Is it possible to place a number of circles inside a square with side 1 cm., such that the sum of radii of all the circles is greater than $2000$ cm., and no two circles have overlapping interiors?

Two isosceles triangles of the same area are located as shown in the figure. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/a/6/f7dbfd267274781b67a5f3d5a9036fb2905156.png[/img]

There is a $\triangle ABC$ which $\angle C=90$, and $\bar{AB}=36$ On the circumcircle of $\triangle ABC$, there is $\overarc{BC}$ which does not include point $A$. D is on $\overarc{BC}$. It satisfies $2\times\angle CAD = \angle BAD $ $E: \bar{AD}\cap\bar{BC} $ $ \bar{AE}=20 $ Find $ \bar{BD}^2 $

$S,T$ are two trees without vertices of degree 2. To each edge is associated a positive number which is called length of this edge. Distance between two arbitrary vertices $v,w$ in this graph is defined by sum of length of all edges in the path between $v$ and $w$. Let $f$ be a bijective function from leaves of $S$ to leaves of $T$, such that for each two leaves $u,v$ of $S$, distance of $u,v$ in $S$ is equal to distance of $f(u), f(v)$ in $T$. Prove that there is a bijective function $g$ from vertices of $S$ to vertices of $T$ such that for each two vertices $u,v$ of $S$, distance of $u,v$ in $S$ is equal to distance of $g(u)$ and $g(v)$ in $T$.

Find all pairs $(p,q)$ of prime numbers such that \[m^{3pq} \equiv m \pmod{3pq} \qquad \forall m \in \mathbb Z.\]

Let $S$ be the sum of integer weights that come with a two pan balance Scale, say $\omega_1 \le \omega_2 \le \omega_3 \le ... \le\omega_n$. Show that all integer-weighted objects in the range $1$ to $S$ can be weighed exactly if and only if $\omega_1=1$ and $$\omega_{j+1} \le 2 \left( \sum_{l=1}^{j} \omega_l\right) +1$$

Let $\omega$ be the circumcircle of triangle $ABC$. A point $T$ on the line $BC$ is such that $AT$ touches $\omega$. The bisector of angle $BAC$ meets $BC$ and $\omega$ at points $L$ and $A_0$ respectively. The line $TA_0$ meets $\omega$ at point $P$. The point $K$ lies on the segment $BC$ in such a way that $BL = CK$. Prove that $\angle BAP = \angle CAK$.

The sequence of positive integers $\{a_n, n\ge 1\}$ is such that $a_n\le a_{n+1}\le a_n+5$ and $a_n$ is divisible by $n$ for all $n \ge 1$. What are the possible values of $a_1$?

Let $a,b,c$ be the sides of triangle $ABC$ and let $\alpha,\beta,\gamma$ be the corresponding angles. (a) If $\alpha=3\beta$, prove that $\left(a^2-b^2\right)(a-b)=bc^2$. (b) Is the converse true?

In triangle $ABC$, the difference between angles $B$ and $C$ is equal to $90^o$, and $AL$ is the angle bisector of triangle $ABC$. The bisector of the exterior angle $A$ of the triangle $ABC$ intersects the line $BC$ at the point $F$. Prove that $AL = AF$. (Alexander Dzyunyak)

[b]p1.[/b] Find the unknown angle $a$ of the triangle inscribed in the square. [img]https://cdn.artofproblemsolving.com/attachments/b/1/4aab5079dea41637f2fa22851984f886f034df.png[/img] [b]p2.[/b] Draw a polygon in the plane and a point outside of it with the following property: no edge of the polygon is completely visible from that point (in other words, the view is obstructed by some other edge). [b]p3.[/b] This problem has two parts. In each part, $2022$ real numbers are given, with some additional property. (a) Suppose that the sum of any three of the given numbers is an integer. Show that the total sum of the $2022$ numbers is also an integer. (b) Suppose that the sum of any five of the given numbers is an integer. Show that 5 times the total sum of the $2022$ numbers is also an integer, but the sum itself is not necessarily an integer. [b]p4.[/b] Replace stars with digits so that the long multiplication in the example below is correct. [img]https://cdn.artofproblemsolving.com/attachments/9/7/229315886b5f122dc0675f6d578624e83fc4e0.png[/img] [b]p5.[/b] Five nodes of a square grid paper are marked (called marked points). Show that there are at least two marked points such that the middle point of the interval connecting them is also a node of the square grid paper [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=\dfrac{8}{3} \\ \dfrac{yz}{y+z}=\dfrac{12}{5} \\\dfrac{xz}{x+z}=\dfrac{24}{7} \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\theta=\frac{2\pi}{2015}$, and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form $\frac{b}{2^a}$, where $a$ is a non-negative integer and $b$ is an odd integer (not necessarily positive). Find $a+b$. [i]2017 CCA Math Bonanza Tiebreaker Round #3[/i]

A number is said to be [i]balanced [/i] if one of its digits is average of the others. How many [i]balanced [/i]$3$-digit numbers are there?

The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles

Welcome to the [b]USAYNO[/b], where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them [b]all[/b] correct, you will receive $\max(0, (n-1)(n-2))$ points. If any of them are wrong (or you leave them all blank), you will receive $0$ points. Your answer should be a six-character string containing 'Y' (for yes), 'N' (for no), or 'B' (for blank). For instance if you think 1, 2, and 6 are 'yes' and 3 and 4 are 'no', you should answer YYNNBY (and receive $12$ points if all five answers are correct, 0 points if any are wrong). (a) $a,b,c,d,A,B,C,$ and $D$ are positive real numbers such that $\frac{a}{b} > \frac{A}{B}$ and $\frac{c}{d} > \frac{C}{D}$. Is it necessarily true that $\frac{a+c}{b+d} > \frac{A+C}{B+D}$? (b) Do there exist irrational numbers $\alpha$ and $\beta$ such that the sequence $\lfloor\alpha\rfloor+\lfloor\beta\rfloor, \lfloor2\alpha\rfloor+\lfloor2\beta\rfloor, \lfloor3\alpha\rfloor+\lfloor3\beta\rfloor, \dots$ is arithmetic? (c) For any set of primes $\mathbb{P}$, let $S_\mathbb{P}$ denote the set of integers whose prime divisors all lie in $\mathbb{P}$. For instance $S_{\{2,3\}}=\{2^a3^b \; | \; a,b\ge 0\}=\{1,2,3,4,6,8,9,12,\dots\}$. Does there exist a finite set of primes $\mathbb{P}$ and integer polynomials $P$ and $Q$ such that $\gcd(P(x), Q(y))\in S_\mathbb{P}$ for all $x,y$? (d) A function $f$ is called [b]P-recursive[/b] if there exists a positive integer $m$ and real polynomials $p_0(n), p_1(n), \dots, p_m(n)$[color = red], not all zero,[/color] satisfying \[p_m(n)f(n+m)=p_{m-1}(n)f(n+m-1)+\dots+p_0(n)f(n)\] for all $n$. Does there exist a P-recursive function $f$ satisfying $\lim_{n\to\infty} \frac{f(n)}{n^{\sqrt{2}}}=1$? (e) Does there exist a [b]nonpolynomial[/b] function $f: \mathbb{Z}\to\mathbb{Z}$ such that $a-b$ divides $f(a)-f(b)$ for all integers $a\neq b$? (f) Do there exist periodic functions $f, g:\mathbb{R}\to\mathbb{R}$ such that $f(x)+g(x)=x$ for all $x$? [color = red]A clarification was issued for problem 33(d) during the test. I have included it above.[/color]

Let $A,B\in\mathcal{M}_3(\mathbb{R})$ de matrices such that $A^2+B^2=O_3.$ Prove that $\det(aA+bB)=0$ for any real numbers $a$ and $b.$

For each real number $x$, let $f(x)$ be the minimum of the numbers $4x+1$, $x+2$, and $-2x+4$. Then the maximum value of $f(x)$ is $\text{(A)} \ \frac 13 \qquad \text{(B)} \ \frac 12 \qquad \text{(C)} \ \frac 23 \qquad \text{(D)} \ \frac 52 \qquad \text{(E)} \ \frac 83$

A lame rook lies on a $9\times 9$ chessboard. It can move one cell horizontally or vertically. The rook made $n{}$ moves, visited each cell at most once, and did not make two moves consecutively in the same direction. What is the largest possible value of $n{}$? [i]From the folklore[/i]

Let $ABCD$ be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at $E$. The midpoints of $AB$ and $CD$ are $F$ and $G$ respectively, and $\ell$ is the line through $G$ parallel to $AB$. The feet of the perpendiculars from E onto the lines $\ell$ and $CD$ are $H$ and $K$, respectively. Prove that the lines $EF$ and $HK$ are perpendicular.

Evaluate $\sum_{n=1}^{\infty}\frac{1}{(2n - 1)(3n - 1)}$.

Is it possible to complete the following square knowning that each row and column make an aritmetic progression?

Let $a,b,c\in \mathbb{R}_{\ge 0}$ satisfy $a+b+c=1$. Prove the inequality : \[ \frac{3-b}{a+1}+\frac{a+1}{b+1}+\frac{b+1}{c+1}\ge 4 \]

Solve the equation $(x + 1)^3(x - 2)^3 + (x -1)^3(x + 2)^3 = 8(x^2 -2)^3.$

The numbers from $1$ to $2025$ are arranged in some order in the cells of the $1 \times 2025$ strip. Let's call a [i]flip[/i] an operation that takes two arbitrary cells of a strip and swaps the numbers written in them, but only if the larger of these numbers is located to the left of the smaller one. A [i]flop[/i] is a set of several flips that do not contain common cells that are executed simultaneously. (For example, a simultaneous flip between the 2nd and 8th cells and a flip between the 5th and 101st cells.) Prove that there exists a sequence of $66$ flops such that for any initial arrangement, applying this sequence of flops to it will result in the numbers being ordered from left to right in ascending order.

In the right triangle shown the sum of the distances $ BM$ and $ MA$ is equal to the sum of the distances $ BC$ and $ CA$. If $ MB \equal{} x$, $ CB \equal{} h$, and $ CA \equal{} d$, then $ x$ equals: [asy]size(200); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; draw((0,0)--(8,0)--(0,5)--cycle); label("C",(0,0),SW); label("A",(8,0),SE); label("M",(0,5),N); dot((0,3.5)); label("B",(0,3.5),W); label("$x$",(0,4.25),W); label("$h$",(0,1),W); label("$d$",(4,0),S);[/asy]$ \textbf{(A)}\ \frac {hd}{2h \plus{} d} \qquad \textbf{(B)}\ d \minus{} h \qquad \textbf{(C)}\ \frac {1}{2}d \qquad \textbf{(D)}\ h \plus{} d \minus{} \sqrt {2d} \qquad \textbf{(E)}\ \sqrt {h^2 \plus{} d^2} \minus{} h$