Found problems: 85335
The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours).
Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected.
[i]V. Dolnikov[/i]
[b]EDIT.[/b] It is confirmed by the official solution that the graph is tacitly assumed to be [b]finite[/b].
Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. (See figure.) Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993?
[asy]
int x=101, y=3*floor(x/4);
draw(Arc(origin, 1, 360*(y-3)/x, 360*(y+4)/x));
int i;
for(i=y-2; i<y+4; i=i+1) {
dot(dir(360*i/x));
}
label("3", dir(360*(y-2)/x), dir(360*(y-2)/x));
label("2", dir(360*(y+1)/x), dir(360*(y+1)/x));
label("1", dir(360*(y+3)/x), dir(360*(y+3)/x));[/asy]
Let $P_1,P_2,P_3,P_4$ be the graphs of four quadratic polynomials drawn in the coordinate plane. Suppose that $P_1$ is tangent to $P_2$ at the point $q_2,P_2$ is tangent to $P_3$ at the point $q_3,P_3$ is tangent to $P_4$ at the point $q_4$, and $P_4$ is tangent to $P_1$ at the point $q_1$. Assume that all the points $q_1,q_2,q_3,q_4$ have distinct $x$-coordinates. Prove that $q_1,q_2,q_3,q_4$ lie on a graph of an at most quadratic polynomial.
Let $\mathbb{R}_{\neq 0}$ be the set of nonzero real numbers. Find all functions $f:\mathbb{R}_{\neq 0}\rightarrow\mathbb{R}_{\neq 0}$ such that, for all $x,y\in\mathbb{R}_{\neq 0}$, \[(x-y)f(y^2)+f\left(xy\,f\left(\frac{x^2}{y}\right)\right)=f(y^2f(y)).\]
Consider a $12$-sided regular polygon. If the vertices going clockwise are $A$, $B$, $C$, $D$, $E$, $F$, etc, draw a line between $A$ and $F$, $B$ and $G$, $C$ and $H$, etc. This will form a smaller $12$-sided regular polygon in the center of the larger one. What is the area of the smaller one divided by the area of the larger one?
A circle centred at $(a, b)$ contains the origin $(0,0)$. Denote by $S^+$ the total area of the parts of the circle in the first and third quadrants, and by $S^-$ the total area of the parts of the circle in the second and the fourth quadrants. Compute $S^+ -S^-$.
(G Galperin)
What is the last digit of $ 17^{1996}$?
A. 1
B. 3
C. 5
D. 7
E. 9
Let ABC be an acute-angled triangle. Let D be a point on the segment BC, I the incentre of ABC. The circumcircle of ABD meets BI at P and the circumcircle of ACD meets CI at Q. If the area of PID and the area of QID are equal, prove that PI*QD=QI*PD.
Let $\mathbb{N} = \{1,2, \ldots\}.$ Does there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that $\forall n \in \mathbb{N},$ $f^{1989}(n) = 2 \cdot n$ ?
Let $n\geqslant 2$ be an integer. Prove that for any positive real numbers $a_1, a_2,\ldots, a_n$, \[\frac{1}{2\sqrt{2}}\sum_{i=1}^{n}2^{i}a_i^2 \geqslant\sum_{1 \leqslant i < j \leqslant n}a_i a_j.\][i]Proposed by Andrei Vila[/i]
Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations:
\[a\sqrt{b}=a+c,b\sqrt{c}=b+a,c\sqrt{a}=c+b.\]
a) Let $AB CD$ be a regular tetrahedron. On the sides $AB$, $AC$ and $AD$, the points $M$, $N$ and $P$, are considered. Determine the volume of the tetrahedron $AMNP$ in terms of $x, y, z$, where $x=AM$, $y=AN$, $z=AP$.
b) Show that for any real numbers $x, y, z, t, u, v \in (0, 1)$ :
$$xyz + uv(1- x) + (1- y)(1- v)t + (1- z)(1- w)(1- t) < 1.$$
Call an $n$-gon to be [i]lattice[/i] if its vertices are lattice points. Prove that inside every lattice convex pentagon there exists a lattice point.
[b]p1.[/b] $p, q, r$ are prime numbers such that $p^q + 1 = r$. Find $p + q + r$.
[b]p2.[/b] $2014$ apples are distributed among a number of children such that each child gets a different number of apples. Every child gets at least one apple. What is the maximum possible number of children who receive apples?
[b]p3.[/b] Cathy has a jar containing jelly beans. At the beginning of each minute he takes jelly beans out of the jar. At the $n$-th minute, if $n$ is odd, he takes out $5$ jellies. If n is even he takes out $n$ jellies. After the $46$th minute there are only $4$ jellies in the jar. How many jellies were in the jar in the beginning?
[b]p4.[/b] David is traveling to Budapest from Paris without a cellphone and he needs to use a public payphone. He only has two coins with him. There are three pay-phones - one that never works, one that works half of the time, and one that always works. The first phone that David tries does not work. Assuming that he does not use the same phone again, what is the probability that the second phone that he uses will work?
[b]p5.[/b] Let $a, b, c, d$ be positive real numbers such that
$$a^2 + b^2 = 1$$
$$c^2 + d^2 = 1;$$
$$ad - bc =\frac17$$
Find $ac + bd$.
[b]p6.[/b] Three circles $C_A,C_B,C_C$ of radius $1$ are centered at points $A,B,C$ such that $A$ lies on $C_B$ and $C_C$, $B$ lies on $C_C$ and $C_A$, and $C$ lies on $C_A$ and $C_B$. Find the area of the region where $C_A$, $C_B$, and $C_C$ all overlap.
[b]p7.[/b] Two distinct numbers $a$ and $b$ are randomly and uniformly chosen from the set $\{3, 8, 16, 18, 24\}$. What is the probability that there exist integers $c$ and $d$ such that $ac + bd = 6$?
[b]p8.[/b] Let $S$ be the set of integers $1 \le N \le 2^{20}$ such that $N = 2^i + 2^j$ where $i, j$ are distinct integers. What is the probability that a randomly chosen element of $S$ will be divisible by $9$?
[b]p9.[/b] Given a two-pan balance, what is the minimum number of weights you must have to weigh any object that weighs an integer number of kilograms not exceeding $100$ kilograms?
[b]p10.[/b] Alex, Michael and Will write $2$-digit perfect squares $A,M,W$ on the board. They notice that the $6$-digit number $10000A + 100M +W$ is also a perfect square. Given that $A < W$, find the square root of the $6$-digit number.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Peter is chasing after Rob. Rob is running on the line $y=2x+5$ at a speed of $2$ units a second, starting at the point $(0,5)$. Peter starts running $t$ seconds after Rob, running at $3$ units a second. Peter also starts at $(0,5)$ and catches up to Rob at the point $(17,39)$. What is the value of t?
Show that for any triangle $ABC$, the following inequality is true:
\[ a^2 + b^2 +c^2 > \sqrt{3} max \{ |a^2 - b^2|, |b^2 -c^2|, |c^2 -a^2| \} . \]
In a chess tournament , each of two players have only one game played. After 2 rounds 5 players left the tournament. At the final of tournament was found that the number of total games played is 100. How many players were at the start of the tournament?
In triangle $ABC$, $D$ is the midpoint of side $AB$. $E$ and $F$ are points arbitrarily chosen on segments $AC$ and $BC$, respectively. Show that $[DEF] < [ADE] + [BDF]$.
Let $ABC$ be a triangle. Construct rectangles $BA_1A_2C$, $CB_1B_2A$, and $AC_1C_2B$ outside $ABC$ such that $\angle BCA_1=\angle CAB_1=\angle ABC_1$. Let $A_1B_2$ and $A_2C_1$ intersect at $A'$ and define $B',C'$ similarly. Prove that line $AA'$ bisects $B'C'$.
[i]Linus Tang[/i]
Quasi-rounding is a substitution one of the two closest integers instead of the given number. Given $n$ numbers. Prove that you can quasi-round them in such a way, that a sum of every subset of quasi-rounded numbers will deviate from the sum of the same subset of initial numbers not greater than $(n+1)/4$ .
Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.
For how many integers $k$ does the following system of equations has a solution other than $a=b=c=0$ in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b),\\ b^2+c^2 = ka(b+c),\\ c^2+a^2=kb(c+a).\end{cases}\end{align*}
Let be the points $ K,L,M $ on the sides $ BC,CA,AB, $ respectively, of a triangle $ ABC. $ Show that at least one of the areas of the triangles $ MAL,KBM,LCK $ doesn't surpass a fourth of the area of $ ABC. $
$ ABCD$ is a rectangle (see the accompanying diagram) with $ P$ any point on $ \overline{AB}$. $ \overline{PS} \perp \overline{BD}$ and $ \overline{PR} \perp \overline{AC}$. $ \overline{AF} \perp \overline{BD}$ and $ \overline{PQ} \perp \overline{AF}$. Then $ PR \plus{} PS$ is equal to:
[asy]defaultpen(linewidth(.8pt));
unitsize(3cm);
pair D = origin;
pair C = (2,0);
pair B = (2,1);
pair A = (0,1);
pair P = waypoint(B--A,0.2);
pair S = foot(P,D,B);
pair R = foot(P,A,C);
pair F = foot(A,D,B);
pair Q = foot(P,A,F);
pair T = intersectionpoint(P--Q,A--C);
pair X = intersectionpoint(A--C,B--D);
draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
draw(P--S);
draw(A--F);
draw(P--R);
draw(P--Q);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SE);
label("$D$",D,SW);
label("$P$",P,N);
label("$S$",S,SE);
label("$T$",T,N);
label("$E$",X,SW+SE);
label("$R$",R,SW);
label("$F$",F,SE);
label("$Q$",Q,SW);[/asy]
$ \textbf{(A)}\ PQ\qquad \textbf{(B)}\ AE\qquad \textbf{(C)}\ PT \plus{} AT\qquad \textbf{(D)}\ AF\qquad \textbf{(E)}\ EF$
In $\vartriangle AB$C, $AB = 3$, $AC = 6,$ and $D$ is drawn on $BC$ such that $AD$ is the angle bisector of $\angle BAC$. $D$ is reflected across $AB$ to a point $E$, and suppose that $AC$ and $BE$ are parallel. Compute $CE$.