
This ISL 2005 problem has not been used in any TST I know. A pity, since it is a nice problem, but in its shortlist formulation, it is absolutely incomprehensible. Here is a mathematical restatement of the problem:
Let $k$ be a nonnegative integer.
A forest consists of rooted (i. e. oriented) trees. Each vertex of the forest is either a leaf or has two successors. A vertex $v$ is called an [i]extended successor[/i] of a vertex $u$ if there is a chain of vertices $u_{0}=u$, $u_{1}$, $u_{2}$, ..., $u_{t-1}$, $u_{t}=v$ with $t>0$ such that the vertex $u_{i+1}$ is a successor of the vertex $u_{i}$ for every integer $i$ with $0\leq i\leq t-1$. A vertex is called [i]dynastic[/i] if it has two successors and each of these successors has at least $k$ extended successors.
Prove that if the forest has $n$ vertices, then there are at most $\frac{n}{k+2}$ dynastic vertices.
The altitudes $CC_1$ and $BB_1$ are drawn in the acute triangle $ABC$. The bisectors of angles $\angle BB_1C$ and $\angle CC_1B$ intersect the line $BC$ at points $D$ and $E$, respectively, and meet each other at point $X$.
Prove that the intersection points of circumcircles of the triangles $BEX$ and $CDX$ lie on the line $AX$.
[i](A. Voidelevich)[/i]
Let $x,y,z$ be real numbers from the interval $[0,1]$. Determine the maximum value of expression $$W=y\cdot \sqrt{1-x}+z\cdot\sqrt{1-y}+x\cdot\sqrt{1-z}$$
Suppose that $P(x)=a_1x+a_2x^2+\ldots+a_nx^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)}=b_0+b_1x+b_2x^2+\ldots$ for all $x.$ Prove that $b_k$ is nonzero for all $k \geq 0.$
Let $ABCD$ be a convex quadrilateral and draw regular triangles $ABM, CDP, BCN, ADQ$, the first two outward and the other two inward. Prove that $MN = AC$. What can be said about the quadrilateral $MNPQ$?
A simple graph has $N{}$ vertices and less than $3(N-1)/2$ edges. Prove that its vertices can be divided into two non-empty groups so that each vertex has at most one neighbor in the group it doesn't belong to.
You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2\times 1$ and $3\times 1$. The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible?
(Walther Janous)
In order to determine $\frac{1}{A}$ for $A>0$, one can use the iteration $X_{k+1}=X_{k}(2-AX_{k}),$ where $X_0$ is a selected starting value. Find the limitation, if any, on the starting value $X_0$ so that the above iteration converges to $\frac{1}{A}.$
Let $x_1,x_2,\cdots,x_n$ be non-negative real numbers such that $x_ix_j\le 4^{-|i-j|}$ $(1\le i,j\le n)$. Prove that\[x_1+x_2+\cdots+x_n\le \frac{5}{3}.\]
Let \(ABC\) be an acute triangle with circumcircle \(\omega\). Let \(P\) be a variable point on the arc \(BC\) of \(\omega\) not containing \(A\). Squares \(BPDE\) and \(PCFG\) are constructed such that \(A\), \(D\), \(E\) lie on the same side of line \(BP\) and \(A\), \(F\), \(G\) lie on the same side of line \(CP\). Let \(H\) be the intersection of lines \(DE\) and \(FG\). Show that as \(P\) varies, \(H\) lies on a fixed circle.
[i]Proposed by Karthik Vedula[/i]
Let $ a,b,c,d$ be real numbers with sum 0. Prove the inequality:
\[ (ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd)^2 \plus{} 12\geq 6(abc \plus{} abd \plus{} acd \plus{} bcd).
\]
Let $x_1 \le x_2 \le \dots < x_n$ (with $n \ge 2$) and let $S$ be the set of all the $x_i$. Let $T$ be a randomly chosen subset of $S$. What is the expected value of the indexed alternating sum of $T$ ? Express your answer in terms of the $x_i$.
Note: We define the indexed alternating sum of $T$ as
\[
\sum_{i=1}^{|T|} (-1)^{i+1}(i) T[i],
\]
where $T[i]$ is the ith element of $T$ when listed in increasing order. For example, if $T = \{1, 3, 5\}$
then the indexed alternating sum of $T$ is
\[
1 \cdot 1 - 2 \cdot 3 + 3 \cdot 5 = 10.
\]
Alternating sums of empty sets are defined to be $0$.
On a table there are all 8 possible horizontal bars $1\times3$ such that each $1\times1$ square is either white or gray (see the figure). It is allowed to move them in any direction by any (not necessarily integer) distance. We may not rotate them or turn them over. Is it possible to move the bars so that they do not overlap, all the white points form a polygon bounded by a closed non-self-intersecting broken line and the same is true for all the gray points?
[i]Mikhail Ilyinsky[/i]
Given $\lambda^3 - 2\lambda^2- 1 = 0$ for some real $\lambda$ prove that $[\lambda[\lambda[\lambda n]]] - n$ is odd for any positive integer $n$ .
(I Voronovich)
A polynomial $ P(x)$ with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable $ x$) with integer coefficients.For example,the polynomials $ x^3 \minus{} 1$ and $ 9x^3 \minus{} 3x^2 \plus{} 3x \plus{} 7 \equal{} (x \minus{} 1)^3 \plus{} (2x)^3 \plus{} 2^3$ are good.
a)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7$ good?
b)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7 \plus{} 3x^{2008}$ good?
Justify your answers.
On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1$, $y_2$, $\ldots$, $y_{24}$ on the number line such that for all $k$ such that $1\leq k\leq 12$, we have that $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of \[\textstyle\sum_{n=1}^{24}(|y_n-1|+|y_n+1|).\]
In Prime Land, there are seven major cities, labelled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e. we take the indices modulo $7$. Al initially starts at city $C_0$.
Each minute for ten minutes, Al flips a fair coin. If the coin land heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\tfrac{m}{1024}$, find $m$.
[i]Proposed by Ray Li[/i]
Sixteen points are placed in the centers of a $4 \times 4$ chess table in the following way:
• • • •
• • • •
• • • •
• • • •
(a) Prove that one may choose $6$ points such that no isoceles triangle can be drawn with the vertices at these points.
(b) Prove that one cannot choose $7$ points with the above property.
In the land of Chaina, people pay each other in the form of links from chains. Fiona, originating from Chaina, has an open chain with $2018$ links. In order to pay for things, she decides to break up the chain by choosing a number of links and cutting them out one by one, each time creating $2$ or $3$ new chains. For example, if she cuts the $1111$th link out of her chain first, then she will have $3$ chains, of lengths $1110$, $1$, and $907$. What is the least number of links she needs to remove in order to be able to pay for anything costing from $1$ to $2018$ links using some combination of her chains?
[i]2018 CCA Math Bonanza Individual Round #10[/i]
Let $k$ be the incircle of triangle $ABC$. It touches $AB=c, BC=a, AC=b$ at $C_1, A_1, B_1$, respectively. Suppose that $KC_1$ is a diameter of the incircle. Let $C_1A_1$ intersect $KB_1$ at $N$ and $C_1B_1$ intersect $KA_1$ at $M$. Find the length of $MN$.
Let $ABCD$ be a rectangle with $AB\ge BC$ Point $M$ is located on the side $(AD)$, and the perpendicular bisector of $[MC]$ intersects the line $BC$ at the point $N$. Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot \angle BCQ $, show that the quadrilateral $ABCD$ is a square.
You have a $10\times 10$ grid of squares. You write a number in each square as follows: you write $1$, $2$, $3$ ,$ ...$ ,$10$ from left to right across the top row, then $11$, $12$, $...$, $20$ across the second row, and so on, ending with a $100$ in the bottom right square. You then write a second number in each square, writing $1$, $2$, $3$ ,$ ...$ ,$10$ in the first column (from top to bottom), then $11$, $12$, $...$, $20$ in the second column, and so forth.
When this process is finished, how many squares will have the property that their two numbers sum to $101$?
Find all functions $f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$ such that for all $x, y \in \mathbb{Q}[\sqrt{2}]$,
$$
f(xy) = f(x)f(y) \quad \text{and} \quad f(x + y) = f(x) + f(y),
$$
where $\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$.
[I]Proposed by Stijn Cambie, Belgium[/i]
A number is chosen uniformly at random from the set of all positive integers with at least two digits, none of which are repeated. Find the probability that the number is even.
The product \[\prod^{63}_{k=4} \frac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = \frac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot \frac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot \frac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots \frac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\] is equal to $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$