Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$? $\textbf{(A) }-10\qquad\textbf{(B) }-6\qquad\textbf{(C) }0\qquad\textbf{(D) }6\qquad \textbf{(E) }10$

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ f\left(x^2\right) - yf(y) = f(x + y) (f(x) - y) $$ for all real numbers $x$ and $y$.

Let \[f(x)=\cos(x^3-4x^2+5x-2).\] If we let $f^{(k)}$ denote the $k$th derivative of $f$, compute $f^{(10)}(1)$. For the sake of this problem, note that $10!=3628800$.

Let \(\varepsilon < \frac{1}{2}\) be a positive real number and let \(U_{\varepsilon}\) denote the set of real numbers that differ from their nearest integer by at most \(\varepsilon\). Prove that there exists a positive integer \(m\) such that for any real number \(x\), the sets \(\left\{x, 2x, 3x, . . . , mx\right\}\) and \(U_{\varepsilon}\) have at least one element in common. proposed by the ICMC Problem Committee

A circle with center $ C$ is tangent to the positive $ x$ and $ y$-axes and externally tangent to the circle centered at $ (3,0)$ with radius $ 1$. What is the sum of all possible radii of the circle with center $ C$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Fixed points $B$ and $C$ are on a fixed circle $\omega$ and point $A$ varies on this circle. We call the midpoint of arc $BC$ (not containing $A$) $D$ and the orthocenter of the triangle $ABC$, $H$. Line $DH$ intersects circle $\omega$ again in $K$. Tangent in $A$ to circumcircle of triangle $AKH$ intersects line $DH$ and circle $\omega$ again in $L$ and $M$ respectively. Prove that the value of $\frac{AL}{AM}$ is constant. [i]Proposed by Mehdi E'tesami Fard[/i]

Prove that there is positive integers $N$ such that the equation $$arctan(N)=\sum_{i=1}^{2020} a_i arctan(i),$$ does not hold for any integers $a_{i}.$

In a $ 2\times 7$ board gridded in $1\times 1$ squares, the $24$ points that are vertices of the squares are considered. [img]https://cdn.artofproblemsolving.com/attachments/9/e/841f11ef9d6fc27cdbe7c91bab6d52d12180e8.gif[/img] Juan and Matías play on this board. Juan paints red the same number of points on each of the three horizontal lines. If Matthias can choose three red dots that are vertices of an acute triangle, Matthias wins the game. What is the maximum number of dots Juan can color in to make sure Matías doesn't win? (For the number found, give an example of coloring that prevents Matías from winning and justify why if the number is greater, Matías can always win.)

Suppose $ABC$ is a triangle with $M$ the midpoint of $BC$. Suppose that $AM$ intersects the incircle at $K,L$. We draw parallel line from $K$ and $L$ to $BC$ and name their second intersection point with incircle $X$ and $Y$. Suppose that $AX$ and $AY$ intersect $BC$ at $P$ and $Q$. Prove that $BP=CQ$.

Prove that there are infinitely many pairs of natural numbers $a$ and $b$ such that $a^2 + 1$ is divisible by $b$ and $b^2 + 1$ is divisible by $a$ .

Let \[f(x) = a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}\] be a polynomial with coefficients satisfying the conditions: \[0\le a_{i}\le a_{0},\quad i=1,2,\ldots,n.\] Let $b_{0},b_{1},\ldots,b_{2n}$ be the coefficients of the polynomial \begin{align*}\left(f(x)\right)^{2}&= \left(a_{0}+a_{1}x+a_{2}x^{2}+\cdots a_{n}x^{n}\right)\\ &= b_{0}+b_{1}x+b_{2}x^{2}+\cdots+b_{2n}x^{2n}. \end{align*} Prove that $b_{n+1}\le \frac{1}{2}\left(f(1)\right)^{2}$.

Al and Bill play a game involving a fair six-sided die. The die is rolled until either there is a number less than $5$ rolled on consecutive tosses, or there is a number greater than $4$ on consecutive tosses. Al wins if the last roll is a $5$ or $6$. Bill wins if the last roll is a $2$ or lower. Let $m$ and $n$ be relatively prime positive integers such that $m/n$ is the probability that Bill wins. Find the value of $m+n$.

Let $ABCDEF$ be a convex hexagon with $AB = BC, CD = DE$ and $EF = FA$. Prove that the lines through $C,E,A$ perpendicular to $BD,DF,FB$ are concurrent.

The Nationwide Basketball Society (NBS) has $8001$ teams, numbered $2000$ through $10000$. For each $n$, team $n$ has $n+1$ players, and in a sheer coincidence, this year each player attempted $n$ shots and on team $n$, exactly one player made $0$ shots, one player made $1$ shot, . . ., one player made $n$ shots. A player's [i]field goal percentage[/i] is defined as the percentage of shots the player made, rounded to the nearest tenth of a percent (For instance, $32.45\%$ rounds to $32.5\%$). A player in the NBS is randomly selected among those whose field goal percentage is $66.6\%$. If this player plays for team $k$, the probability that $k\geq 6000$ can be expressed as $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $p+q$.

Let ABCDEF be an equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are the extensions of AB, CD and EF has side lengths 200, 240 and 300 respectively. Find the side length of the hexagon.

Compute the sum: $\sum_{i=0}^{101} \frac{x_i^3}{1-3x_i+3x_i^2}$ for $x_i=\frac{i}{101}$.

In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$. Proposed by Fatemeh Sajadi

$ a,b,c\ge 0,\ \ \ a^3+b^3+c^3+abc=4 $ Prove that $a^3b+b^3c+c^3b \le 3$

Consider a triangle $ABC$ and a point $D$ on its side $\overline{AB}$. Let $I$ be a point inside $\triangle ABC$ on the angle bisector of $ACB$. The second intersections of lines $AI$ and $CI$ with circle $ACD$ are $P$ and $Q$, respectively. Similarly, the second intersection of lines $BI$ and $CI$ with circle $BCD$ are $R$ and $S$, respectively. Show that if $P\neq Q$ and $R\neq S$, then lines $AB$, $PQ$ and $RS$ pass through a point or are parallel.

Determine if there is a positive integer $n$ such that for any $n$ consecutive positive integers, there is [b]one[/b] of them(denote $c$) such that $c$ can be written as sum of consecutive integers(not necessarily all positive) of at most $2020$ distinct ways.

In the right parallelopiped $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$, with $AB=12\sqrt{3}$ cm and $AA^{\prime}=18$ cm, we consider the points $P\in AA^{\prime}$ and $N\in A^{\prime}B^{\prime}$ such that $A^{\prime}N=3B^{\prime}N$. Determine the length of the line segment $AP$ such that for any position of the point $M\in BC$, the triangle $MNP$ is right angled at $N$.

Circles $A$, $B$, and $C$ are externally tangent circles. Line $PQ$ is drawn such that $PQ$ is tangent to $A$ at $P$, tangent to $B$ at $Q$, and does not intersect with $C$. Circle $D$ is drawn such that it passes through the centers of $A$, $B$, and $C$. Let $R$ be the point on $D$ furthest from $PQ$. If $A$, $B$, and $C$ have radii $3$, $2$, and $1$, respectively, the area of triangle $PQR$ can be expressed in the form of $a+b\sqrt{c}$, where $a$, $b$, and $c$ are integers with $c$ not divisible by any prime square. What is $a + b + c$?

Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

Consider the sequence of positive integers defined by $s_1,s_2,s_3, \dotsc $ of positive integers defined by [list] [*]$s_1=2$, and[/*] [*]for each positive integer $n$, $s_{n+1}$ is equal to $s_n$ plus the product of prime factors of $s_n$.[/*] [/list] The first terms of the sequence are $2,4,6,12,18,24$. Prove that the product of the $2019$ smallest primes is a term of the sequence.

The number $27,\,000,\,001$ has exactly four prime factors. Find their sum.