Found problems: 85335
In a country there are $4^9$ schoolchildren living in four cities. At the end of the school year a state examination was held in 9 subjects. It is known that any two students have different marks at least in one subject. However, every two students from the same city got equal marks at least in one subject. Prove that there is a subject such that every two children living in the same city have equal marks in this subject.
[i]Fedor Petrov[/i]
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$ satisfies,
$$2+f(x)f(y)\leq xy+2f(x+y+1).$$
Let $\mathcal L$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell\in\mathcal L$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_1,\ell_2,\ell_3$ passing through $X$, the points $f(\ell_1),f(\ell_2),f(\ell_3)$, and $X$ lie on a circle.
Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$.
[i]Australia[/i]
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$.
[i]Proposed by David Monk, United Kingdom[/i]
There are $100$ points on the circumference of a circle, arbitrarily labelled by $1,2,\ldots,100$. For each three points, call their triangle [b]clockwise[/b] if the increasing order of them is in clockwise order. Prove that it is impossible to have exactly $2017$ [b]clockwise[/b] triangles.
The numbers from $ 51$ to $ 150$ are arranged in a $ 10\times 10$ array. Can this be done in such a way that, for any two horizontally or vertically adjacent numbers $ a$ and $ b$, at least one of the equations $ x^2 \minus{} ax \plus{} b \equal{} 0$ and $ x^2 \minus{} bx \plus{} a \equal{} 0$ has two integral roots?
Let $a$ and $b$ be integers. Prove that if $\sqrt[3]{a}+\sqrt[3]{b}$ is a rational number, then both $a$ and $b$ are perfect cubes.
If $x > y > 0$, then $\frac{x^{y}y^{x}}{y^{y}x^{x}} = $
$ \textbf{(A)}\ (x - y)^{y/x}\qquad\textbf{(B)}\ \left(\frac{x}{y}\right)^{x-y}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \left(\frac{x}{y}\right)^{y-x}\qquad\textbf{(E)}\ (x - y)^{x/y} $
The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.
Estimate the value of $$\sum^{2023}_{n=1} \left(1+ \frac{1}{n} \right)^n$$ to $3$ decimal places.
Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the number in each group will be the same. In how many ways can this be done?
$\textbf{(A) }0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$
Suppose $ABC$ is a triangle inscribed in circle $\omega$ . Let $A'$ be the point on $\omega$ so that $AA'$ is a diameter, and let $G$ be the centroid of $ABC$. Given that $AB = 13$, $BC = 14$, and $CA = 15$, let $x$ be the area of triangle $AGA'$ . If $x$ can be expressed in the form $m/n$ , where m and n are relatively prime positive integers, compute $100n + m$.
Passing through the origin of the coordinate plane are $180$ lines, including the coordinate axes,
which form $1$ degree angles with one another at the origin. Determine the sum of the x-coordinates
of the points of intersection of these lines with the line $y = 100-x$
Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?
[asy]
draw((0,0)--(4,4*sqrt(3)));
draw((1,-sqrt(3))--(5,3*sqrt(3)));
draw((2,-2*sqrt(3))--(6,2*sqrt(3)));
draw((3,-3*sqrt(3))--(7,sqrt(3)));
draw((4,-4*sqrt(3))--(8,0));
draw((8,0)--(4,4*sqrt(3)));
draw((7,-sqrt(3))--(3,3*sqrt(3)));
draw((6,-2*sqrt(3))--(2,2*sqrt(3)));
draw((5,-3*sqrt(3))--(1,sqrt(3)));
draw((4,-4*sqrt(3))--(0,0));
draw((3,3*sqrt(3))--(5,3*sqrt(3)));
draw((2,2*sqrt(3))--(6,2*sqrt(3)));
draw((1,sqrt(3))--(7,sqrt(3)));
draw((-1,0)--(9,0));
draw((1,-sqrt(3))--(7,-sqrt(3)));
draw((2,-2*sqrt(3))--(6,-2*sqrt(3)));
draw((3,-3*sqrt(3))--(5,-3*sqrt(3)));[/asy]
$ \text{(A)}\ 4\qquad\text{(B)}\ 5\qquad\text{(C)}\ 6\qquad\text{(D)}\ 7\qquad\text{(E)}\ 9 $
If $p$ is a prime number such that there exist positive integers $a$ and $b$ such that $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$ then $p$ is
(A): $3$, (B): $5$, (C): $11$, (D): $7$, (E) None of the above.
In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular. \\
(b) Show that $FM$ and $BM$ are perpendicular.
$A$ and $B$ move uniformly along two straight paths intersecting at right angles in point $O$. When $A$ is at $O$, $B$ is $500$ yards short of $O$. In $2$ minutes, they are equidistant from $O$, and in $8$ minutes more they are again equidistant from $O$. Then the ratio of $A'$s speed to $B'$s speed is:
$\textbf{(A)}\ 4:5 \qquad\textbf{(B)}\ 5:6 \qquad\textbf{(C)}\ 2:3 \qquad\textbf{(D)}\ 5:8 \qquad\textbf{(E)}\ 1:2$
Given the graph $G$ and cycle $C$ in it, we can perform the following operation: add another vertex $v$ to the graph, connect it to all vertices in $C$ and erase all the edges from $C$. Prove that we cannot perform the operation indefinitely on a given graph.
The [i]fractional distance[/i] between two points $(x_1,y_1)$ and $(x_2,y_2)$ is defined as \[ \sqrt{ \left\| x_1 - x_2 \right\|^2 + \left\| y_1 - y_2 \right\|^2},\]where $\left\| x \right\|$ denotes the distance between $x$ and its nearest integer. Find the largest real $r$ such that there exists four points on the plane whose pairwise fractional distance are all at least $r$.
Let $a$ and $n$ be positive integers such that:
1. $a^{2^n}-a$ is divisible by $n$,
2. $\sum\limits_{k=1}^{n} k^{2024}a^{2^k}$ is [i]not[/i] divisible by $n$.
Prove that $n$ has a prime factor [i]smaller[/i] than $2024$.
[i]Proposed by Shantanu Nene[/i]
On a tournament with $ n \ge 3$ participants, every two participants played exactly one match and there were no draws. A three-element set of participants is called a [i]draw-triple[/i] if they can be enumerated so that the first defeated the second, the second defeated the third, and the third defeated the first. Determine the largest possible number of draw-triples on such a tournament.
Problems $15, 16$, and $17$ all refer to the following:
In the very center of the Irenic Sea lie the beautiful Nisos Isles. In $1998$ the number of people on these islands is only 200, but the population triples every $25$ years. Queen Irene has decreed that there must be at least $1.5$ square miles for every person living in the Isles. The total area of the Nisos Isles is $24,900$ square miles.
15. Estimate the population of Nisos in the year $2050$.
$ \text{(A)}\ 600\qquad\text{(B)}\ 800\qquad\text{(C)}\ 1000\qquad\text{(D)}\ 2000\qquad\text{(E)}\ 3000 $
Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i\geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.
Suppose $x,y,z$ are positive numbers such that $x+y+z=1$. Prove that
(a) $xy+yz+xz\ge 9xyz$;
(b) $xy+yz+xz<\frac{1}{4}+3xyz$;
Let $ A$ and $ B$ be the endpoints of a semicircular arc of radius $ 2$. The arc is divided into seven congruent arcs by six equally spaced points $ C_1,C_2,\ldots,C_6$. All chords of the form $ \overline{AC_i}$ or $ \overline{BC_i}$ are drawn. Let $ n$ be the product of the lengths of these twelve chords. Find the remainder when $ n$ is divided by $ 1000$.