A week ago, Sandy’s seasonal Little League batting average was $360$. After five more at bats this week, Sandy’s batting average is up to $400$. What is the smallest number of hits that Sandy could have had this season?

A splitting of a planar polygon is a finite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer $n \ge 3$, determine the largest integer $m$ such that no planar $n$-gon splits into less than $m$ triangles.

Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, define $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Let $a_0,a_1,a_2,\dots$ be a sequence of nonnegative integers such that $a_2=5$, $a_{2014}=2015$, and $a_n=a_{a_{n-1}}$ for all positive integers $n$. Find all possible values of $a_{2015}$.

Suppose two regular pyramids with the same base $ABC$: $P-ABC$ and $Q-ABC$ are circumscribed by the same sphere. If the angle formed by one of the lateral face and the base of pyramid $P-ABC$ is $\frac{\pi}{4}$, find the tangent value of the angle formed by one of the lateral face and the base of the pyramid $Q-ABC$.

Find all real solutions of the system of equations $$\begin{cases} (x_3 + x_4 + x_5)^5 = 3x_1 \\ (x_4 + x_5 + x_1)^5 = 3x_2\\ (x_5 + x _1 + x_2)^5 = 3x_3\\ (x_1 + x_2 + x_3)^5 = 3x_4\\ (x_2 + x_3 + x_4)^5 = 3x_5 \end{cases}$$ (L. Tumescu , Romania)

We are given a non-obtuse $\Delta ABC$ $(BC>AC)$ with an altitude $CD$ $(D\in AB)$, center $O$ of its circumscribed circle, and a middle point $M$ of its side $AB$. Point $E$ lies on the ray $\overrightarrow{BA}$ in such way that $AE.BE=DE.ME$. If the line $OE$ bisects the area of $\Delta ABC$ and $CO=CD.cos\angle ACB$, determine the angles of $\Delta ABC$.

What number is nine more than four times the answer to this question?

Let $f_1, f_2, \ldots$ be continuous real functions on the real line. Is it true that if the series $\sum_{n=1}^{\infty} f_n(x)$ is divergent for every $x$, then this holds also true for any typical choice of the signs in the sum (i.e. the set of those $\{ \epsilon _n\}_{n=1}^{\infty} \in \{ +1, -1\}^{\mathbb{N}}$ sequences, for which there series $\sum_{n=1}^{\infty} \epsilon_nf_n(x)$ is convergent at least at one point $x$, forms a subset of first category within the set $\{+1,-1\}^{\mathbb{N}} $)? (translated by L. Erdős)

[b]p1. [/b] Evaluate $S$. $$S =\frac{10000^2 - 1}{\sqrt{10000^2 - 19999}}$$ [b]p2. [/b] Starting on a triangular face of a right triangular prism and allowing moves to only adjacent faces, how many ways can you pass through each of the other four faces and return to the first face in five moves? [b]p3.[/b] Given that $$(a + b) + (b + c) + (c + a) = 18$$ $$\frac{1}{a + b}+\frac{1}{b + c}+ \frac{1}{c + a}=\frac59,$$ determine $$\frac{c}{a + b}+\frac{a}{b + c}+\frac{b}{c + a}.$$ [b]p4.[/b] Find all primes $p$ such that $2^{p+1} + p^3 - p^2 - p$ is prime. [b]p5.[/b] In right triangle $ABC$ with the right angle at $A$, $AF$ is the median, $AH$ is the altitude, and $AE$ is the angle bisector. If $\angle EAF = 30^o$ , find $\angle BAH$ in degrees. [b]p6.[/b] For which integers $a$ does the equation $(1 - a)(a - x)(x- 1) = ax$ not have two distinct real roots of $x$? [b]p7. [/b]Given that $a^2 + b^2 - ab - b +\frac13 = 0$, solve for all $(a, b)$. [b]p8. [/b] Point $E$ is on side $\overline{AB}$ of the unit square $ABCD$. $F$ is chosen on $\overline{BC}$ so that $AE = BF$, and $G$ is the intersection of $\overline{DE}$ and $\overline{AF}$. As the location of $E$ varies along side $\overline{AB}$, what is the minimum length of $\overline{BG}$? [b]p9.[/b] Sam and Susan are taking turns shooting a basketball. Sam goes first and has probability $P$ of missing any shot, while Susan has probability $P$ of making any shot. What must $P$ be so that Susan has a $50\%$ chance of making the first shot? [b]p10.[/b] Quadrilateral $ABCD$ has $AB = BC = CD = 7$, $AD = 13$, $\angle BCD = 2\angle DAB$, and $\angle ABC = 2\angle CDA$. Find its area. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\mathbb{F}_p$ be the set of integers modulo $p$. Call a function $f : \mathbb{F}_p^2 \to \mathbb{F}_p$ [i]quasiperiodic[/i] if there exist $a,b \in \mathbb{F}_p$, not both zero, so that $f(x + a, y + b) = f(x, y)$ for all $x,y \in \mathbb{F}_p$. Find the number of functions $\mathbb{F}_p^2 \to \mathbb{F}_p$ that can be written as the sum of some number of quasiperiodic functions.

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 $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other

Let $n\ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties: (i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets containing exactly $k$ elements; (ii) for any two sets $A, B\in \mathcal{F}$, their union $A\cup B$ also belongs to $\mathcal{F}$. Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements. (Proposed by Fedor Petrov, St. Petersburg State University)

A wall made of mirrors has the shape of $\triangle ABC$, where $AB = 13$, $BC = 16$, and $CA = 9$. A laser positioned at point $A$ is fired at the midpoint $M$ of $BC$. The shot reflects about $BC$ and then strikes point $P$ on $AB$. If $\tfrac{AM}{MP} = \tfrac{m}{n}$ for relatively prime positive integers $m, n$, compute $100m+n$. [i]Proposed by Michael Tang[/i]

The domain of a function $f$ is $\mathbb{N}$ (The set of natural numbers). The function is defined as follows : $$f(n)=n+\lfloor\sqrt{n}\rfloor$$ where $\lfloor k\rfloor$ denotes the nearest integer smaller than or equal to $k$. Prove that, for every natural number $m$, the following sequence contains at least one perfect square $$m,~f(m),~f^2(m),~f^3(m),\cdots$$ The notation $f^k$ denotes the function obtained by composing $f$ with itself $k$ times.

Find all triples of integers $(a, b, c)$ satisfying $a^2 + b^2 + c^2 =3(ab + bc + ca).$

In the Olympic school the exams are graded with whole numbers, the lowest possible grade is $0$, and the highest is $10$. In the arithmetic class the teacher takes two exams. This year he has $15$ students. When one of his students gets less than $3$ on the first exam and more than $7$ on the second exam, he calls him an overachieving student. The teacher, at the end of correcting the exams, averaged the $30$ grades and obtained $8$. What is the largest number of students who passed this class could have had?

In the following sum: $1 + 2 + 3 + 4 + 5 + 6$, if we remove the first two “+” signs, we obtain the new sum $123 + 4 + 5 + 6 = 138$. By removing three “$+$” signs, we can obtain $1 + 23 + 456 = 480$. Let us now consider the sum $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13$, in which some “$+$” signs are to be removed. What are the three smallest multiples of $100$ that we can get in this way?

Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the least number of squares in a fleet to which no new ship can be added.

The points $A_1$,$B_1$,$C_1$ are middle points of the arcs $\widehat{BC}, \widehat{CA}, \widehat{AB}$ of the circumscribed circle of $\Delta ABC$, respectively. The points $I_a,I_b,I_c$ are the reflections in the middle points of $BC,CA,AB$ of the center $I$ of the inscribed circle in the triangle. Prove that $I_a A_1,I_b B_1$, and $I_c C_1$ are concurrent.

Consider the sequence $(a_n)_{n\ge 1}$ defined by $a_n=n$ for $n\in \{1,2,3.4,5,6\}$, and for $n \ge 7$: $$a_n={\lfloor}\frac{a_1+a_2+...+a_{n-1}}{2}{\rfloor}$$ where ${\lfloor}x{\rfloor}$ is the greatest integer less than or equal to $x$. For example : ${\lfloor}2.4{\rfloor} = 2, {\lfloor}3{\rfloor} = 3$ and ${\lfloor}\pi {\rfloor}= 3$. For all integers $n \ge 2$, let $S_n = \{a_1,a_1,...,a_n\}- \{r_n\}$ where $r_n$ is the remainder when $a_1 + a_2 + ... + a_n$ is divided by $3$. The minus $-$ denotes the ''[i]remove it if it is there[/i]'' notation. For example : $S_4 = {2,3,4}$ because $r_4= 1$ so $1$ is removed from $\{1,2,3,4\}$. However $S_5= \{1,2,3,4,5\}$ betawe $r_5 = 0$ and $0$ is not in the set $\{1,2,3,4,5\}$. 1. Determine $S_7,S_8,S_9$ and $S_{10}$. 2. We say that a set $S_n$ for $n\ge 6$ is well-balanced if it can be partitioned into three pairwise disjoint subsets with equal sum. For example : $S_6 = \{1,2,3,4,5,6\} =\{1,6\}\cup \{2,5\}\cup \{3,4\}$ and $1 +6 = 2 + 5 = 3 + 4$. Prove that $S_7,S_8,S_9$ and $S_{10}$ are well-balanced . 3. Is the set $S_{2019}$ well-balanced? Justify your answer.

In the plane, the segments $AB$ and $CD$ are given, while the lines $AB$ and $CD$ intersect. Prove that the set of all points $P$ in the plane such that triangles $ABP$ and $CDP$ have equal areas , form two lines intersecting at the intersection of the lines $AB$ and $CD$.

Let $a, b$, and $c$ be positive real numbers. Prove that \[\frac{a^4 + 3}{b} + \frac{b^4 + 3}{c} + \frac{c^4 + 3}{a} \ge 12.\] When does equality hold? [i]Proposed by Petar Filipovski[/i]

Determine all real numbers $x$ which satisfy the inequality: \[ \sqrt{3-x}-\sqrt{x+1}>\dfrac{1}{2} \]