There is a reunion of $201$ people from $5$ different countries. It is known that in each group of $6$ people, at least two have the same age. Show that there must be at least $5$ people with the same country, age and sex.

Consider a rectangle whose lengths of sides are natural numbers. If someone places as many squares as possible, each with area $3$, inside of the given rectangle, such that the sides of the squares are parallel to the rectangle sides, then the maximal number of these squares fill exactly half of the area of the rectangle. Determine the dimensions of all rectangles with this property.

The function f has the following properties : $f(x + y) = f(x) + f(y) + xy$ for all real $x$ and $y$ $f(4) = 10$ Calculate $f(2001)$.

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

$\mathbb{N}$ is the set of positive integers and $a\in\mathbb{N}$. We know that for every $n\in\mathbb{N}$, $4(a^n+1)$ is a perfect cube. Prove that $a=1$.

Let $\triangle ABC$ be an acute-angled triangle with altitudes $AD$ and $BE$ meeting at $H$. Let $M$ be the midpoint of segment $AB$, and suppose that the circumcircles of $\triangle DEM$ and $\triangle ABH$ meet at points $P$ and $Q$ with $P$ on the same side of $CH$ as $A$. Prove that the lines $ED, PH,$ and $MQ$ all pass through a single point on the circumcircle of $\triangle ABC$.

The families of second degree functions $f_m, g_m: R\to R, $ are considered , $f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1$, $g_m (x) = m^2x^2 + mx - 1$, where $m$ is a real nonzero parameter. Show that, for any function $h$ of the second degree with the property that $g_m (x) \le h (x) \le f_m (x)$ for any real $x$, there exists $\lambda \in [0, 1]$ which verifies the condition $h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)$, whatever real $x$ is.

How many functions $f : \{1,2,3,4,5,6\} \to \{1,2,3,4,5,6\}$ have the property that $f(f(x))+f(x)+x$ is divisible by $3$ for all $x \in \{1,2,3,4,5,6\}?$ [i]Proposed by Kyle Lee[/i]

Black and white checkers are placed on an $8 \times 8$ chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones?

Each of Mahdi and Morteza has drawn an inscribed $93$-gon. Denote the first one by $A_1A_2…A_{93}$ and the second by $B_1B_2…B_{93}$. It is known that $A_iA_{i+1} // B_iB_{i+1}$ for $1 \le i \le 93$ ($A_{93} = A_1, B_{93} = B_1$). Show that $\frac{A_iA_{i+1} }{ B_iB_{i+1}}$ is a constant number independent of $i$. by Morteza Saghafian

Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$: (i) move the last digit of $a$ to the first position to obtain the numb er $b$; (ii) square $b$ to obtain the number $c$; (iii) move the first digit of $c$ to the end to obtain the number $d$. (All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.) Find all numbers $a$ for which $d\left( a\right) =a^2$. [i]Proposed by Zoran Sunic, USA[/i]

Prove that the product of the sides of a quadrilateral inscribed in a circle with radius $1$ does not exceed $4$.

a) Pupils of the $8$-th form are standing in a row. There is the pupil of the $7$-th form in before each, and he is smaller (in height) than the elder. Prove that if you will sort the pupils in each of rows with respect to their height, every 8-former will still be taller than the $7$-former standing before him. b) An infantry detachment soldiers stand in the rectangle, being arranged in columns with respect to their height. Prove that if you rearrange them with respect to their height in every separate row, they will still be staying with respect to their height in columns.

The digits $2$, $0$, $2$, and $3$ are placed in the expression below, one digit per box. What is the maximum possible value of the expression? [asy] // Diagram by TheMathGuyd. I can compress this later size(5cm); real w=2.2; pair O,I,J; O=(0,0);I=(1,0);J=(0,1); path bsqb = O--I; path bsqr = I--I+J; path bsqt = I+J--J; path bsql = J--O; path lsqb = shift((1.2,0.75))*scale(0.5)*bsqb; path lsqr = shift((1.2,0.75))*scale(0.5)*bsqr; path lsqt = shift((1.2,0.75))*scale(0.5)*bsqt; path lsql = shift((1.2,0.75))*scale(0.5)*bsql; draw(bsqb,dashed); draw(bsqr,dashed); draw(bsqt,dashed); draw(bsql,dashed); draw(lsqb,dashed); draw(lsqr,dashed); draw(lsqt,dashed); draw(lsql,dashed); label(scale(3)*"$\times$",(w,1/3)); draw(shift(1.3w,0)*bsqb,dashed); draw(shift(1.3w,0)*bsqr,dashed); draw(shift(1.3w,0)*bsqt,dashed); draw(shift(1.3w,0)*bsql,dashed); draw(shift(1.3w,0)*lsqb,dashed); draw(shift(1.3w,0)*lsqr,dashed); draw(shift(1.3w,0)*lsqt,dashed); draw(shift(1.3w,0)*lsql,dashed); [/asy]$\textbf{(A) } 0\qquad\textbf{(B) } 8\qquad\textbf{(C) } 9\qquad\textbf{(D) } 16\qquad\textbf{(E) } 18$

Let $ABCD$ be a square inscribed in circle $K$. Let $P$ be a point on the small arc $CD$ of circle $K$. The line $PB$ intersects $AC$ in $E$. The line $PA$ intersects $DB$ in $F$. The circle circumscribed to triangle $PEF$ intersects for second time $K$ in $Q$. Prove that $PQ$ is parallel to $CD$.

Kevin colors three distinct squares in a $3\times 3$ grid red. Given that there exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line, find the number of ways he could have colored the original three squares.

In a triangle $ABC$, two lines are drawn that together trisect the angle at $A$. These intersect the side $BC$ at points $P$ and $Q$ so that $P$ is closer to $B$ and $Q$ is closer to $C$. Determine the smallest constant $k$ such that $| P Q | \le k (| BP | + | QC |)$, for all such triangles. Determine if there are triangles for which equality applies.

If $b_k \geq a_k \geq 0$ $(k = 1, 2, 3)$ and $\alpha \geq 1$ then$$(\alpha+3)\sum \limits_{cyc}(b_1-a_1)\left((b_2+b_3)^{\alpha+2}+(a_2+a_3)^{\alpha+2}-(a_2+b_3)^{\alpha+1}-(b_2+a_3)^{\alpha+1}\right)$$ $$\leq (\alpha+2)(\alpha+3)\sum \limits_{cyc}(b_1-a_1)(b_2-a_2)(b_3^{\alpha+1}-a_3^{\alpha+1})$$ $$+ (b_3 + b_2 + a_1)^{\alpha+3}+(b_3 + a_2 + a_1)^{\alpha+3}+(a_3 + b_2 + a_1)^{\alpha+3}+(a_3 + a_2 + b_1)^{\alpha+3}$$ $$-(b_3 + b_2 + b_1)^{\alpha+3}-(b_3 + a_2 + a_1)^{\alpha+3}-(a_3 + b_2 + b_1)^{\alpha+3}-(a_3 + a_2 + a_1)^{\alpha+3}$$

Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\] \[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$

A foreman noticed an inspector checking a $ 3"$-hole with a $ 2"$-plug and a $ 1"$-plug and suggested that two more gauges be inserted to be sure that the fit was snug. If the new gauges are alike, then the diameter, $ d$, of each, to the nearest hundredth of an inch, is: $ \textbf{(A)}\ .87 \qquad \textbf{(B)}\ .86 \qquad \textbf{(C)}\ .83 \qquad \textbf{(D)}\ .75 \qquad \textbf{(E)}\ .71$

In the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$. Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$. Find $\angle OPQ$ in degrees. [asy] pathpen = rgb(0,0,0.6)+linewidth(0.7); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6)+ linewidth(0.7) + linetype("4 4"); real n = 10, start = 360/n*6-15; pair O=(0,0), A=dir(start), B=dir(start+360/n), C=dir(start+2*360/n), D=dir(start+3*360/n), P=(A+C)/2, Q=(O+D)/2; D(D("O",O,NE)--D("A",A,W)--D("B",B,SW)--D("C",C,S)--D("D",D,SE)--O--D("P",P,1.6*dir(95))--D("Q",Q,NE)); D(A--C); D(A--(A+dir(start-360/n))/2, dd); D(D--(D+dir(start+4*360/n))/2, dd); [/asy]

In triangle $ABC$, let $O$ be the circumcenter and let $G$ be the centroid. The line perpendicular to $OG$ at $O $ intersects $BC$ at $M$ such that $M$, $G$, and $A$ are collinear and $OM = 3$. Compute the area of $ABC$, given that $OG = 1$.

A triangle is given. Its side a is of length $20$ cm, and its area is $125$ cm$^2$. It is also known that one of the angles lying on side a is twice as large as the other one. We cut the triangle into two parts at the median belonging to side a. Then we move the so-obtained two parts towards each other, such that the two segments of side a remain on the same line (i.e., the line initially occupied by side a). We move the two parts towards each other until we first reach a moment when the common part of the two segments is of length $4$ cm. What is the area of the so-obtained shape in cm$^2$? The so-obtained shape is the union of the two parts, which is a heptagon. [img]https://cdn.artofproblemsolving.com/attachments/3/0/3d45e2df6a0043dfa4fe5ccf64865da8879b42.png[/img]

[b]p1.[/b] Consider a cube with side length $2$. Take any one of its vertices and consider the three midpoints of the three edges emanating from that vertex. What is the distance from that vertex to the plane formed by those three midpoints? [b]p2.[/b] Digits $H$, $M$, and $C$ satisfy the following relations where $\overline{ABC}$ denotes the number whose digits in base $10$ are $A$, $B$, and $C$. $$\overline{H}\times \overline{H} = \overline{M}\times \overline{C} + 1$$ $$\overline{HH}\times \overline{H} = \overline{MC}\times \overline{C} + 1$$ $$\overline{HHH}\times \overline{H} = \overline{MCC}\times \overline{C} + 1$$ Find $\overline{HMC}$. [b]p3.[/b] Two players play the following game on a table with fair two-sided coins. The first player starts with one, two, or three coins on the table, each with equal probability. On each turn, the player flips all the coins on the table and counts how many coins land heads up. If this number is odd, a coin is removed from the table. If this number is even, a coin is added to the table. A player wins when he/she removes the last coin on the table. Suppose the game ends. What is the probability that the first player wins? [b]p4.[/b] Cyclic quadrilateral $[BLUE]$ has right $\angle E$. Let $R$ be a point not in $[BLUE]$. If $[BLUR] =[BLUE]$, $\angle ELB = 45^o$, and $\overline{EU} = \overline{UR}$, find $\angle RUE$. [b]p5.[/b] There are two tracks in the $x, y$ plane, defined by the equations $$y =\sqrt{3 - x^2}\,\,\, \text{and} \,\,\,y =\sqrt{4- x^2}$$ A baton of length $1$ has one end attached to each track and is allowed to move freely, but no end may be picked up or go past the end of either track. What is the maximum area the baton can sweep out? [b]p6.[/b] For integers $1 \le a \le 2$, $1 \le b \le 10$,$ 1 \le c \le 12$, $1 \le d \le 18$, let $f(a, b, c, d)$ be the unique integer between $0$ and $8150$ inclusive that leaves a remainder of a when divided by $3$, a remainder of $b$ when divided by $11$, a remainder of $c$ when divided by $13$, and a remainder of $d$ when divided by $19$. Compute $$\sum_{a+b+c+d=23}f(a, b, c, d).$$ [b]p7.[/b] Compute $\cos ( \theta)$ if $$\sum^{\infty}_{n=0} \frac{ \cos (n\theta)}{3^n} = 1.$$ [b]p8.[/b] How many solutions does this equation $$\left(\frac{a+b}{2}\right)^2=\left(\frac{b+c}{2019}\right)^2$$ have in positive integers $a, b, c$ that are all less than $2019^2$? [b]p9.[/b] Consider a square grid with vertices labeled $1, 2, 3, 4$ clockwise in that order. Fred the frog is jumping between vertices, with the following rules: he starts at the vertex label $1$, and at any given vertex he jumps to the vertex diagonally across from him with probability $\frac12$ and the vertices adjacent to him each with probability $\frac14$ . After $2019$ jumps, suppose the probability that the sum of the labels on the last two vertices he has visited is $3$ can be written as $2^{-m} -2^{-n}$ for positive integers $m,n$. Find $m + n$. [b]p10.[/b] The base ten numeral system uses digits $0-9$ and each place value corresponds to a power of $10$. For example, $$2019 = 2 \cdot 10^3 + 0 \cdot 10^2 + 1 \cdot 10^1 + 9 \cdot 10^0.$$ Let $\phi =\frac{1 +\sqrt5}{2}$. We can define a similar numeral system, base , where we only use digits $0$ and $1$, and each place value corresponds to a power of . For example, $$11.01 = 1 \cdot \phi^1 + 1 \cdot \phi^0 + 0 \cdot \phi^{-1} + 1 \cdot \phi^{-2}$$ Note that base  representations are not unique, because, for example, $100_{\phi} = 11_{\phi}$. Compute the base $\phi$ representation of $7$ with the fewest number of $1$s. [b]p11.[/b] Let $ABC$ be a triangle with $\angle BAC = 60^o$ and with circumradius $1$. Let $G$ be its centroid and $D$ be the foot of the perpendicular from $A$ to $BC$. Suppose $AG =\frac{\sqrt6}{3}$ . Find $AD$. [b]p12.[/b] Let $f(a, b)$ be a function with the following properties for all positive integers $a \ne b$: $$f(1, 2) = f(2, 1)$$ $$f(a, b) + f(b, a) = 0$$ $$f(a + b, b) = f(b, a) + b$$ Compute: $$\sum^{2019}_{i=1} f(4^i - 1, 2^i) + f(4^i + 1, 2^i)$$ [b]p13.[/b] You and your friends have been tasked with building a cardboard castle in the two-dimensional Cartesian plane. The castle is built by the following rules: 1. There is a tower of height $2^n$ at the origin. 2. From towers of height $2^i \ge 2$, a wall of length $2^{i-1}$ can be constructed between the aforementioned tower and a new tower of height $2^{i-1}$. Walls must be parallel to a coordinate axis, and each tower must be connected to at least one other tower by a wall. If one unit of tower height costs $\$9$ and one unit of wall length costs $\$3$ and $n = 1000$, how many distinct costs are there of castles that satisfy the above constraints? Two castles are distinct if there exists a tower or wall that is in one castle but not in the other. [b]p14.[/b] For $n$ digits, $(a_1, a_2, ..., a_n)$ with $0 \le a_i < n$ for $i = 1, 2,..., n$ and $a_1 \ne 0$ define $(\overline{a_1a_2 ... a_n})_n$ to be the number with digits $a_1$, $a_2$, $...$, $a_n$ written in base $n$. Let $S_n = \{(a_1, a_2, a_3,..., a_n)| \,\,\, (n + 1)| (\overline{a_1a_2 ... a_n})_n, a_1 \ge 1\}$ be the set of $n$-tuples such that $(\overline{a_1a_2 ... a_n})_n$ is divisible by $n + 1$. Find all $n > 1$ such that $n$ divides $|S_n| + 2019$. [b]p15.[/b] Let $P$ be the set of polynomials with degree $2019$ with leading coefficient $1$ and non-leading coefficients from the set $C = \{-1, 0, 1\}$. For example, the function $f = x^{2019} - x^{42} + 1$ is in $P$, but the functions $f = x^{2020}$, $f = -x^{2019}$, and $f = x^{2019} + 2x^{21}$ are not in $P$. Define a [i]swap [/i]on a polynomial $f$ to be changing a term $ax^n$ to $bx^n$ where $b \in C$ and there are no terms with degree smaller than $n$ with coefficients equal to $a$ or $b$. For example, a swap from $x^{2019} + x^{17} - x^{15} + x^{10}$ to $x^{2019} + x^{17} - x^{15} - x^{10}$ would be valid, but the following swaps would not be valid: $$x^{2019} + x^3 \,\,\, \text{to} \,\,\, x^{2019}$$ $$x^{2019} + x^3 \,\,\, \text{to} \,\,\, x^{2019} + x^3 + x^2$$ $$x^{2019} + x^2 + x + 1 \,\,\, \text{to} \,\,\, x^{2019} - x^2 - x - 1$$ Let $B$ be the set of polynomials in $P$ where all non-leading terms have the same coefficient. There are $p$ polynomials that can be reached from each element of $B$ in exactly $s$ swaps, and there exist $0$ polynomials that can be reached from each element of $B$ in less than $s$ swaps. Compute $p \cdot s$, expressing your answer as a prime factorization. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The $KLMN$ tetrahedron (triangle pyramid) vertices are situated inside or on the faces or on the edges of the $ABCD$ tetrahedron. Prove that perimeter of $KLMN$ is less than $4/3$ perimeter of $ABCD$.