Find all real numbers $a$ such that there exist $f:\mathbb{R} \to \mathbb{R}$ with$$f(x+f(y))=f(x)+a\lfloor y \rfloor $$for all $x,y\in \mathbb{R}$

Let $\mathcal P$ be the parabola $y^2=2px$, and let $T_0$ be a point on it. Point $T_0'$ is such that the midpoint of the segment $T_0T_0'$ lies on the axis of the parabola. For a variable point $T$ on $\mathcal P$, the perpendicular from $T_0'$ to the line $T_0T$ intersects the line through $T$ parallel to the axis of $\mathcal P$ at a point $T'$. Find the locus of $T'$.

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