Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

One day, Rabbit was about to go for a meeting with Donkey, but Winnie the Pooh and Duck unexpectedly came to his home. Being well-bred, Rabbit offered the guests some refreshments. Pooh tied Duck’s mouth by a napkin and ate $10$ pots of honey and $22$ cups of condensed milk alone, whereby he needed two minutes for each pot of honey and $1$ minute for each cup of milk. Knowing that there was nothing sweet left in the house, Pooh released the Duck. Afflicted Rabbit observed that he wouldn’t have been late for the meeting with Donkey if Pooh had shared the refreshments with Duck. Knowing that Duck needs $5$ minutes for a pot of honey and $3$ minutes for a cup of milk, he computed the time the guests would have needed to devastate his supplies. What is that time?

Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition : 1) $f(-1)\geq f(1).$ 2) $x+f(x)$ is non decreasing function. 3) $\int_{-1}^ 1 f(x)\ dx=0.$ Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

We can obviously put 100 unit balls in a $10\times10\times1$ box. How can one put $105$ unit balls in? How can we put $106$ unit balls in?

If an integer $n$ is such that $7n$ is the form $a^2 +3b^2$, prove that $n$ is also of that form.

$6$ cats can eat $6$ fish in $1$ day, and $c$ cats can eat $91$ fish in $d$ days. Given that $c$ and $d$ are both whole numbers, and the number of cats, $c$, is more than $1$ but less than $10$, find $c + d$.

Without using a calculator, find out which number is greater: $$29^{200}\cdot 2^{151} \,\,\, or \,\,\, 5^{279} \cdot 3^{300}$$

Brennan chooses a set $A = \{a, b,c, d, e \}$ of five real numbers with $a \leq b \leq c \leq d \leq e.$ Delaney determines the subsets of $A$ containing three numbers and adds up the numbers in these subsets. She obtains the sums $0, 3; 4, 8; 9, 10, 11, 12, 14, 19.$ What are the five numbers in Brennan's set?

Let $OX, OY$ and $OZ$ be three rays in the space, and $G$ a point "[i]between these rays[/i]" (i. e. in the interior of the part of the space bordered by the angles $Y OZ, ZOX$ and $XOY$). Consider a plane passing through $G$ and meeting the rays $OX, OY$ and $OZ$ in the points $A, B, C$, respectively. There are infinitely many such planes; construct the one which minimizes the volume of the tetrahedron $OABC$.

A rod moves freely between the horizontal floor and the slanted wall. When the end in contact with the floor is moving at v, what is the speed of the end in contact with the wall? $\textbf{(A)} v\frac{\sin{\theta}}{\cos(\alpha-\theta)}$ $\textbf{(B)}v\frac{\sin(\alpha - \theta)}{\cos(\alpha + \theta)} $ $\textbf{(C)}v\frac{\cos(\alpha - \theta)}{\sin(\alpha + \theta)}$ $\textbf{(D)}v\frac{\cos(\theta)}{\cos(\alpha - \theta)}$ $\textbf{(E)}v\frac{\sin(\theta)}{\cos(\alpha + \theta)}$

An [i]exterior [/i] angle is the supplementary angle to an interior angle in a polygon. What is the sum of the exterior angles of a triangle and dodecagon ($12$-gon), in degrees?

Consider the parallelogram $ABCD$. A circle is drawn that passes through $A$ and intersects side $AD$ at $N$, side $AB$ at $M$ and diagonal $AC$ in $P$ such that points $A, M, N, P$ are different. Prove that $$AP\cdot AC = AM \cdot AB + AN \cdot AD.$$

Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose $P,Q,R$ are points on segments $AC, AB, BC$ respectively such that $AP = QB$, $\angle PBC = 90^\circ - \angle BAC$ and $RP = RQ$. Let $O_1, O_2$ be the circumcenters of $\triangle APQ$ and $\triangle CRP$. Prove that $BR = O_1O_2$. [i]Proposed by Atul Shatavart Nadig[/i]

Do there exist two polynomials $P$ and $Q$ with integer coefficient such that i) both $P$ and $Q$ have a coefficient with absolute value bigger than $2021$, ii) all coefficients of $P \cdot Q$ by absolute value are at most $1$.

$\frac{2}{10}+\frac{4}{100}+\frac{6}{1000} =$ $\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$

A real number $\alpha \geq 0$ is given. Find the smallest $\lambda = \lambda (\alpha ) > 0$, such that for any complex numbers ${z_1},{z_2}$ and $0 \leq x \leq 1$, if $\left| {{z_1}} \right| \leq \alpha \left| {{z_1} - {z_2}} \right|$, then $\left| {{z_1} - x{z_2}} \right| \leq \lambda \left| {{z_1} - {z_2}} \right|$.

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

$x, y, z$ are positive reals which their product is not smaller then their sum. Prove the inequality: $$\sqrt{2x^2+yz}+\sqrt{2y^2+zx}+\sqrt{2z^2+xy} \geq 9$$

Let $C$ be a closed convex curve with a continuously turning tangent and let $O$ be a point inside $C.$ For each point $P$ on $C$ we define $T(P)$ as follows: Draw the tangent to $C$ at $P$ and from $O$ drop the perpendicular to that tangent. Then $T(P)$ is the point at which $C$ intersects this perpendicular. Starting now with a point $P_{0}$ on $C$, define points $P_n$ by $P_n =T(P_{n-1}).$ Prove that the points $P_{n}$ approach a limit and characterize all possible limit points. (You may assume that $T$ is continuous.)

A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB = CD = EF$ and diagonals $AD$, $BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. Prove that $CP/PE = (AC/CE)^2$.

Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying: i) $f(1)=f(2)=1$; ii)$f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$. For each integer $m\ge 2$, find the value of $f(2^m)$.

In triangle $ABC$ we have $\angle ABC = 2 \angle ACB$. Prove that a) $AC^2 = AB^2 + AB \cdot BC$; b) $AB+BC < 2 \cdot AC$.

A given convex quadrilateral $ABCD$ is such that $\angle ABD + \angle ACD > \angle BAC + \angle BDC.$ Prove that \[S_{ABD}+S_{ACD} > S_{BAC}+S_{BDC}.\]

$.4+.02+.006=$ $\text{(A)}\ .012 \qquad \text{(B)}\ .066 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .24 \qquad \text{(E)} .426$

Let $k$ be a circle with diameter $AB$ and centre $O$. Let C be an arbitrary point on the circle different from $A$ and $B$. Let $D$ be the point for which $O$, $B$, $D$ and $C$ (in this order) are the four vertices of a parallelogram. Let $E$ be the intersection of the line $BD$ and the circle $k$, and let $F$ be the orthocenter of the triangle $OAC$. Prove that the points $O, D, E, C, F$ lie on a circle.