For any quadrilateral with the side lengths $a,$ $b,$ $c,$ $d$ and the area $S,$ prove the inequality$S\leq \frac{a+c}{2}\cdot \frac{b+d}{2}.$

If $ y\equal{}x\plus{}\frac{1}{x}$, then $ x^4\plus{}x^3\minus{}4x^2\plus{}x\plus{}1\equal{}0$ becomes: $ \textbf{(A)}\ x^2(y^2\plus{}y\minus{}2)\equal{}0 \qquad\textbf{(B)}\ x^2(y^2\plus{}y\minus{}3)\equal{}0\\ \textbf{(C)}\ x^2(y^2\plus{}y\minus{}4)\equal{}0 \qquad\textbf{(D)}\ x^2(y^2\plus{}y\minus{}6)\equal{}0\\ \textbf{(E)}\ \text{none of these}$

Solve: \[ \begin{cases}x_{2}x_{3}x_{4}\cdots x_{n}=a_{1}x_{1}\\ x_{1}x_{3}x_{4}\cdots x_{n}=a_{2}x_{2}\\x_{1}x_{2}x_{4}\cdots x_{n}=a_{3}x_{3}\\ \ldots\\x_{1}x_{2}x_{3}\cdots x_{n-1}=a_{n-1}x_{n-1} \end{cases} \]

Let $ABC$ be a triangle and let $D$ be the foot of the altitude from $A$. Let points $E$ and $F$ on a line through $D$ such that $AE$ is perpendicular to $BE$, $AF$ is perpendicular to $CF$, where $E$ and $F$ are points other than the point $D$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. Prove that $AN$ is perpendicular to $NM$.

Find all positive integers $n$ such that the number \[ \frac{3 + \sqrt{4n + 9}}{2} \] is the sixth smallest positive divisor of $n$.

Define a positive integer $n$ to be [i]squarish[/i] if either $n$ is itself a perfect square or the distance from $n$ to the nearest perfect square is a perfect square. For example, $2016$ is squarish, because the nearest perfect square to $2016$ is $45^2=2025$ and $2025-2016=9$ is a perfect square. (Of the positive integers between $1$ and $10,$ only $6$ and $7$ are not squarish.) For a positive integer $N,$ let $S(N)$ be the number of squarish integers between $1$ and $N,$ inclusive. Find positive constants $\alpha$ and $\beta$ such that \[\lim_{N\to\infty}\frac{S(N)}{N^{\alpha}}=\beta,\] or show that no such constants exist.

If $P$ is the product of $n$ quantities in Geometric Progression, $S$ their sum, and $S'$ the sum of their reciprocals, then $P$ in terms of $S$, $S'$, and $n$ is $\textbf{(A) }(SS')^{\frac{1}{2}n}\qquad\textbf{(B) }(S/S')^{\frac{1}{2}n}\qquad\textbf{(C) }(SS')^{n-2}\qquad\textbf{(D) }(S/S')^n\qquad \textbf{(E) }(S/S')^{\frac{1}{2}(n-1)}$

In the following diagram, let $ABCD$ be a square and let $M,N,P$ and $Q$ be the midpoints of its sides. Prove that \[S_{A'B'C'D'} = \frac 15 S_{ABCD}.\] [asy] import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttzz = rgb(0,0.2,0.6); pen qqzzff = rgb(0,0.6,1); draw((0,4)--(4,4),qqttzz+linewidth(1.6pt)); draw((4,4)--(4,0),qqttzz+linewidth(1.6pt)); draw((4,0)--(0,0),qqttzz+linewidth(1.6pt)); draw((0,0)--(0,4),qqttzz+linewidth(1.6pt)); draw((0,4)--(2,0),qqzzff+linewidth(1.2pt)); draw((2,4)--(4,0),qqzzff+linewidth(1.2pt)); draw((0,2)--(4,4),qqzzff+linewidth(1.2pt)); draw((0,0)--(4,2),qqzzff+linewidth(1.2pt)); dot((0,4),ds); label("$A$", (0.07,4.12), NE*lsf); dot((0,0),ds); label("$D$", (-0.27,-0.37), NE*lsf); dot((4,0),ds); label("$C$", (4.14,-0.39), NE*lsf); dot((4,4),ds); label("$B$", (4.08,4.12), NE*lsf); dot((2,4),ds); label("$M$", (2.08,4.12), NE*lsf); dot((4,2),ds); label("$N$", (4.2,1.98), NE*lsf); dot((2,0),ds); label("$P$", (1.99,-0.49), NE*lsf); dot((0,2),ds); label("$Q$", (-0.48,1.9), NE*lsf); dot((0.8,2.4),ds); label("$A'$", (0.81,2.61), NE*lsf); dot((2.4,3.2),ds); label("$B'$", (2.46,3.47), NE*lsf); dot((3.2,1.6),ds); label("$C'$", (3.22,1.9), NE*lsf); dot((1.6,0.8),ds); label("$D'$", (1.14,0.79), NE*lsf); clip((-4.44,-11.2)--(-4.44,6.41)--(16.48,6.41)--(16.48,-11.2)--cycle); [/asy] [$S_{X}$ denotes area of the $X.$]

Let $ n\ge 2 $ be a positive integer and $ S= \{1,2,\cdots ,n\} $. Let two functions $ f:S \rightarrow \{1,-1\} $ and $ g:S \rightarrow S $ satisfy: i) $ f(x)f(y)=f(x+y) , \forall x,y \in S $ \\ ii) $ f(g(x))=f(x) , \forall x \in S $\\ iii) $f(x+n)=f(x) ,\forall x \in S$\\ iv) $ g $ is bijective.\\ Find the number of pair of such functions $ (f,g)$ for every $n$.

Find all primes $ p,q $ such that $ \alpha^{3pq} -\alpha \equiv 0 \pmod {3pq} $ for all integers $ \alpha $.

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$. Let $M$ be the midpoint of $BC$. A variable point $P$ is selected in the line segment $AM$. The circumcircles of triangles $BPM$ and $CPM$ intersect $\Gamma$ again at points $D$ and $E$, respectively. The lines $DP$ and $EP$ intersect (a second time) the circumcircles to triangles $CPM$ and $BPM$ at $X$ and $Y$, respectively. Prove that as $P$ varies, the circumcircle of $\triangle AXY$ passes through a fixed point $T$ distinct from $A$.

X is a compact T2 space such that every subspace of cardinality $\aleph_1$ is first countable. Prove that X is first countable.

Let $\Gamma$ and $\Gamma_1$ be two circles internally tangent at $A$, with centers $O$ and $O_1$ and radii $r$ and $r_1$, respectively ($r>r_1$). $B$ is a point diametrically opposed to $A$ in $\Gamma$, and $C$ is a point on $\Gamma$ such that $BC$ is tangent to $\Gamma_1$ at $P$. Let $A'$ the midpoint of $BC$. Given that $O_1A'$ is parallel to $AP$, find the ratio $r/r_1$.

The figure consists of alternating light and dark squares. The number of dark squares exceeds the number of light squares by $ \text{(A)}\ 7\qquad\text{(B)}\ 8\qquad\text{(C)}\ 9\qquad\text{(D)}\ 10\qquad\text{(E)}\ 11 $ [asy] unitsize(12); for(int a=0; a<7; ++a) { fill((2a,0)--(2a+1,0)--(2a+1,1)--(2a,1)--cycle,black); draw((2a+1,0)--(2a+2,0)); } for(int b=7; b<15; ++b) { fill((b,14-b)--(b+1,14-b)--(b+1,15-b)--(b,15-b)--cycle,black); } for(int c=1; c<7; ++c) { fill((c,c)--(c+1,c)--(c+1,c+1)--(c,c+1)--cycle,black); } for(int d=1; d<6; ++d) { draw((2d+1,1)--(2d+2,1)); } fill((6,4)--(7,4)--(7,5)--(6,5)--cycle,black); draw((5,4)--(6,4)); fill((7,5)--(8,5)--(8,6)--(7,6)--cycle,black); draw((7,4)--(8,4)); fill((8,4)--(9,4)--(9,5)--(8,5)--cycle,black); draw((9,4)--(10,4)); label("same",(6.3,2.45),N); label("pattern here",(7.5,1.4),N);[/asy]

Equilateral triangle $ABC$ has been creased and folded so that vertex $A$ now rests at $A'$ on $\overline{BC}$ as shown. If $BA' = 1$ and $A'C = 2$ then the length of crease $\overline{PQ}$ is [asy] size(170); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=(1.5,3*sqrt(3)/2), C=(3,0), D=(1,0), P=B+1.6*dir(B--A), Q=C+1.2*dir(C--A); draw(B--P--D--B^^P--Q--D--C--Q); draw(Q--A--P, linetype("4 4")); label("$A$", A, N); label("$B$", B, W); label("$C$", C, E); label("$A'$", D, S); label("$P$", P, W); label("$Q$", Q, E); [/asy] $ \textbf{(A)}\ \frac{8}{5}\qquad\textbf{(B)}\ \frac{7}{20}\sqrt{21}\qquad\textbf{(C)}\ \frac{1+\sqrt{5}}{2}\qquad\textbf{(D)}\ \frac{13}{8}\qquad\textbf{(E)}\ \sqrt{3} $

[b]p1.[/b] $\vartriangle DEF$ is constructed from equilateral $\vartriangle ABC$ by choosing $D$ on $AB$, $E$ on $BC$ and $F$ on $CA$ so that $\frac{DB}{AB}=\frac{EC}{BC}=\frac{FA}{CA}=a$, where $a$ is a number between $0$ and $1/2$. (a) Show that $\vartriangle DEF$ is also equilateral. (b) Determine the value of $a$ that makes the area of $\vartriangle DEF$ equal to one half the area of $\vartriangle ABC$. [b]p2.[/b] A bowl contains some red balls and some white balls. The following operation is repeated until only one ball remains in the bowl: Two balls are drawn at random from the bowl. If they have different colors, then the red one is discarded and the white one is returned to the bowl. If they have the same color, then both are discarded and a red ball (from an outside supply of red balls) is added to the bowl. (Note that this operation—in either case—reduces the number of balls in the bowl by one.) (a) Show that if the bowl originally contained exactly $1$ red ball and $ 2$ white balls, then the color of the ball remaining at the end (i.e., after two applications of the operation) does not depend on chance, and determine the color of this remaining ball. (b) Suppose the bowl originally contained exactly $1986$ red balls and $1986$ white balls. Show again that the color of the ball remaining at the end does not depend on chance and determine its color. [b]p3.[/b] Let $a, b$, and $c$ be three consecutive positive integers, with $a < b < c.$ (a) Show that $ab$ cannot be the square of an integer. (b) Show that $ac$ cannot be the square of an integer. (c) Show that $abc$ cannot be the square of an integer. [b]p4.[/b] Consider the system of equations $$\sqrt{x}+\sqrt{y}=2$$ $$ x^2+y^2=5$$ (a) Show (algebraically or graphically) that there are two or more solutions in real numbers $x$ and $y$. (b) The graphs of the two given equations intersect in exactly two points. Find the equation of the straight line passing through these two points of intersection. [b]p5.[/b] Let $n$ and $m$ be positive integers. An $n \times m $ rectangle is tiled with unit squares. Let $r(n, m)$ denote the number of rectangles formed by the edges of these unit squares. Thus, for example, $r(2, 1) = 3$. (a) Find $r(2, 3)$. (b) Find $r(n, 1)$. (c) Find, with justification, a formula for $r(n, m)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $N$ be an odd number, $N\geq 3$. $N$ tennis players take part in a championship. Before starting the championship, a commission puts the players in a row depending on how good they think the players are. During the championship, every player plays with every other player exactly once, and each match has a winner. A match is called [i]suprising[/i] if the winner was rated lower by the commission. At the end of the tournament, players are arranged in a line based on the number of victories they have achieved. In the event of a tie, the commission's initial order is used to decide which player will be higher. It turns out that the final order is exactly the same as the commission's initial order. What is the maximal number of suprising matches that could have happened.

Compute $$\lim_{t\rightarrow0}\int_0^t\frac{x^2+4x+4}{\sqrt{t^2-x^2}}dx.$$

We are given the sequence $a_1,a_2,a_3,\ldots$, for which: $$a_n=\frac{a^2_{n-1}+c}{a_{n-2}}\enspace\text{for all }n>2.$$ Prove that the numbers $a_1$, $a_2$ and $\frac{a_1^2+a_2^2+c}{a_1a_2}$ are whole numbers.

An $8\times8$ array consists of the numbers $1,2,...,64$. Consecutive numbers are adjacent along a row or a column. What is the minimum value of the sum of the numbers along the diagonal?

The point $(a,b)$ lies on the circle $x^2+y^2=1$. The tangent to the circle at this point meets the parabola $y=x^2+1$ at exactly one point. Find all such points $(a,b)$.

Five girls and five boys took part in a competition. Suppose that we can number the boys and girls $1, 2, 3, 4, 5$ such that for each $1 \leq i,j \leq 5$, there are exactly $|i-j|$ contestants that the girl numbered $i$ and the boy numbered $j$ both know. Let $a_i$ and $b_i$ be the number of contestants that the girl numbered $i$ knows and the number of contestants that the boy numbered $i$ knows respectively. Find the minimum value of $\max(\sum\limits_{i=1}^5a_i, \sum\limits_{i=1}^5b_i)$. (Note that for a pair of contestants $A$ and $B$, $A$ knowing $B$ doesn't mean that $B$ knows $A$ and a contestant cannot know themself.)

Prove that among the numbers of the form $ 50^n \plus{} (50n\plus{}1)^{50}$, where $ n$ is a natural number, there exist infinitely many composite numbers.

Let $P$ be an interior point of the parallelogram $ABCD$. Prove that $\angle APB+ \angle CPD = 180^\circ$ if and only if $\angle PDC = \angle PBC$.

The real number $a>1$ is given. Suppose that $r$, $s$ and $t$ are different positive integer numbers such that $\{a^r\}=\{a^s\}=\{a^t\}$. Prove that $\{a^r\}=\{a^s\}=\{a^t\}=0$.