A team of four students goes to LMT, and each student brings a lunch. However, on the bus, the students’ lunches get mixed up, and during lunch time, each student chooses a random lunch to eat (no two students may eat the same lunch). What is the probability that each student chooses his or her own lunch correctly?

A triangular colony area is divided into four fields of varying size as shown in the figure below shows. The only other thing we know is that the distances $AF$, $FD$, $BF$ and $FE$ have the lengths $5$, $2$, $4$ and $2$ respectively (in $10$s of m). When the lots are distributed, Joar gets to choose first. Which lot should he choose to get the one with the largest area? [img]https://cdn.artofproblemsolving.com/attachments/9/5/073e2699d54c8ee3a4dd7d23b69c2a894fd93e.png[/img]

Are there integers $ x$, $ y$, not both divisible by $ 5$, such that $ x^2 \plus{} 19y^2 \equal{} 198\cdot 10^{1989}$?

Let $ABCD$ be a cyclic quadrilateral, $O$ be its circumcenter, $P$ be a common points of its diagonals, and $M , N$ be the midpoints of $AB$ and $CD$ respectively. A circle $OPM$ meets for the second time segments $AP$ and $BP$ at points $A_1$ and $B_1$ respectively and a circle $OPN$ meets for the second time segments $CP$ and $DP$ at points $C_1$ and $D_1$ respectively. Prove that the areas of quadrilaterals $AA_1B_1B$ and $CC_1D_1D$ are equal.

Prove that for any integer $k > 2$, there exist infinitely many positive integers $n$ such that the least common multiple of $n$, $n + 1$,$...$, $n + k - 1$ is greater than the least common multiple of $n + 1$,$n + 2$,$...$, $n + k$.

Determine all real numbers $a$ for which the system of equations \begin{align*} 3x^2+2y^2+2z^2&=a\\ 4x^2+4y^2+5z^2&=1-a \end{align*} has at least one solution $(x,y,z)$ in the real numbers.

Let be three real numbers $ x,y,z>1 $ that satisfy $ xyz=8. $ Prove that: $$ \left( \sqrt{\log_2 x} +\sqrt{\log_2 y} \right)\cdot \left( \sqrt{\log_2 y} +\sqrt{\log_2 z} \right)\cdot \left( \sqrt{\log_2 z} +\sqrt{\log_2 x} \right)\le 8 $$ [i]Gabriela Constantinescu[/i]

Consider the sequence $a(n)$ defined by the following conditions: $$a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...$$ Prove that the sequence contains an infinite number of perfect squares. (Note: $[x]$ means the integer part of $x$, that is the greatest integer not greater than $x$.) (A Andjans)

Let $n$ be a positive integers. Equilateral triangle with sides of length $n$ is split into equilateral triangles with side lengths $1$, forming a triangular lattice. Call an equilateral triangle with vertices in the lattice "important". Let $p_k$ be the number of unordered pairs of vertices in the lattice which participate in exactly $k$ important triangles. Find (as a function of $n$) (a) $p_0+p_1+p_2$ (b) $p_1+2p_2$

Which natural numbers can be expressed as the difference of squares of two integers?

Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.

Let $k$ be a positive integer. Find all positive integers $n$, such that it is possible to mark $n$ points on the sides of a triangle (different from its vertices) and connect some of them with a line in such a way that the following conditions are satisfied: 1) there is at least $1$ marked point on each side, 2) for each pair of points $X$ and $Y$ marked on different sides, on the third side there exist exactly $k$ marked points which are connected to both $X$ and $Y$ and exactly k points which are connected to neither $X$ nor $Y$

The $30$ edges of a regular icosahedron are distinguished by labeling them $1,2,\dots,30.$ How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color?

The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, $\,\ldots, \,$ red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides are red, blue, red, blue, red, blue, $\, \ldots, \,$ red, yellow, blue?

Determine whether or not there exists a positive integer $k$ such that $p = 6k+1$ is a prime and \[\binom{3k}{k} \equiv 1 \pmod{p}.\]

A farmer buys a batch of trees, which he wishes to plant in a square grid. For example, if he had $25$ trees, then he could plant them as shown below. [asy] size(3cm,0); dot((0,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((1,0)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((2,0)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((3,0)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((4,0)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); [/asy] However, the farmer finds that he cannot plant his trees in a square grid. If he had $20$ more trees, or if he had $39$ fewer trees, then he could plant his trees in a square grid. How many trees did the farmer buy?

Let $I, C, \omega$ and $\Omega$ be the incenter, circumcenter, incircle and circumcircle, respectively, of the scalene triangle $XYZ$ with $XZ > YZ > XY$. The incircle $\omega$ is tangent to the sides $YZ, XZ$ and $XY$ at the points $D, E$ and $F$. Let $S$ be the point on $\Omega$ such that $XS, CI$ and $YZ$ are concurrent. Let $(XEF) \cap \Omega = R$, $(RSD) \cap (XEF) = U$, $SU \cap CI = N$, $EF \cap YZ = A$, $EF \cap CI = T$ and $XU \cap YZ = O$. Prove that $NARUTO$ is cyclic.

Let $\vartriangle BCD$ be an equilateral triangle and $A$ be a point on the circumcircle of $\vartriangle BCD$ such that $A$ is on the minor arc $BD$. Then, let $P$ be the intersection of $\overline{AB}$ with $\overline{CD}$, $Q$ be the intersection of $\overline{AC}$ with $\overline{DB}$, and $R$ be the intersection of $\overline{AD}$ with $\overline{BC}$. Finally, let $X$, $Y$ , and $Z$ be the feet of the altitudes from $P$, $Q$, and $R$, respectively, in triangle $\vartriangle PQR$. Given $BQ = 3 -\sqrt5$ and $BC = 2$, compute the product of the areas $[\vartriangle XCD] \cdot [\vartriangle Y DB] \cdot [\vartriangle ZBC]$.

In an acute triangle $ABC$ let $K,L,$ and $M$ be the midpoints of sides $AB,BC,$ and $CA,$ respectively. From each of $K,L,$ and $M$ drop two perpendiculars to the other two sides of the triangle; e.g., drop perpendiculars from $K$ to sides $BC$ and $CA,$ etc. The resulting $6$ perpendiculars intersect at points $Q,S,$ and $T$ as in the figure to form a hexagon $KQLSMT$ inside triangle $ABC.$ Prove that the area of this hexagon $KQLSMT$ is half of the area of the original triangle $ABC.$ [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra; diagram by adihaya*/ import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = 11.888712276357234, xmax = 17.841346447833423, ymin = 10.61620970860601, ymax = 15.470685507068502; /* image dimensions */ pen zzttqq = rgb(0.6,0.2,0.); pen qqwuqq = rgb(0.,0.39215686274509803,0.); pair A = (12.488234161849352,12.833838721895551), B = (16.50823416184936,15.093838721895553), C = (16.28823416184936,11.353838721895551), K = (14.498234161849355,13.963838721895552), L = (16.39823416184936,13.223838721895552), M = (14.388234161849356,12.093838721895551), D = (13.615830174638527,13.467760858438725), F = (15.75135711740064,11.562938202365055), G = (15.625830174638523,14.597760858438724), H = (16.435061748056253,13.849907687412797), T = (14.02296781802369,12.74356027153236), Q = (16.032967818023693,13.873560271532357), O = (16.325061748056253,11.979907687412794); draw(A--B--C--cycle, zzttqq); draw((13.426050287639166,13.361068683160477)--(13.532742462917415,13.171288796161116)--(13.722522349916774,13.277980971439364)--D--cycle, qqwuqq); draw((14.054227993863618,12.223925334689998)--(14.133240861538676,12.426796211152979)--(13.930369985075695,12.505809078828037)--(13.851357117400637,12.302938202365056)--cycle, qqwuqq); draw((16.337846386707046,12.19724654447628)--(16.12050752964356,12.210031183127075)--(16.107722890992765,11.992692326063588)--O--cycle, qqwuqq); draw((15.830369985075697,11.765809078828037)--(15.627499108612716,11.844821946503092)--(15.54848624093766,11.641951070040111)--F--cycle, qqwuqq); draw((15.436050287639164,14.491068683160476)--(15.542742462917412,14.301288796161115)--(15.73252234991677,14.407980971439365)--G--cycle, qqwuqq); draw((16.217722890992764,13.86269232606359)--(16.20493825234197,13.645353469000101)--(16.42227710940546,13.63256883034931)--H--cycle, qqwuqq); Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax, Ticks(laxis, Step = 1., Size = 2, NoZero),EndArrow(6), above = true); yaxis(ymin, ymax, Ticks(laxis, Step = 1., Size = 2, NoZero),EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */ /* draw figures */ draw(A--B, zzttqq); draw(B--C, zzttqq); draw(C--A, zzttqq); draw(M--D); draw(K--(13.851357117400637,12.302938202365056)); draw(F--L); draw(L--G); draw(K--H); draw(M--O); /* dots and labels */ dot(A,dotstyle); label("$A$", (12.52502834296331,12.93568440300881), NE * labelscalefactor); dot(B,dotstyle); label("$B$", (16.548187989892043,15.193580123223922), NE * labelscalefactor); dot(C,dotstyle); label("$C$", (16.332661580235147,11.457789022504372), NE * labelscalefactor); dot(K,linewidth(3.pt) + dotstyle); label("$K$", (14.536608166427676,14.02357961365791), NE * labelscalefactor); dot(L,linewidth(3.pt) + dotstyle); label("$L$", (16.43529320388129,13.28463192340569), NE * labelscalefactor); dot(M,linewidth(3.pt) + dotstyle); label("$M$", (14.433976542781535,12.155684063298134), NE * labelscalefactor); dot(D,linewidth(3.pt) + dotstyle); dot((13.851357117400637,12.302938202365056),linewidth(3.pt) + dotstyle); dot(F,linewidth(3.pt) + dotstyle); dot(G,linewidth(3.pt) + dotstyle); dot(H,linewidth(3.pt) + dotstyle); dot((15.922967818023695,12.003560271532354),linewidth(3.pt) + dotstyle); label("$S$", (15.96318773510904,12.063315602016607), NE * labelscalefactor); dot(T,linewidth(3.pt) + dotstyle); label("$T$", (14.064502697655428,12.802263292268826), NE * labelscalefactor); dot(Q,linewidth(3.pt) + dotstyle); label("$Q$", (16.076082521119794,13.931211152376383), NE * labelscalefactor); dot(O,linewidth(3.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]

If $a_1, a_2, \ldots , a_n$ are positive integers and $k = \max\{a_1, \ldots, a_n\}$, $t = \min\{a_1,\ldots, a_n\}$, prove the inequality \[\left(\frac{a_1^2+a_2^2+\cdots+a_n^2}{a_1+a_2+\cdots+a_n}\right)^{\frac{kn}{t}} \geq a_1a_2\cdots a_n.\] When does equality hold?

Let $ABC$ be a triangle, and erect three rectangles $ABB_1A_2$, $BCC_1B_2$, $CAA_1C_2$ externally on its sides $AB$, $BC$, $CA$, respectively. Prove that the perpendicular bisectors of the segments $A_1A_2$, $B_1B_2$, $C_1C_2$ are concurrent.

Let $n\geq 2$ be a given integer. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that \[f(x-f(y))=f(x+y^n)+f(f(y)+y^n), \qquad \forall x,y \in \mathbb R.\]

Source: 2017 Canadian Open Math Challenge, Problem A4 ----- Three positive integers $a$, $b$, $c$ satisfy $$4^a \cdot 5^b \cdot 6^c = 8^8 \cdot 9^9 \cdot 10^{10}.$$ Determine the sum of $a + b + c$.

[u]Round 5 [/u] [b]p13.[/b] How many ways can we form a group with an odd number of members (plural) from $99$ people? Express your answer in the form $a^b + c$, where $a, b$, and $c$ are integers and $a$ is prime. [b]p14.[/b] A cube is inscibed in a right circular cone such that the ratio of the height of the cone to the radius is $2:1$. Compute the fraction of the cone’s volume that the cube occupies. [b]p15.[/b] Let $F_0 = 1$, $F_1 = 1$ and $F_k = F_{k-1} + F_{k-2}$. Let $P(x) = \sum^{99}_{k=0} x^{F_k}$ . The remainder when $P(x)$ is divided by $x^3 - 1$ can be expressed as $ax^2 + bx + c$. Find $2a + b$. [u]Round 6 [/u] [b]p16.[/b] Ankit finds a quite peculiar deck of cards in that each card has n distinct symbols on it and any two cards chosen from the deck will have exactly one symbol in common. The cards are guaranteed to not have a certain symbol which is held in common with all the cards. Ankit decides to create a function f(n) which describes the maximum possible number of cards in a set given the previous constraints. What is the value of $f(10)$? [b]p17.[/b] If $|x| <\frac14$ and $$X = \sum^{\infty}_{N=0} \sum^{N}_{n=0} {N \choose n}x^{2n}(2x)^{N-n}.$$ then write $X$ in terms of $x$ without any summation or product symbols (and without an infinite number of ‘$+$’s, etc.). [b]p18.[/b] Dietrich is playing a game where he is given three numbers $a, b, c$ which range from $[0, 3]$ in a continuous uniform distribution. Dietrich wins the game if the maximum distance between any two numbers is no more than $1$. What is the probability Dietrich wins the game? [u]Round 7 [/u] [b]p19.[/b] Consider f defined by $$f(x) = x^6 + a_1x^5 + a_2x^4 + a_3x^3 + a_4x^2 + a_5x + a_6.$$ How many tuples of positive integers $(a_1, a_2, a_3, a_4, a_5, a_6)$ exist such that $f(-1) = 12$ and $f(1) = 30$? [b]p20.[/b] Let $a_n$ be the number of permutations of the numbers $S = \{1, 2, ... , n\}$ such that for all $k$ with $1 \le k \le n$, the sum of $k$ and the number in the $k$th position of the permutation is a power of $2$. Compute $a_1 + a_2 + a_4 + a_8 + ... + a_{1048576}$. [b]p21.[/b] A $4$-dimensional hypercube of edge length $1$ is constructed in $4$-space with its edges parallel to the coordinate axes and one vertex at the origin. Its coordinates are given by all possible permutations of $(0, 0, 0, 0)$,$(1, 0, 0, 0)$,$(1, 1, 0, 0)$,$(1, 1, 1, 0)$, and $(1, 1, 1, 1)$. The $3$-dimensional hyperplane given by $x+y+z+w = 2$ intersects the hypercube at $6$ of its vertices. Compute the 3-dimensional volume of the solid formed by the intersection. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2781335p24424563]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. His drink contains $45\%$ of orange juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $60\%$ of orange juice? [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] For one particular number $a > 0$ the function f satisfies the equality $f(x + a) =\frac{1 + f(x)}{1 - f(x)}$ for all $x$. Show that $f$ is a periodic function. (A function $f$ is periodic with the period $T$ if $f(x + T) = f(x)$ for any $x$.) [b]p4.[/b] If $a, b, c, x, y, z$ are numbers so that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}= 1$ and $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}= 0$. Show that $\frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} = 1$ [b]p5.[/b] Is it possible that a four-digit number $AABB$ is a perfect square? (Same letters denote the same digits). [b]p6.[/b] A finite number of arcs of a circle are painted black (see figure). The total length of these arcs is less than $\frac15$ of the circumference. Show that it is possible to inscribe a square in the circle so that all vertices of the square are in the unpainted portion of the circle. [img]https://cdn.artofproblemsolving.com/attachments/2/c/bdfa61917a47f3de5dd3684627792a9ebf05d5.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].