Let $ a $ and $ b $ be natural numbers with property $ gcd(a,b)=1 $ . Find the least natural number $ k $ such that for every natural number $ r \ge k $ , there exist natural numbers $ m,n >1 $ in such a way that the number $ m^a n^b $ has exactly $ r+1 $ positive divisors.

Let $p$ be a prime number. Positive integers numbers $a$ and $b$ are such $\frac{p}{a}+\frac{p}{b}=1$ and $a+b$ is divisible by $p$. What values can an expression $\frac{a+b}{p}$ take? [i]Yu.A.Karpenko[/i]

A cube-shaped container has vertices $A$, $B$, $C$, and $D$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of the faces of the cube. Vertex $A$ of the cube is set on a horizontal plane $\mathcal P$ so that the plane of the rectangle $ABCD$ is perpendicular to $\mathcal P$, vertex $B$ is $2$ meters above $\mathcal P$, vertex $C$ is $8$ meters above $\mathcal P$, and vertex $D$ is $10$ meters above $\mathcal P$. The cube contains water whose surface is $7$ meters above $\mathcal P$. The volume of the water is $\tfrac mn$ cubic meters, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(250); defaultpen(linewidth(0.6)); pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B, D = B+C, W = B+Y; pair P1 = 0.8*C+0.2*Y, P2 = 2/3*C+1/3*X, P3 = 0.2*D+0.8*Z, P4 = 0.63*D+0.37*W; pair E = (-20,6), F = (-6,-5), G = (18,-2), H = (9,8); filldraw(E--F--G--H--cycle,rgb(0.98,0.98,0.2)); fill(A--Y--P1--P4--P3--Z--B--cycle,rgb(0.35,0.7,0.9)); draw(A--B--Z--X--A--Y--C--X^^C--D--Z); draw(P1--P2--P3--P4--cycle^^D--P4); dot("$A$",A,S); dot("$B$",B,S); dot("$C$",C,N); dot("$D$",D,N); label("$\mathcal P$",(-13,4.5)); [/asy]

Let $a$ and $c$ be fixed positive numbers. For each real number $t$ let $(x_t, y_t)$ be the vertex of the parabola $y = ax^2+bx+c$. If the set of vertices $(x_t, y_t)$ for all real values of $t$ is graphed in the plane, the graph is A. a straight line B. a parabola C. part, but not all, of a parabola D. one branch of a hyperbola E. None of these

Determine all positive integers $k$ and $n$ satisfying the equation $$k^2 - 2016 = 3^n$$ (Stephan Wagner)

On the diagonals $AC$ and $BD$ of the inscribed quadrilateral A$BCD$, the points $X$ and $Y$ are marked, respectively, so that the quadrilateral $ABXY$ is a parallelogram. Prove that the circumscribed circles of triangles $BXD$ and $CYA$ have equal radii. (Vyacheslav Yasinsky)

Chitoge is painting a cube; she can paint each face either black or white, but she wants no vertex of the cube to be touching three faces of the same color. In how many ways can Chitoge paint the cube? Two paintings of a cube are considered to be the same if you can rotate one cube so that it looks like the other cube.

A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following 2 statements is false? (1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$. (2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by $L$-shaped tetrominoes.

[b]6.1[/b] The capacities of cubic vessels are in the ratio 1:8:27 and the volumes of liquid poured into them are 1: 2: 3. After this, from the first to a certain amount of liquid was poured into the second vessel, and then from the second in the third so that in all three vessels the liquid level became the same. After this, 128 4/7 liters were poured from the first vessel into the second, and from the second in the first back so much that the height of the liquid column in the first vessel became twice as large as in the second. It turned out that in the first vessel there were 100 fewer liters than at first. How much liquid was initially in each vessel? [b]6.2[/b] How many times a day do all three hands on a clock coincide, including the second hand? [b]6.3.[/b] Prove that in Leningrad there are two people who have the same number of familiar Leningraders. [b]6.4 / 7.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same. [b]6.5 / 7.6[/b] The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

In an acute-angled triangle, one of the angles is $60^o$. Prove that the line passing through the center of the circumcircle and the intersection point of the medians of the triangle cuts off an equilateral triangle from it. (A. Zaslavsky)

Construct a triangle $ABC$, given the length $c$ of its side $AB$, the radius $r$ of its inscribed circle, and the radius $r_c$ of its ex-circle tangent to the side $AB$ and the extensions of $BC$ and $CA$.

Find all positive integers $n$ such that $\phi (n)$ is a divisor of $n^2+3$.

You have a pencil, paper and an angle of $19$ degrees made out of two equal very thin sticks. Can you construct an angle of $1$ degree using only these tools?

Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $x^{2016}+y> 1-\frac{1}{100}$

Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches. What is the largest possible number of matches?

A subset $S$ of the set of integers 0 - 99 is said to have property $A$ if it is impossible to fill a crossword-puzzle with 2 rows and 2 columns with numbers in $S$ (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ with the property $A.$

Points $D$ and $E$ are taken on $[BC]$ and $[AC]$ of acute angled triangle $ABC$ such that $BD$ and $CE$ are angle bisectors. Projections of $D$ onto $BC$ and $BA$ are $P$ and $Q$, projections of $E$ onto $CA$ and $CB$ are $R$ and $S$. Let $AP \cap CQ=X$, $AS \cap BR=Y$ and $BX \cap CY=Z$. Show that $AZ \perp BC$.

Let $f(x,y)$ be a function defined for all pairs of nonnegative integers $(x, y),$ such that $f(0,k)=f(k,0)=2^k$ and \[f(a,b)+f(a+1,b+1)=f(a+1,b)+f(a,b+1)\] for all nonnegative integers $a, b.$ Determine the number of positive integers $n\leq2016$ for which there exist two nonnegative integers $a, b$ such that $f(a,b)=n$. [i]Proposed by Michael Ren[/i]

Let $a,b,c,d$ be four positive integers such that $a>b>c>d$. Given that $ab+bc+ca+d^2|(a+b)(b+c)(c+a)$. Find the minimal value of $ \Omega (ab+bc+ca+d^2)$. Here $ \Omega(n)$ denotes the number of prime factors $n$ has. e.g. $\Omega(12)=3$

Let $\theta_{i}\in(0,\frac{\pi}{2})(i=1,2,\cdots,n)$. Prove: $$(\sum_{i=1}^n\tan\theta_{i})(\sum_{i=1}^n\cot\theta_{i})\geq(\sum_{i=1}^n\sin\theta_{i})^2+(\sum_{i=1}^n\cos\theta_{i})^2.$$

Find all pairs $(n,k)$ of positive integers such that $\binom nk=2002$.

Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 15 $

The natural number $n>1$ is called “heavy”, if it is coprime with the sum of its divisors. What’s the maximal number of consecutive “heavy” numbers?

Solve in real number the system $x^3=\frac{z}{y}-\frac{2y}{z}, y^3=\frac{x}{z}-\frac{2z}{x}, z^3=\frac{y}{x}-\frac{2x}{y}$

Consider an acute triangle $ABC$ with circumcircle $\mathcal{C}$. Let $H$ be the orthocenter of $ABC$ and $M$ the midpoint of $BC$. Lines $AH$, $BH$ and $CH$ cut $\mathcal{C}$ again at points $D$, $E$, and $F$ respectively; line $MH$ cuts $\mathcal{C}$ at $J$ such that $H$ lies between $J$ and $M$. Let $K$ and $L$ be the incenters of triangles $DEJ$ and $DFJ$ respectively. Prove $KL$ is parallel to $BC$.