Found problems: 85335
Let the line $\ell$ intersects the sides $AC,BC$ of the triangle $ABC$ respectively at the points $E$ and $F$. Prove that the line $\ell$ is passing through the incenter of the triangle $ABC$ if and only if the following equality is true:
$$BC\cdot\frac{AE}{CE}+AC\cdot\frac{BF}{CF}=AB.$$
[i]H. Lesov[/i]
Prove that there exists a four-coloring of the set $M = \{1, 2, \cdots, 1987\}$ such that any arithmetic progression with $10$ terms in the set $M$ is not monochromatic.
[b][i]Alternative formulation[/i][/b]
Let $M = \{1, 2, \cdots, 1987\}$. Prove that there is a function $f : M \to \{1, 2, 3, 4\}$ that is not constant on every set of $10$ terms from $M$ that form an arithmetic progression.
[i]Proposed by Romania[/i]
Let $a$ and $b$ be the positive integers. Show that at least one of the numbers $a$, $b$, $a+b$ can be expressed as the difference of the squares of two integers.
How many ways are there to tile a $3 \times 3$ square with $4$ dominoes of size $1 \times 2$ and $1$ domino of size $1 \times 1$?
Tilings that can be obtained from each other by rotating the square are considered different. Dominoes of the same size are completely identical
[b]p1.[/b] An Egyptian fraction has the form $1/n$, where $n$ is a positive integer. In ancient Egypt, these were the only fractions allowed. Other fractions between zero and one were always expressed as a sum of distinct Egyptian fractions. For example, $3/5$ was seen as $1/2 + 1/10$, or $1/3 + 1/4 + 1/60$. The preferred method of representing a fraction in Egypt used the "greedy" algorithm, which at each stage, uses the Egyptian fraction that eats up as much as possible of what is left of the original fraction. Thus the greedy fraction for $3/5$ would be $1/2 + 1/10$.
a) Find the greedy Egyptian fraction representations for $2/13$.
b) Find the greedy Egyptian fraction representations for $9/10$.
c) Find the greedy Egyptian fraction representations for $2/(2k+1)$, where $k$ is a positive integer.
d) Find the greedy Egyptian fraction representations for $3/(6k+1)$, where $k$ is a positive integer.
[b]p2.[/b] a) The smaller of two concentric circles has radius one unit. The area of the larger circle is twice the area of the smaller circle. Find the difference in their radii.
[img]https://cdn.artofproblemsolving.com/attachments/8/1/7c4d81ebfbd4445dc31fa038d9dc68baddb424.png[/img]
b) The smaller of two identically oriented equilateral triangles has each side one unit long. The smaller triangle is centered within the larger triangle so that the perpendicular distance between parallel sides is always the same number $d$. The area of the larger triangle is twice the area of the smaller triangle. Find $d$.
[img]https://cdn.artofproblemsolving.com/attachments/8/7/1f0d56d8e9e42574053c831fa129eb40c093d9.png[/img]
[b]p3.[/b] Suppose that the domain of a function $f$ is the set of real numbers and that $f$ takes values in the set of real numbers. A real number $x_0$ is a fixed point of f if $f(x_0) = x_0$.
a) Let $f(x) = m x + b$. For which $m$ does $f$ have a fixed point?
b) Find the fixed point of f$(x) = m x + b$ in terms of m and b, when it exists.
c) Consider the functions $f_c(x) = x^2 - c$.
i. For which values of $c$ are there two different fixed points?
ii. For which values of $c$ are there no fixed points?
iii. In terms of $c$, find the value(s) of the fixed point(s).
d) Find an example of a function that has exactly three fixed points.
[b]p4.[/b] A square based pyramid is made out of rubber balls. There are $100$ balls on the bottom level, 81 on the next level, etc., up to $1$ ball on the top level.
a) How many balls are there in the pyramid?
b) If each ball has a radius of $1$ meter, how tall is the pyramid?
c) What is the volume of the solid that you create if you place a plane against each of the four sides and the base of the balls?
[b]p5.[/b] We wish to consider a general deck of cards specified by a number of suits, a sequence of denominations, and a number (possibly $0$) of jokers. The deck will consist of exactly one card of each denomination from each suit, plus the jokers, which are "wild" and can be counted as any possible card of any suit. For example, a standard deck of cards consists of $4$ suits, $13$ denominations, and $0$ jokers.
a) For a deck with $3$ suits $\{a, b, c\}$ and $7$ denominations $\{1, 2, 3, 4, 5, 6, 7\}$, and $0$ jokers, find the probability that a 3-card hand will be a straight. (A straight consists of $3$ cards in sequence, e.g., $1 \heartsuit$ ,$2 \spadesuit$ , $3\clubsuit$ , $2\diamondsuit$ but not $6 \heartsuit$ ,$7 \spadesuit$ , $1\diamondsuit$).
b) For a deck with $3$ suits, $7$ denominations, and $0$ jokers, find the probability that a $3$-card hand will consist of $3$ cards of the same suit (i.e., a flush).
c) For a deck with $3$ suits, $7$ denominations, and $1$ joker, find the probability that a $3$-card hand dealt at random will be a straight and also the probability that a $3$-card hand will be a flush.
d) Find a number of suits and the length of the denomination sequence that would be required if a deck is to contain $1$ joker and is to have identical probabilities for a straight and a flush when a $3$-card hand is dealt. The answer that you find must be an answer such that a flush and a straight are possible but not certain to occur.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $a$, $b$ and $c$ be positive real numbers satisfying $abc=1$. Prove that the following inequality holds:
$$\frac{a+b+c}{3}\geq\frac{a}{a^2b+2}+\frac{b}{b^2c+2}+\frac{c}{c^2a+2}.$$
The figure shows a rectangle, its circumscribed circle and four semicircles, which have the rectangle’s sides as diameters. Prove that the combined area of the four dark gray crescentshaped regions is equal to the area of the light gray rectangle.
[img]https://1.bp.blogspot.com/-gojv6KfBC9I/XzT9ZMKrIeI/AAAAAAAAMVU/NB-vUldjULI7jvqiFWmBC_Sd8QFtwrc7wCLcBGAsYHQ/s0/2013%2BMohr%2Bp3.png[/img]
In isosceles triangle $ABC$ with base side $AB$, on side $BC$ it is given point $M$. Let $O$ be a circumcenter and $S$ incenter of triangle $ABC$. Prove that $$ SM \mid \mid AC \Leftrightarrow OM \perp BS$$
Let $ABC$ be an isosceles triangle with $AB=AC=20$ . Let $P$ be a point inside the triangle $ABC$ such that the sum of the distances of $P$ to $AB$ and $AC$ is $1$ . Describe the locus of all such points inside triangle $ABC$.
Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$.
A line segment of length 1 is given on the plane. Show that a line segment of length $\sqrt{2010}$ can be constructed using only a straightedge and a compass.
Given any $n$ points on a unit circle show that at most $\frac{n^2}{3}$ of the segments joining two points have length $> \sqrt{2}$.
Thirty-four countries participated in a jury session of the IMO, each represented by the leader and the deputy leader of the team. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with his deputy. After the meeting, the leader of the Illyrian team asked every other participant the number of people they had shaken hands with, and all the answers she got were different. How many people did the deputy leader of the Illyrian team greet ?
For any natural number $n$, expressed in base $10$, let $S(n)$ denote the sum of all digits of $n$. Find all positive integers $n$ such that $n^3 = 8S(n)^3+6S(n)n+1$.
Find all ordered pairs of positive integers $\left(a,b\right)$ such that $$\frac{1}{a}+\frac{a}{b}+\frac{1}{ab}=1.$$
Prove that if the two altitudes of a tetrahedron intersect, then the other two atltitudes intersect also.
For $ f(x)\equal{}e^{\frac{x}{2}}\cos \frac{x}{2}$, evaluate $ \sum_{n\equal{}0}^{\infty} \int_{\minus{}\pi}^{\pi}f(x)f(x\minus{}2n\pi)dx\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.
$1999-999+99$ equals
$\textbf{(A)}\ 901 \qquad \textbf{(B)}\ 1099 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 199 \qquad \textbf{(E)}\ 99$
Let be three real numbers $ u,v,t $ under the condition $ u+v+t=0. $ Prove that for any positive real number $ a\neq 1 $ the following inequality is true with equality only and only if $ u=v=t=0: $
$$ a^u/a^v+a^v/a^t+a^{v+t}\ge a^u+a^v+1 $$
[i]Ion Tecu[/i]
Prove that if you choose $10^{100}$ points on a circle and arrange numbers from $1$ to $10^{100}$ on them in some order, then you can choose $100$ pairwise disjoint chords with ends at the selected points such that the sums of the numbers at the ends of all of them are equal to each other.
Let $ABC$ be an acute triangle such that $AH = HD$, where $H$ is the orthocenter of $ABC$ and $D \in BC$ is the foot of the altitude from the vertex $A$. Let $\ell$ denote the line through $H$ which is tangent to the circumcircle of the triangle $BHC$. Let $S$ and $T$ be the intersection points of $\ell$ with $AB$ and $AC$, respectively. Denote the midpoints of $BH$ and $CH$ by $M$ and $N$, respectively. Prove that the lines $SM$ and $TN$ are parallel.
Find the condition of $a,b$ for which the function $f(x)\ (0\leq x\leq 2\pi)$ satisfying the following equality can be determined uniquely,then determine $f(x)$, assuming that $f(x) $ is a continuous function at $0\leq x\leq 2\pi$.
\[f(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x\]
As shown below, convex pentagon $ ABCDE$ has sides $ AB \equal{} 3$, $ BC \equal{} 4$, $ CD \equal{} 6$, $ DE \equal{} 3$, and $ EA \equal{} 7$. The pentagon is originally positioned in the plane with vertex $ A$ at the origin and vertex $ B$ on the positive $ x$-axis. The pentagon is then rolled clockwise to the right along the $ x$-axis. Which side will touch the point $ x \equal{} 2009$ on the $ x$-axis?
[asy]size(250);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair A=(0,0), Ep=7*dir(105), B=3*dir(0);
pair D=Ep+B;
pair C=intersectionpoints(Circle(D,6),Circle(B,4))[1];
pair[] ds={A,B,C,D,Ep};
dot(ds);
draw(B--C--D--Ep--A);
draw((6,6)..(8,4)..(8,3),EndArrow(3));
xaxis("$x$",-8,14,EndArrow(3));
label("$E$",Ep,NW);
label("$D$",D,NE);
label("$C$",C,E);
label("$B$",B+(.2,.1),ENE);
label("$A$",A+(-.1,.1),WNW);
label("$(0,0)$",A,S);
label("$3$",midpoint(A--B),N);
label("$4$",midpoint(B--C),NW);
label("$6$",midpoint(C--D),NE);
label("$3$",midpoint(D--Ep),S);
label("$7$",midpoint(Ep--A),W);[/asy]$ \textbf{(A)}\ \overline{AB} \qquad \textbf{(B)}\ \overline{BC} \qquad \textbf{(C)}\ \overline{CD} \qquad \textbf{(D)}\ \overline{DE} \qquad \textbf{(E)}\ \overline{EA}$
Let $ABC$ be a triangle, and let $D, E$ and $F$ be the feet of the altitudes from $A, B$ and $C,$ respectively. A circle $\omega_A$ through $B$ and $C$ crosses the line $EF$ at $X$ and $X'$. Similarly, a circle $\omega_B$ through $C$ and $A$ crosses the line $FD$ at $Y$ and $Y',$ and a circle $\omega_C$ through $A$ and $B$ crosses the line $DE$ at $Z$ and $Z'$. Prove that $X, Y$ and $Z$ are collinear if and only if $X', Y'$ and $Z'$ are collinear.
[i]Vlad Robu[/i]
On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game.
Alice starts and they alternately play the following moves:
$\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$
$\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$.
Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps.
Prove that Alice has a winning strategy.
(Daniel Holmes)