Found problems: 85335
A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Prove that we can divide students into $n-k+1$ parts taht students in each part are not friends.
Altitudes $AA_1,CC_1$ of acute-angles $ABC$ meet at point $H$ ; $B_0$ is the midpoint of $AC$. A line passing through $B$ and parallel to $AC$ meets $B_0A_1 , B_0C_1$ at points $A',C'$ respectively. Prove that $AA',CC'$ and $BH$ concur.
$l,m$ are skew lines. Three points $A,B,C$ on line $l$ satisfy that $AB=BC$. Projection of $A,B,C$ on $m$ are $D,E,F$. If $|AD|=\sqrt{15},|BE|=\frac{7}{2}|CF|=\sqrt{10}$, find the distance between $l$ and $m$.
Given two integers $h \geq 1$ and $p \geq 2$, determine the minimum number of pairs of opponents an $hp$-member parliament may have, if in every partition of the parliament into $h$ houses of $p$ member each, some house contains at least one pair of opponents.
Let $ABC$ be an acute triangle, $D$ be the midpoint of $BC$. Bisectors of angles $ADB$ and $ADC$ intersect the circles circumscribed around the triangles $ADB$ and $ADC$ at points $E$ and $F$, respectively. Prove that $EF\perp AD$.
find all pairs of non-negative integer pairs $(m,n)$,satisfies
$107^{56}(m^{2}-1)+2m+3=\binom{113^{114}}{n}$
Points $A, B, C,$ and $D{}$ lie on the surface of a sphere with diameter 1. Determine the maximum possible volume of tetrahedron $ABCD.$
[i]Proposed by Fredy Yip[/i]
In $\vartriangle ABC, PQ // BC$ where $P$ and $Q$ are points on $AB$ and $AC$ respectively. The lines $PC$ and $QB$ intersect at $G$. It is also given $EF//BC$, where $G \in EF, E \in AB$ and $F\in AC$ with $PQ = a$ and $EF = b$. Find value of $BC$.
Point $P$ and equilateral triangle $ABC$ satisfy $|AP|=2$, $|BP|=3$. Maximize $|CP|$.
Let $\mathbb R$ be the real numbers. Set $S = \{1, -1\}$ and define a function $\operatorname{sign} : \mathbb R \to S$ by
\[ \operatorname{sign} (x) =
\begin{cases}
1 & \text{if } x \ge 0; \\
-1 & \text{if } x < 0.
\end{cases}
\] Fix an odd integer $n$. Determine whether one can find $n^2+n$ real numbers $a_{ij}, b_i \in S$ (here $1 \le i, j \le n$) with the following property: Suppose we take any choice of $x_1, x_2, \dots, x_n \in S$ and consider the values \begin{align*}
y_i &= \operatorname{sign} \left( \sum_{j=1}^n a_{ij} x_j \right), \quad \forall 1 \le i \le n; \\
z &= \operatorname{sign} \left( \sum_{i=1}^n y_i b_i \right)
\end{align*} Then $z=x_1 x_2 \dots x_n$.
Let $ABC$ be an acute triangle. Its incircle touches the sides $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$, respectively. Let $P$, $Q$ and $R$ be the circumcenters of triangles $AEF$, $BDF$ and $CDE$, respectively. Prove that triangles $ABC$ and $PQR$ are similar.
On circle $ O$, points $ C$ and $ D$ are on the same side of diameter $ \overline{AB}$, $ \angle AOC \equal{} 30^\circ$, and $ \angle DOB \equal{} 45^\circ$. What is the ratio of the area of the smaller sector $ COD$ to the area of the circle?
[asy]unitsize(6mm);
defaultpen(linewidth(0.7)+fontsize(8pt));
pair C = 3*dir (30);
pair D = 3*dir (135);
pair A = 3*dir (0);
pair B = 3*dir(180);
pair O = (0,0);
draw (Circle ((0, 0), 3));
label ("$C$", C, NE);
label ("$D$", D, NW);
label ("$B$", B, W);
label ("$A$", A, E);
label ("$O$", O, S);
label ("$45^\circ$", (-0.3,0.1), WNW);
label ("$30^\circ$", (0.5,0.1), ENE);
draw (A--B);
draw (O--D);
draw (O--C);[/asy]$ \textbf{(A)}\ \frac {2}{9} \qquad \textbf{(B)}\ \frac {1}{4} \qquad \textbf{(C)}\ \frac {5}{18} \qquad \textbf{(D)}\ \frac {7}{24} \qquad \textbf{(E)}\ \frac {3}{10}$
Let $a, b, c$ be reals such that some two of them have difference greater than $\frac{1}{2 \sqrt{2}}$. Prove that there exists an integer $x$, such that $$x^2-4(a+b+c)x+12(ab+bc+ca)<0.$$
Let $x, y, z$ be positive integers such that $xy=z^2+2$. Prove that there exist integers $a, b, c, d$ such that the following equalities are satisfied:
\begin{eqnarray*} x=a^2+2b^2\\
y=c^2+d^2\\
z=ac+2bd\\
\end{eqnarray*}
[i]Proposed by Isaac Jiménez[/i]
A polyhedron has $12$ faces and is such that:
[b][i](i)[/i][/b] all faces are isosceles triangles,
[b][i](ii)[/i][/b] all edges have length either $x$ or $y$,
[b][i](iii)[/i][/b] at each vertex either $3$ or $6$ edges meet, and
[b][i](iv)[/i][/b] all dihedral angles are equal.
Find the ratio $x/y.$
Find the number of ordered triples $(a, b)$ of positive integers with $a < b$ and $100 \leq a, b \leq 1000$ satisfy $\gcd(a, b) : \mathrm{lcm}(a, b) = 1 : 495$?
Let $ABCD$ be a convex quadrilateral, and let $K$ be a point on side$ AB$ such that $\angle KDA = \angle BCD$. Let $L$ be a point on the diagonal $AC$ such that $KL \parallel BC$. Prove that $\angle KDB = \angle LDC$.
Prove that if $n$ and $k$ are positive integers such that $1<k<n-1$,Then the binomial coefficient $\binom nk$ is divisible by at least two different primes.
Find the sum of all integers $0\le a \le124$ so that $a^3-2$ is a multiple of $125$.
Find all pairs $(m,p)$ of natural numbers , such that $p$ is a prime and \[2^mp^2+27\] is the third power of a natural numbers
Do there exist integers $x$ and $y$ such that $19^{19} = x^3 +y^4$ ? Justify your answer.
[b]p1.[/b] Three camps are located in the vertices of an equilateral triangle. The roads connecting camps are along the sides of the triangle. Captain America is inside the triangle and he needs to know the distances between camps. Being able to see the roads he has found that the sum of the shortest distances from his location to the roads is 50 miles. Can you help Captain America to evaluate the distances between the camps?
[b]p2.[/b] $N$ regions are located in the plane, every pair of them have a non-empty overlap. Each region is a connected set, that means every two points inside the region can be connected by a curve all points of which belong to the region. Iron Man has one charge remaining to make a laser shot. Is it possible for him to make the shot that goes through all $N$ regions?
[b]p3.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills.
(a) What is the smallest amount of money you need to buy a slice of pizza that costs $\$1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do since the pizza man can only give you $\$5$ back.
(b) Vending machines in Wonderland accept only exact payment (do not give back change). List all positive integer numbers which CANNOT be used as prices in such vending machines. (That is, find the sums of money that cannot be paid by exact change.)
[b]p4.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line.
(b) Do the same with $6$ points.
[b]p5.[/b] Numbers $1,2,3,…,100$ are randomly divided in two groups $50$ numbers in each. In the first group the numbers are written in increasing order and denoted $a_1,a_2, ..., a_{50}$. In the second group the numberss are written in decreasing order and denoted $b_1,b_2, ..., b_{50}$. Thus $a_1<a_2<...<a_{50}$ and $ b_1>b_2>...>b_{50}$. Evaluate $|a_1-b_1|+|a_2-b_2|+...+|a_{50}-b_{50}|$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
For every triple $(a,b,c)$ of non-zero real numbers, form the number \[ \frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}. \] The set of all numbers formed is
$\textbf{(A)}\ {0} \qquad
\textbf{(B)}\ \{-4,0,4\} \qquad
\textbf{(C)}\ \{-4,-2,0,2,4\} \qquad
\textbf{(D)}\ \{-4,-2,2,4\} \qquad
\textbf{(E)}\ \text{none of these}$
Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be?
Let $ABCD$ be a unit square and let $E$ and $F$ be points inside $ABCD$ such that the line containing $\overline{EF}$ is parallel to $\overline{AB}$. Point $E$ is closer to $\overline{AD}$ than point $F$ is to $\overline{AD}$. The line containing $\overline{EF}$ also bisects the square into two rectangles of equal area. Suppose $[AEF B] = [DEFC] = 2[AED] = 2[BFC]$. The length of segment $\overline{EF}$ can be expressed as $m/n$ , where m and $n$ are relatively prime positive integers. Compute $m + n$.