A positive integer $n$ is [i]magical[/i] if $\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor=\lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil$. Find the number of magical integers between $1$ and $10,000$ inclusive.

Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold: $$d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1$$ [i]Proposed by Mykyta Kharin[/i]

In quadrilateral $ABCD$, $AB=7, BC=24, CD=15, DA=20,$ and $AC=25$. Let segments $AC$ and $BD$ intersect at $E$. What is the length of $EC$? [i]Proposed by James Lin[/i]

We wish to write $n$ distinct real numbers $(n\geq3)$ on the circumference of a circle in such a way that each number is equal to the product of its immediate neighbors to the left and right. Determine all of the values of $n$ such that this is possible.

Is it possible to put distinct positive integers less than $1991$ in the cells of a $9\times 9$ table so that the products of all the numbers in every column and every row are equal to each other? (N.B. Vasiliev, Moscow)

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T = aL + bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimate it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet? $\textbf{(A) } 240 \qquad \textbf{(B) } 246 \qquad \textbf{(C) } 252 \qquad \textbf{(D) } 258 \qquad \textbf{(E) } 264$

The first step in finding the product $ (3x \plus{} 2)(x \minus{} 5)$ by use of the distributive property in the form $ a(b \plus{} c) \equal{} ab \plus{} ac$ is: $ \textbf{(A)}\ 3x^2 \minus{} 13x \minus{} 10 \qquad \textbf{(B)}\ 3x(x \minus{} 5) \plus{} 2(x \minus{} 5)\qquad \\\textbf{(C)}\ (3x \plus{} 2)x \plus{} (3x \plus{} 2)( \minus{} 5)\qquad \textbf{(D)}\ 3x^2 \minus{} 17x \minus{} 10\qquad \textbf{(E)}\ 3x^2 \plus{} 2x \minus{} 15x \minus{} 10$

Let $ABCD$ be a convex quadrilateral with $AB=CD$. From a random point $P$ of it's diagonal $BD$, we draw a line parallel to $AB$ that intersects $AD$ at point $M$ and a line parallel to $CD$ that intersects $BC$ at point $N$. Prove that: a) The sum $PM+PN$ is constant, independent of the position of $P$ on the diagonal $BD$. b) $MN\le BD$. When the equality holds?

For positive real numbers $x$ and $y$ define $x*y=\frac{x\cdot y}{x+y}$; then $ \textbf{(A)}\ \text{"*" is commutative but not associative} \\ \qquad\textbf{(B)}\ \text{"*" is associative but not commutative} \\ \qquad\textbf{(C)}\ \text{"*" is neither commutative nor associative} \\ \qquad\textbf{(D)}\ \text{"*" is commutative and associative} \\ \qquad\textbf{(E)}\ \text{none of these} $

Anselmo and Bonifacio start a game where they alternatively substitute a number written on a board. In each turn, a player can substitute the written number by either the number of divisors of the written number or by the difference between the written number and the number of divisors it has. Anselmo is the first player to play, and whichever player is the first player to write the number $0$ is the winner. Given that the initial number is $1036$, determine which player has a winning strategy and describe that strategy. Note: For example, the number of divisors of $14$ is $4$, since its divisors are $1$, $2$, $7$, and $14$.

A unit sphere is centered at $(0, 0, 1)$. There is a point light source located at $(1, 0, 4)$ that sends out light uniformly in every direction but is blocked by the sphere. What is the area of the sphere’s shadow on the $x-y$ plane? (A point $(a, b, c)$ denotes the point in three dimensions with $x$-coordinate $a$, $y$-coordinate $b$, and $z$-coordinate $c$)

Let the incircle of the triangle $ABC$ touch $[BC]$ at $D$ and $I$ be the incenter of the triangle. Let $T$ be midpoint of $[ID]$. Let the perpendicular from $I$ to $AD$ meet $AB$ and $AC$ at $K$ and $L$, respectively. Let the perpendicular from $T$ to $AD$ meet $AB$ and $AC$ at $M$ and $N$, respectively. Show that $|KM|\cdot |LN|=|BM|\cdot|CN|$.

Let $ABC$ be a triangle with $AB<AC$. Let $D$ be the point where the bisector of angle $\angle BAC$ touches $BC$ and let $D'$ be the reflection of $D$ in the midpoint of $BC$. Let $X$ be the intersection of the bisector of angle $\angle BAC$ with the line parallel to $AB$ that passes through $D'$. Prove that the line $AC$ is tangent with the circumscribed circle of triangle $XCD'$

[b]p1.[/b] Let $X,Y ,Z$ be nonzero real numbers such that the quadratic function $X t^2 - Y t + Z = 0$ has the unique root $t = Y$ . Find $X$. [b]p2.[/b] Let $ABCD$ be a kite with $AB = BC = 1$ and $CD = AD =\sqrt2$. Given that $BD =\sqrt5$, find $AC$. [b]p3.[/b] Find the number of integers $n$ such that $n -2016$ divides $n^2 -2016$. An integer $a$ divides an integer $b$ if there exists a unique integer $k$ such that $ak = b$. [b]p4.[/b] The points $A(-16, 256)$ and $B(20, 400)$ lie on the parabola $y = x^2$ . There exists a point $C(a,a^2)$ on the parabola $y = x^2$ such that there exists a point $D$ on the parabola $y = -x^2$ so that $ACBD$ is a parallelogram. Find $a$. [b]p5.[/b] Figure $F_0$ is a unit square. To create figure $F_1$, divide each side of the square into equal fifths and add two new squares with sidelength $\frac15$ to each side, with one of their sides on one of the sides of the larger square. To create figure $F_{k+1}$ from $F_k$ , repeat this same process for each open side of the smallest squares created in $F_n$. Let $A_n$ be the area of $F_n$. Find $\lim_{n\to \infty} A_n$. [img]https://cdn.artofproblemsolving.com/attachments/8/9/85b764acba2a548ecc61e9ffc29aacf24b4647.png[/img] [b]p6.[/b] For a prime $p$, let $S_p$ be the set of nonnegative integers $n$ less than $p$ for which there exists a nonnegative integer $k$ such that $2016^k -n$ is divisible by $p$. Find the sum of all $p$ for which $p$ does not divide the sum of the elements of $S_p$ . [b]p7. [/b] Trapezoid $ABCD$ has $AB \parallel CD$ and $AD = AB = BC$. Unit circles $\gamma$ and $\omega$ are inscribed in the trapezoid such that circle $\gamma$ is tangent to $CD$, $AB$, and $AD$, and circle $\omega$ is tangent to $CD$, $AB$, and $BC$. If circles $\gamma$ and $\omega$ are externally tangent to each other, find $AB$. [b]p8.[/b] Let $x, y, z$ be real numbers such that $(x+y)^2+(y+z)^2+(z+x)^2 = 1$. Over all triples $(x, y, z)$, find the maximum possible value of $y -z$. [b]p9.[/b] Triangle $\vartriangle ABC$ has sidelengths $AB = 13$, $BC = 14$, and $CA = 15$. Let $P$ be a point on segment $BC$ such that $\frac{BP}{CP} = 3$, and let $I_1$ and $I_2$ be the incenters of triangles $\vartriangle ABP$ and $\vartriangle ACP$. Suppose that the circumcircle of $\vartriangle I_1PI_2$ intersects segment $AP$ for a second time at a point $X \ne P$. Find the length of segment $AX$. [b]p10.[/b] For $1 \le i \le 9$, let Ai be the answer to problem i from this section. Let $(i_1,i_2,... ,i_9)$ be a permutation of $(1, 2,... , 9)$ such that $A_{i_1} < A_{i_2} < ... < A_{i_9}$. For each $i_j$ , put the number $i_j$ in the box which is in the $j$th row from the top and the $j$th column from the left of the $9\times 9$ grid in the bonus section of the answer sheet. Then, fill in the rest of the squares with digits $1, 2,... , 9$ such that $\bullet$ each bolded $ 3\times 3$ grid contains exactly one of each digit from $ 1$ to $9$, $\bullet$ each row of the $9\times 9$ grid contains exactly one of each digit from $ 1$ to $9$, and $\bullet$ each column of the $9\times 9$ grid contains exactly one of each digit from $ 1$ to $9$. PS. You had better use hide for answers.

Prove that the set $\{1, 2, . . . , 1986\}$ can be partitioned into $27$ disjoint sets so that no one of these sets contains an arithmetic triple (i.e., three distinct numbers in an arithmetic progression).

Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$.Points $X,Y$ lie on $AB,AC$,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$.Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$.

Out of a sample of $100$ people, $24$ do not like red or blue, $40$ like both red and blue, and $50$ people like red. How many people like blue but not red? $\textbf{(A) }24\qquad\textbf{(B) }26\qquad\textbf{(C) }48\qquad\textbf{(D) }64\qquad\textbf{(E) }76$

Determine the integers $x$ and $y$ for which $\sqrt{4^x + 5^y}$ is rational.

Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$

Three numbers are initially written on the board: 2023, 2024, and 2025. In each move, you can increase any two of these numbers by 1 and decrease the third one by 2. a) Determine whether it is possible to perform a sequence of operations such that the board eventually contains two numbers that are equal. b) Calculate the number of all possible ordered triples of positive integers that can be obtained by performing such operations some number of times. [i]Proposed by Giorgi Arabidze, Georgia [/i]

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i] [b][i]Original Formulation[/i][/b] Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. [i]Proposed by Soviet Union. [/i]

Let $F(x)$ be a real valued function defined for all real $x$ except for $x=0$ and $x=1$ and satisfying the functional equation $F(x)+F\{(x-1)/x\}=1+x.$ Find all functions $F(x)$ satisfying these conditions.

Does there exist a number $\alpha, 0 < \alpha < 1$ such that there is an infinite sequence $\{a_n\}$ of positive numbers satisfying \[ 1 + a_{n+1} \leq a_n + \frac{\alpha}{n} \cdot \alpha_n, n = 1,2, \ldots? \]

Peter and Basil play the following game on a horizontal table $1\times{2019}$. Initially Peter chooses $n$ positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by $s$ cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by $s$ cells neither to the left, nor to the right, the coin stays in the current cell. Find the least $n$ such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays.