Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.

Find $$\lim_{n\to \infty} \frac{1}{n^5 } \sum_{h=1}^{n} \sum_{k=1}^{n} (5h^4 -18h^2 k^2 +5k^4).$$

Let $P(x)$ be a polynomial with real coefficients. Let $\alpha_1 , \dots , \alpha_k$ be the distinct real roots of $P(x)=0$. If $P'$ is the derivative of $P$, show that for each $i=1,\dots , k$ \[\lim_{x\to \alpha_i} \frac{(x-\alpha_i)P'(x)}{P(x)} = r_i, \] for some positive integer $r_i$.

Equilateral triangle $ABC$ is inscribed in a circle. Chord $AD$ meets $BC$ at $E$. If $DE = 2013$, how many scenarios exist such that both $DB$ and $DC$ are integers (two scenarios are different if $AB$ is different or $AD$ is different)?

Bart wrote the digit "$1$" $2024$ times in a row. Then, Lisa wrote an additional $2024$ digits to the right of the digits Bart wrote, such that the resulting number is a square of an integer. Find all possibilities for the digits Lisa wrote.

A sparrow, flying horizontally in a straight line, is $50$ feet directly below an eagle and $100$ feet directly above a hawk. Both hawk and eagle fly directly toward the sparrow, reaching it simultaneously. The hawk flies twice as fast as the sparrow. How far does each bird fly? At what rate does the eagle fly?

Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition: \[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]

In space are given $n$ segments $A_iB_i$ and a point $O$ not lying on any segment, such that the sum of the angles $A_iOB_i$ is less than $180^o$ . Prove that there exists a plane passing through $O$ and not intersecting any of the segments.

Given an acute triangle whose circumcenter is $O$.let $K$ be a point on $BC$,different from its midpoint.$D$ is on the extension of segment $AK,BD$ and $AC$,$CD$and$AB$intersect at $N,M$ respectively.prove that $A,B,D,C$ are concyclic.

Let $ f(x) \equal{} x^3 \plus{} x \plus{} 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) \equal{} \minus{} 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$.

[b]p1.[/b] One day, Granny Smith bought a certain number of apples at Horock’s Farm Market. When she returned the next day she found that the price of the apples was reduced by $20\%$. She could therefore buy more apples while spending the same amount as the previous day. How many percent more? [b]p2.[/b] You are asked to move several boxes. You know nothing about the boxes except that each box weighs no more than $10$ tons and their total weight is $100$ tons. You can rent several trucks, each of which can carry no more than $30$ tons. What is the minimal number of trucks you can rent and be sure you will be able to carry all the boxes at once? [b]p3.[/b] The five numbers $1, 2, 3, 4, 5$ are written on a piece of paper. You can select two numbers and increase them by $1$. Then you can again select two numbers and increase those by $1$. You can repeat this operation as many times as you wish. Is it possible to make all numbers equal? [b]p4.[/b] There are $15$ people in the room. Some of them are friends with others. Prove that there is a person who has an even number of friends in the room. [u]Bonus Problem [/u] [b]p5.[/b] Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ A $ be a nonempty set of real numbers, and let be two functions $ f,g:A\longrightarrow A $ having the following properties: $ \text{(i)} f $ is increasing $ \text{(ii)} f-g $ is nonpositive everywhere $ \text{(iii)} f(A)\subset g(A) $ [b]a)[/b] Prove that $ f=g $ if $ A $ is the set of all nonnegative integers. [b]b)[/b] Is true that $ f=g $ if $ A $ is the set of all integers? [i]Dorel Miheț[/i]

Determine all triplets of real numbers $(x, y, z)$ that satisfy the equation $4xyz = x^4 + y^4 + z^4 + 1$.

Let $SABC$ be the pyramid where$ m(\angle ASB) = m(\angle BSC) = m(\angle CSA) = 90^o$. Show that, whatever the point $M \in AS$ is and whatever the point $N \in BC$ is, holds the relation $$\frac{1}{MN^2} \le \frac{1}{SB^2} + \frac{1}{SC^2}.$$

Inside a convex $100$-gon are selected $k$ points, $2 \leq k \leq 50$. Show that one can choose $2k$ vertices of the $100$-gon so that the convex $2k$-gon determined by these vertices contains all the selected points.

A block wall $100$ feet long and $7$ feet high will be constructed using blocks that are $1$ foot high and either $2$ feet long or $1$ foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall? [asy] draw((0,0)--(6,0)--(6,1)--(5,1)--(5,2)--(0,2)--cycle); draw((0,1)--(5,1)); draw((1,1)--(1,2)); draw((3,1)--(3,2)); draw((2,0)--(2,1)); draw((4,0)--(4,1)); [/asy] $\text{(A)}\ 344 \qquad \text{(B)}\ 347 \qquad \text{(C)}\ 350 \qquad \text{(D)}\ 353 \qquad \text{(E)}\ 356$

All positive divisors of a positive integer $N$ are written on a blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In the firt move, the player $A$ erases $N$. If the last erased number is $d$, then the next player erases either a divisor of $d$ or a multiple of $d$. The player who cannot make a move loses. Determine all numbers $N$ for which $A$ can win independently of the moves of $B$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 2)[/i]

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

G is a complete geometric graph such that for any 4-coloring of its edges, we can find n edges which are pairwise disjoint and have the same color. Prove that the minimum number of vertices of G is 6n-4. [hide=idea]a graph with 6n-4 vertices has 2n-1 pairwise disjoint edges with 1 of 2 colors. by PP, there exist n pairwise disjoint edges of the same color. [/hide]

Consider the square $ABCD$ and the point $E$ on the side $AB$. The line $DE$ intersects the line $BC$ at point $F$, and the line $CE$ intersects the line $AF$ at point $G$. Prove that the lines $BG$ and $DF$ are perpendicular.

Let $\omega$ is tangent to the sides of an acute angle with vertex $A$ at points $B$ and $C$. Let $D$ be an arbitrary point onn the major arc $BC$ of the circle $\omega$. Points $E$ and $F$ are chosen inside the angle $DAC$ so that quadrilaterals $ABDF$ and $ACED$ are inscribed and the points $A,E,F$ lie on the same straight line. Prove that lines $BE$ and $CF$ intersectat $\omega$.

There was a rook at some square of a $10 \times 10{}$ chessboard. At each turn it moved to a square adjacent by side. It visited each square exactly once. Prove that for each main diagonal (the diagonal between the corners of the board) the following statement is true: in the rook’s path there were two consecutive steps at which the rook first stepped away from the diagonal and then returned back to the diagonal. [i]Alexandr Gribalko[/i]

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that $f^{k}((n,m))=(m,n)$,where $f^{k}(x)=f(f(f(...f(x))))$,$f$ being composed with itself $k$ times.

Given a finite set of polygons in the plane. Every two of them have a common point. Prove that there exists a straight line, that crosses all the polygons.

$a,b,c$ are positive real numbers such that $a^2+b^2=c^2$ and $ab=c$. Determine the value of $\left\lvert{\frac{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}{c^2}}\right\rvert$