Point $A$ is located at $(0,0)$. Point $B$ is located at $(2,3)$ and is the midpoint of $AC$. Point $D$ is located at $(10,0)$. What are the coordinates of the midpoint of segment $CD$?

Let $f(x)=x^3+7x^2+9x+10$. Which value of $p$ satisfies the statement \[ f(a) \equiv f(b) \ (\text{mod } p) \Rightarrow a \equiv b \ (\text{mod } p) \] for every integer $a,b$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 17 $

Jeff has a deck of $12$ cards: $4$ $L$s, $4$ $M$s, and $4$ $T$s. Armaan randomly draws three cards without replacement. The probability that he takes $3$ $L$s can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m +n$.

The bisectors $AA_1, CC_1$ of a triangle $ABC$ with $\angle B = 60^{\circ}$ meet at point $I$. The circumcircles of triangles $ABC, A_1IC_1$ meet at point $P$. Prove that the line $PI$ bisects the side $AC$.

Let $O$ be a point in the plane and let $F$ be a (not necessary convex) polygon. Let $P$ be the perimeter of $F$, let $D$ be sum of the distances of the point $O$ from the vertices of $F$, and let $H$ be sum of the distances of the point $O$ from the lines that pass through the vertices of $F$. Show that \[D^2-H^2 \geq \frac{P^2}{4}.\]

In an acute triangle $\triangle{ABC}$, let $AE$ and $BF$ be highs of it, and $H$ its orthocenter. The symmetric line of $AE$ with respect to the angle bisector of $\sphericalangle{A}$ and the symmetric line of $BF$ with respect to the angle bisector of $\sphericalangle{B}$ intersect each other on the point $O$. The lines $AE$ and $AO$ intersect again the circuncircle to $\triangle{ABC}$ on the points $M$ and $N$ respectively. Let $P$ be the intersection of $BC$ with $HN$; $R$ the intersection of $BC$ with $OM$; and $S$ the intersection of $HR$ with $OP$. Show that $AHSO$ is a paralelogram.

Today was the 5th Kettering Olympiad - and here are the problems, which are very good intermediate problems. 1. Find all real $x$ so that $(1+x^2)(1+x^4)=4x^3$ 2. Mark and John play a game. They have $100$ pebbles on a table. They take turns taking at least one at at most eight pebbles away. The person to claim the last pebble wins. Mark goes first. Can you find a way for Mark to always win? What about John? 3. Prove that $\sin x + \sin 3x + \sin 5x + ... + \sin 11 x = (1-\cos 12 x)/(2 \sin x)$ 4. Mark has $7$ pieces of paper. He takes some of them and splits each into $7$ pieces of paper. He repeats this process some number of times. He then tells John he has $2000$ pieces of paper. John tells him he is wrong. Why is John right? 5. In a triangle $ABC$, the altitude, angle bisector, and median split angle $A$ into four equal angles. Find the angles of $ABC.$ 6. There are $100$ cities. There exist airlines connecting pairs of cities. a) Find the minimal number of airlines such that with at most $k$ plane changes, one can go from any city to any other city. b) Given that there are $4852$ airlines, show that, given any schematic, one can go from any city to any other city.

There are $2007$ boys and $2007$ girls in a middle school,every student can attend no more than $100$ academic meetings,if we know any pair of students with different sex attend at least one common meeting.prove that there must be a meeting with at least $11$ boys and $11$ girls attend.

Prove that: the polynomial$$(x(x+1)(x+2)(x+3))^{2^{2023}}+1$$is irreducible in $\mathbb{Q}[x].$

[b]p1.[/b] Archimedes, Euclid, Fermat, and Gauss had a math competition. Archimedes said, “I did not finish $1$st or $4$th.” Euclid said, “I did not finish $4$th.” Fermat said, “I finished 1st.” Gauss said, “I finished $4$th.” There were no ties in the competition, and exactly three of the mathematicians told the truth. Who finished first and who finished last? Justify your answers. [b]p2.[/b] Find the area of the set in the xy-plane defined by $x^2 - 2|x| + y^2 \le 0$. Justify your answer. [b]p3.[/b] There is a collection of $2004$ circular discs (not necessarily of the same radius) in the plane. The total area covered by the discs is $1$ square meter. Show that there is a subcollection $S$ of discs such that the discs in S are non-overlapping and the total area of the discs in $S$ is at least $1/9$ square meter. [b]p4.[/b] Let $S$ be the set of all $2004$-digit integers (in base $10$) all of whose digits lie in the set $\{1, 2, 3, 4\}$. (For example, $12341234...1234$ is in $S$.) Let $n_0$ be the number of $s \in S$ such that $s$ is a multiple of $3$, let $n_1$ be the number of $s \in S$ such that $s$ is one more than a multiple of $3$, and let $n_2$ be the number of $s \in S$ such that $s$ is two more than a multiple of $3$. Determine which of $n_0$, $n_1$, $n_2$ is largest and which is smallest (and if there are any equalities). Justify your answers. [b]p5.[/b] There are $6$ members on the Math Competition Committee. The problems are kept in a safe. There are $\ell$ locks on the safe and there are $k$ keys, several for each lock. The safe does not open unless all of the locks are unlocked, and each key works on exactly one lock. The keys should be distributed to the $6$ members of the committee so that each group of $4$ members has enough keys to open all of the $\ell$ locks. However, no group of $3$ members should be able to open all of the $\ell$ locks. (a) Show that this is possible with $\ell = 20$ locks and $k = 60$ keys. That is, it is possible to use $20$ locks and to choose and distribute 60 keys in such a way that every group of $4$ can open the safe, but no group of $3$ can open the safe. (b) Show that we always must have $\ell \ge 20$ and $k\ge60$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

From an initial triangle $\Delta A_0B_0C_0$, a sequence of triangles $\Delta A_1B_1C_1$, $A_2B_2C_2$, ... is formed such that, at each stage, $A_{k + 1}$, $B_{k + 1}$ and $C_{k + 1}$ are the points where the incircle of $\Delta A_kB_kC_k$ touches the sides $B_kC_k$, $C_kA_k$ and $A_kB_k$ respectively. (a) Express $\angle A_{k + 1}B_{k + 1}C_{k + 1}$ in terms of $\angle A_kB_kC_k$. (b) Deduce that, as $k$ increases, $\angle A_kB_kC_k$ tends to $60^{\circ}$.

The angle bisector of the acute angle formed at the origin by the graphs of the lines $y=x$ and $y=3x$ has equation $y=kx$. What is $k$? $\textbf{(A)} \: \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)} \: \frac{1+\sqrt{7}}{2} \qquad \textbf{(C)} \: \frac{2+\sqrt{3}}{2} \qquad \textbf{(D)} \: 2\qquad \textbf{(E)} \: \frac{2+\sqrt{5}}{2}$

A sphere is the set of points at a fixed positive distance $r$ from its center. Let $\mathcal{S}$ be a set of $2010$-dimensional spheres. Suppose that the number of points lying on every element of $\mathcal{S}$ is a finite number $n$. Find the maximal possible value of $n$.

Prove that the polynomial $x^{200} y^{200} +1$ cannot be represented in the form $f(x)g(y)$, where $f$ and $g$ are polynomials of only $x$ and $y$, respectively.

Let $x,y$ be positive numbers, $s$ -- the least of $$\{ x, (y+ 1/x), 1/y\}$$ What is the greatest possible value of $s$? To what $x$ and $y$ does it correspond?

Give a prime number $p>2023$. Let $r(x)$ be the remainder of $x$ modulo $p$. Let $p_1<p_2< \ldots <p_m$ be all prime numbers less that $\sqrt[4]{\frac{1}{2}p}$. Let $q_1, q_2, \ldots, q_n$ be the inverses modulo $p$ of $p_1, p_2, \ldots p_n$. Prove that for every integers $0 < a,b < p$, the sets $$\{r(q_1), r(q_2), \ldots, r(q_m)\}, ~~ \{r(aq_1+b), r(aq_2+b), \ldots, r(aq_m+b)\}$$ have at most $3$ common elements.

Let $H$ be the orthocentre of an acute triangle $ABC$ with $BC > AC$, inscribed in a circle $\Gamma$. The circle with centre $C$ and radius $CB$ intersects $\Gamma$ at the point $D$, which is on the arc $AB$ not containing $C$. The circle with centre $C$ and radius $CA$ intersects the segment $CD$ at the point $K$. The line parallel to $BD$ through $K$, intersects $AB$ at point $L$. If $M$ is the midpoint of $AB$ and $N$ is the foot of the perpendicular from $H$ to $CL$, prove that the line $MN$ bisects the segment $CH$.

Let $A_1,A_2,...,A_n$ be $n$ finite subsets of a set $X, n \ge 2$, such that (i) $|A_i| \ge 2, 1 \le i \le n$, (ii) $ |A_i \cap A_j | \ne 1, j \le i < j \le n$. Prove that the elements of $A_1 \cup A_2 \cup ... \cup A_n$ may be colored with $2$ colors so that no $A_i$ is colored by the same color.

A $100 \times 100 \times 100$ cube is divided into a million unit cubes and in each small cube there is a light bulb. Three faces $100 \times 100$ of the large cube having a common vertex are painted: one in red, one in blue and the other in green. Call a $\textit{column}$ a set of $100$ cubes forming a block $1 \times 1 \times 100$. Each of the $30 000$ columns have one painted end cell, on which there is a switch. After pressing a switch, the states of all light bulbs of this column are changed. Petya pressed several switches, getting a situation with exactly $k$ lamps on. Prove that Vasya can press several switches so that all lamps are off, but by using no more than $\frac {k} {100}$ switches on the red face.

There is a square of checkered paper measuring $102 \times 102$ squares and a connected figure of unknown shape, consisting of 101 cells. What is the largest number of such figures that can be cut from this square with a guarantee? A figure made up of cells is called [i]connected [/i] if any two its cells can be connected by a chain of its cells in which any two adjacent cells have a common side.

Let the operation * be defined by $ a * b \equal{} ab \plus{} a \minus{} b$. Which of the expression below is wrong, or are they all correct? A. $ a * a \equal{} a^2$ B. $ a * b \equal{} (\minus{}b) * (\minus{}a)$ C. $ (a * b) * a \equal{} a^2 * b$ D. $ (a * b) * c \equal{} (a * c) * b$ E. All four are correct

Find all prime numbers $p$ and $q$ such that $2^2+p^2+q^2$ is also prime. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

Find all the pairs $(n, m)$ of positive integers which fulfil simultaneously the conditions: i) the number $n$ is composite; ii) if the numbers $d_1, d_2, ..., d_k, k \in N^*$ are all the proper divisors of $n$, then the numbers $d_1 + 1, d_2 + 1, . . . , d_k + 1$ are all the proper divisors of $m$.

A die is an unitary cube with numbers from $1$ to $6$ written on its faces, so that each number appears once and the sum of the numbers on any two opposite faces is $7$. We construct a large $3 \cdot 3 \cdot 3$ cube using$ 27$ dice. Find all possible values of the sum of numbers which can be seen on the faces of the large cube.

Let $x$ satisfy $x^4+x^3+x^2+x+1=0$. Compute the value of $(5x+x^2)(5x^2+x^4)(5x^3+x^6)(5x^4+x^8)$. [i]Proposed by Andrew Zhao[/i]