$\forall k\in N \,\,\, \exists n(k) \in N, a(k):0<a(k)<1 [(1+\sqrt2)^{2k+1}=n(k)+a(k)]$ Prove: $(n(k) + a(k))a(k) = 1$, for all $k \in N$, and calculate $\lim_{k \to \infty }a(k)$

Let $ABC$ be an acute angle triangle. Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$. Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$. Prove that \[ 2(PQ+QR+RP)\geq DE+EF+FD \]

Find the maximum value of $ \dfrac{x}{y} $ if $ x $ and $ y $ are real numbers such that $ x^2 + y^2 - 8x - 6y + 20 = 0 $.

Determine all integer solutions of the equation $x^n+y^n=1994$ where $n\geq 2$

Let $a, b, c, d$ be integers. Show that the product \[(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)\] is divisible by $12$.

Prove that for any three distinct positive real numbers $a, b$ and $c$: $$\frac{(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3}{(a - b)^3 + (b - c)^3 + (c - a)^3}> 8abc.$$

Consider $111$ distinct points which lie on or in the internal of a circle with radius 1.Prove that there are at least $1998$ segments formed by these points with length $\leq \sqrt{3}$

If $ \frac{m}{n}\equal{}\frac{4}{3}$ and $ \frac{r}{t}\equal{}\frac{9}{14}$, the value of $ \frac{3mr\minus{}nt}{4nt\minus{}7mr}$ is: $ \textbf{(A)}\ \minus{}5 \frac{1}{2} \qquad \textbf{(B)}\ \minus{}\frac{11}{14} \qquad \textbf{(C)}\ \minus{}1\frac{1}{4} \qquad \textbf{(D)}\ \frac{11}{14} \qquad \textbf{(E)}\ \minus{}\frac{2}{3}$

Call a positive integer [i]balanced [/i] if the number of its distinct prime factors is equal to the number of its digits in the decimal representation; for example, the number $385 = 5 \cdot 7 \cdot 11$ is balanced, while $275 = 5^2 \cdot 11$ is not. Prove that there exist only a finite number of balanced numbers.

$n\geq 4$ is an integer number. For any permutation $x_1,x_2,\cdots,x_n$ of the numbers $1,2 \cdots,n$ we write the number $$ x_1+2x_2+\cdots+nx_n $$ on the board. Compute the number of total distinct numbers written on the board.

Let $ABCD$ be a parallelogram and let $E$, $F$, $G$, and $H$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$, respectively. If $BH \cap AC = I$, $BD \cap EC = J$, $AC \cap DF = K$, and $AG \cap BD = L$, prove that the quadrilateral $IJKL$ is a parallelogram.

(a) An infinite sheet is divided into squares by two sets of parallel lines. Two players play the following game: the first player chooses a square and colours it red, the second player chooses a non-coloured square and colours it blue, the first player chooses a non-coloured square and colours it red, the second player chooses a non-coloured square and colours it blue, and so on. The goal of the first player is to colour four squares whose vertices form a square with sides parallel to the lines of the two parallel sets. The goal of the second player is to prevent him. Can the first player win? (b) What is the answer to this question if the second player is permitted to colour two squares at once? (DG Azov) PS. (a) for Juniors, (a),(b) for Seniors

A convex quadrilateral $ABCD$ is constructed out of metal rods with negligible thickness. The side lengths are $AB=BC=CD=5$ and $DA=3$. The figure is then deformed, with the angles between consecutive rods allowed to change but the rods themselves staying the same length. The resulting figure is a convex polygon for which $\angle{ABC}$ is as large as possible. What is the area of this figure? $\text{(A) }6\qquad\text{(B) }8\qquad\text{(C) }9\qquad\text{(D) }10\qquad\text{(E) }12$

In the expression $ \frac{x \plus{} 1}{x \minus{} 1}$ each $ x$ is replaced by $ \frac{x \plus{} 1}{x \minus{} 1}$. The resulting expression, evaluated for $ x \equal{} \frac{1}{2}$, equals: $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ \minus{}3\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \minus{}1\qquad \textbf{(E)}\ \text{none of these}$

What is the coefficient for $\text{O}_2$ when the following reaction $\_\text{As}_2\text{S}_3+\_\text{O}_2 \rightarrow \_\text{As}_2\text{O}_3+\_\text{SO}_2$ is correctly balanced with the smallest integer coefficients? $ \textbf{(A)} 5 \qquad\textbf{(B)} 6 \qquad\textbf{(C)} 8 \qquad\textbf{(D)} 9 \qquad $

Let $ABC$ be an acute triangle. $PQRS$ is a rectangle with $P$ on $AB$, $Q$ and $R$ on $BC$, and $S$ on $AC$ such that $PQRS$ has the largest area among all rectangles $TUVW$ with $T$ on $AB$, $U$ and $V$ on $BC$, and $W$ on $AC$. If $D$ is the point on $BC$ such that $AD\perp BC$, then $PQ$ is the harmonic mean of $\frac{AD}{DB}$ and $\frac{AD}{DC}$. What is $BC$? Note: The harmonic mean of two numbers $a$ and $b$ is the reciprocal of the arithmetic mean of the reciprocals of $a$ and $b$. [i]2017 CCA Math Bonanza Lightning Round #4.4[/i]

2025 IMO leaders are discussing $100$ problems in a meeting. For each $i = 1, 2,\ldots , 100$, each leader will talk about the $i$-th problem for $i$-th minutes. The chair can assign one leader to talk about a problem of his choice, but he would have to wait for the leader to complete the entire talk of that problem before assigning the next leader and problem. The next leader can be the same leader. The next problem can be a different problem. Each leader’s longest idle time is the longest consecutive time that he is not talking. Find the minimum positive integer $T$ so that the chair can ensure that the longest idle time for any leader does not exceed $T$. [i]Proposed by usjl[/i]

Let $AD$ be the altitude $ABC$ of an acute triangle. On the line $AD$ are chosen different points $E$ and $F$ so that $|DE |= |DF|$ and point $E$ is in the interior of triangle $ABC$. The circumcircle of triangle $BEF$ intersects $BC$ and $BA$ for second time at points $K$ and $M$ respectively. The circumcircle of the triangle $CEF$ intersects the $CB$ and $CA$ for the second time at points $L$ and $N$ respectively. Prove that the lines $AD, KM$ and $LN$ intersect at one point.

Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.

For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$. (We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.) [i](United Kingdom) Luke Betts[/i]

Consider points $D,E$ and $F$ on sides $BC,AC$ and $AB$, respectively, of a triangle $ABC$, such that $AD, BE$ and $CF$ concurr at a point $G$. The parallel through $G$ to $BC$ cuts $DF$ and $DE$ at $H$ and $I$, respectively. Show that triangles $AHG$ and $AIG$ have the same areas.

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

Let $n$ be a given positive integer. The sequence of real numbers $a_1, a_2, a_3, \cdots, a_n$ satisfy for each $m \leq n$, $$\left|\sum_{k=1}^m\frac{a_k}k\right| \leq 1.$$ Given this information, find the greatest possible value of $\left|\sum_{k=1}^n a_k\right|$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 13)[/i]

If $p$ and $p^2+2$ are prime numbers, at most how many prime divisors can $p^3+3$ have? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

We say that a positive integer N is [i]nice[/i] if it satisfies the following conditions: $\bullet$ All of its digits are $1$ or $2$ $\bullet$ All numbers formed by $3$ consecutive digits of $N$ are distinct. For example, $121222$ is nice, because the $4$ numbers formed by $3$ consecutive digits of $121222$, which are $121,212,122$ and $222$, are distinct. However, $12121$ is not nice. What is the largest quantity possible number of numbers that a nice number can have? What is the greatest nice number there is?