Show that: a) There are infinitely many positive integers $n$ such that there exists a square equal to the sum of the squares of $n$ consecutive positive integers (for instance, $2$ is one such number as $5^2=3^2+4^2$). b) If $n$ is a positive integer which is not a perfect square, and if $x$ is an integer number such that $x^2+(x+1)^2+...+(x+n-1)^2$ is a perfect square, then there are infinitely many positive integers $y$ such that $y^2+(y+1)^2+...+(y+n-1)^2$ is a perfect square.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that (i): For different reals $x,y$, $f(x) \not= f(y)$. (ii): For all reals $x,y$, $f(x+f(f(-y)))=f(x)+f(f(y))$

In convex hexagon $ABCDEF$, we know that $\angle BCA = \angle DEC = \angle AFB = \angle CBD = \angle EDF.$ Prove that $AB = CD = EF.$

Find all quadruples of real numbers such that each of them is equal to the product of some two other numbers in the quadruple.

Determine whether or not there exists a sequence of integers $a_1,a_2,\dots, a_{19}, a_{20}$ such that, the sum of all the terms is negative, and the sum of any three consecutive terms is positive.

In the $\triangle ABC$ shown, $D$ is some interior point, and $x, y, z, w$ are the measures of angles in degrees. Solve for $x$ in terms of $y, z$ and $w$. [asy] draw((0,0)--(10,0)--(2,7)--cycle); draw((0,0)--(4,3)--(10,0)); label("A", (0,0), SW); label("B", (10,0), SE); label("C", (2,7), W); label("D", (4,3), N); label("x", (2.25,6)); label("y", (1.5,2), SW); label("$z$", (7.88,1.5)); label("w", (4,2.85), S); [/asy] $ \textbf{(A)}\ w-y-z \qquad\textbf{(B)}\ w-2y-2z \qquad\textbf{(C)}\ 180-w-y-z \\ \qquad\textbf{(D)}\ 2w-y-z \qquad\textbf{(E)}\ 180-w+y+z $

In acute triangle $ABC (AB>AC)$, $M$ is the midpoint of minor arc $BC$, $O$ is the circumcenter of $(ABC)$ and $AK$ is its diameter. The line parallel to $AM$ through $O$ meets segment $AB$ at $D$, and $CA$ extended at $E$. Lines $BM$ and $CK$ meet at $P$, lines $BK$ and $CM$ meet at $Q$. Prove that $\angle OPB+\angle OEB =\angle OQC+\angle ODC$.

Find the largest $p_n$ such that $p_n+\sqrt{p_{n-1}+\sqrt{p_{n-2}+\sqrt{\ldots+\sqrt{p_1}}}}\leq 100$, where $p_n$ denotes the $n^{\text{th}}$ prime number.

Let $1 < x_1 < 2$ and, for $n = 1$, 2, $\dots$, define $x_{n + 1} = 1 + x_n - \frac{1}{2} x_n^2$. Prove that, for $n \ge 3$, $|x_n - \sqrt{2}| < 2^{-n}$.

Find all $6$ digit natural numbers, which consist of only the digits $1,2,$ and $3$, in which $3$ occurs exactly twice and the number is divisible by $9$.

Given six irrational numbers, will it be possible to choose three such that the sum of any two of these three is irrational?

Find the number of second-degree polynomials $ f(x)$ with integer coefficients and integer zeros for which $ f(0)\equal{}2010$.

Let $f(x)=|2\{x\} -1|$ where $\{x\}$ denotes the fractional part of $x$. The number $n$ is the smallest positive integer such that the equation $$nf(xf(x)) = x$$ has at least $2012$ real solutions $x$. What is $n$? $\textbf{Note:}$ the fractional part of $x$ is a real number $y= \{x\}$, such that $ 0 \le y < 1$ and $x-y$ is an integer. $ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 31\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 62\qquad\textbf{(E)}\ 64 $

The [I]minkowski sausage[/I] is constructed as follows. $M_0$ is the line segment from $(0,0)$ to $(1,0).$ $M_{I+1}$ is constructed by replacing each segment in $M_i$ with eight segments, each of length $1/4_{I+1}$ (see figure below, where we have provided $M_0$ through $M_3$). Let $M_{\infty}$ denote the limiting shape of $M_0, M_1, \ldots.$ The area of the smallest convex polygon which encloses $M_{\infty}$ can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b.$ Find $a+b.$ [center] [img]https://cdn.artofproblemsolving.com/attachments/1/e/25c9980469584ce7ae4ab2ccb4ce80f3e5dfee.png[/img] [/center]

In how many ways can $6$ juniors and $6$ seniors form $3$ disjoint teams of $4$ people so that each team has $2$ juniors and $2$ seniors? $ \textbf{(A) }720 \qquad \textbf{(B) }1350 \qquad \textbf{(C) }2700 \qquad \textbf{(D) }3280 \qquad \textbf{(E) }8100 \qquad $

Suppose we have two identical cardboard polygons. We placed one polygon upon the other one and aligned. Then we pierced polygons with a pin at a point. Then we turned one of the polygons around this pin by $25^o 30'$. It turned out that the polygons coincided (aligned again). What is the minimal possible number of sides of the polygons?

$30$ people sit around a table, some of which are Yale students. Each person is asked if the person to their right is a Yale student. Yale students will always answer correctly, but non-Yale students will answer randomly. Find the smallest possible number of Yale students such that, after hearing everyone’s answers and knowing the number of Yale students, it is possible to identify for certain at least one Yale student.

Evaluate $\int_{\frac 12}^1 \sqrt{1-x^2}\ dx.$

Which of these five numbers is the largest? $ \text{(A)}\ 13579+\frac{1}{2468}\qquad\text{(B)}\ 13579-\frac{1}{2468}\qquad\text{(C)}\ 13579\times\frac{1}{2468} $ $ \text{(D)}\ 13579\div\frac{1}{2468}\qquad\text{(E)}\ 13579.2468 $

Let $ABCD$ be a parallelogram with $\angle DAB < 90$ Let $E$ be the point on the line $BC$ such that $AE = AB$ and let $F$ be the point on the line $CD$ such that $AF = AD$. The circumcircle of the triangle $CEF$ intersects the line $AE$ again in $P$ and the line $AF$ again in $Q$. Let $X$ be the reflection of $P$ over the line $DE$ and $Y$ the reflection of $Q$ over the line $BF$. Prove that $A, X, Y$ lie on the same line.

Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that the numbers \[n, f(n),f(f(n)),\dots,f^{m-1}(n)\] are distinct modulo $m$ for all integers $n,m$ with $m>1$. (Here $f^k$ is defined by $f^0(n)=n$ and $f^{k+1}(n)=f(f^{k}(n))$ for $k \ge 0$.)

[b]7.1 / 6.1[/b] There are $8$ rooks on the chessboard such that no two of them they don't hit each other. Prove that the black squares contain an even number of rooks. [b]7.2[/b] The sides of triangle $ABC$ are extended as shown in the figure. At this $AA' = 3 AB$,, $BB' = 5BC$ , $CC'= 8 CA$. How many times is the area of the triangle $ABC$ less than the area of the triangle $A'B'C' $? [img]https://cdn.artofproblemsolving.com/attachments/9/f/06795292291cd234bf2469e8311f55897552f6.png[/img] [url=https://artofproblemsolving.com/community/c893771h1860178p12579333]7.3[/url] Prove the equality $$\frac{2}{x^2-1}+\frac{4}{x^2-4} +\frac{6}{x^2-9}+...+\frac{20}{x^2-100} =\frac{11}{(x-1)(x+10)}+\frac{11}{(x-2)(x+9)}+...+\frac{11}{(x-10)(x+1)}$$ [url=https://artofproblemsolving.com/community/c893771h1861966p12597273]7.4* / 8.4 *[/url] (asterisk problems in separate posts) [b]7.5 [/b]. The collective farm consists of $4$ villages located in the peaks of square with side $10$ km. It has the means to conctruct 28 kilometers of roads . Can a collective farm build such a road system so that was it possible to get from any village to any other? [b]7.6 / 6.6[/b] Two brilliant mathematicians were told in natural terms number and were told that these numbers differ by one. After that they take turns asking each other the same question: “Do you know my number?" Prove that sooner or later one of them will answer positively. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].

Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$

Given an integer $n>1$, let $S_n$ be the group of permutations of the numbers $1,\;2,\;3,\;\ldots,\;n$. Two players, A and B, play the following game. Taking turns, they select elements (one element at a time) from the group $S_n$. It is forbidden to select an element that has already been selected. The game ends when the selected elements generate the whole group $S_n$. The player who made the last move loses the game. The first move is made by A. Which player has a winning strategy? [i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]

Find the greatest possible sum of integers $a$ and $b$ such that $\frac{2021!}{20^a\cdot 21^b}$ is a positive integer. [i]Proposed by Aidan Duncan[/i]