How many $3$-inch-by-$5$-inch photos will it take to completely cover the surface of a $3$-foot-by-$5$-foot poster? $\text{(A) }24\qquad\text{(B) }114\qquad\text{(C) }160\qquad\text{(D) }172\qquad\text{(E) }225$

Given real numbers $x,y,z,a$ satisfying: $$ x+y+z = a$$ $$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z} = \frac{1}{a} $$ Prove that at least one of the numbers $x,y,z$ is equal to $a$.

Find the sum of the fiftieth powers of all sides and diagonals of a regular $100$-gon inscribed in a circle of radius $R.$

After viewing the John Harvard statue, a group of tourists decides to estimate the distances of nearby locations on a map by drawing a circle, centered at the statue, of radius $\sqrt{n}$ inches for each integer $2020\leq n \leq 10000$, so that they draw $7981$ circles altogether. Given that, on the map, the Johnston Gate is $10$-inch line segment which is entirely contained between the smallest and the largest circles, what is the minimum number of points on this line segment which lie on one of the drawn circles? (The endpoint of a segment is considered to be on the segment.)

Let $f$ be a polynomial function with integer coefficients and $p$ be a prime number. Suppose there are at least four distinct integers satisfying $f(x) = p$. Show that $f$ does not have integer zeros.

A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$ ? [asy] draw((0,2)--(2,2)--(2,0)--(0,0)--cycle); draw((0,0.3)--(0.3,2)--(2,1.7)--(1.7,0)--cycle); label("$a$",(-0.1,0.15)); label("$b$",(-0.1,1.15)); [/asy] $\textbf{(A)}\hspace{.05in}\dfrac15 \qquad \textbf{(B)}\hspace{.05in}\dfrac25 \qquad \textbf{(C)}\hspace{.05in}\dfrac12 \qquad \textbf{(D)}\hspace{.05in}1 \qquad \textbf{(E)}\hspace{.05in}4 $

I’m given a square of side length $7,$ and I want to make a regular tetrahedron from it. Specifically, my strategy is to cut out a net. If I cut out a parallelogram-shaped net that yields the biggest regular tetrahedron, what is the surface area of the resulting tetrahedron?

Point $A$, $B$, $C$, and $D$ form a rectangle in that order. Point $X$ lies on $CD$, and segments $\overline{BX}$ and $\overline{AC}$ intersect at $P$. If the area of triangle $BCP$ is 3 and the area of triangle $PXC$ is 2, what is the area of the entire rectangle?

If $x^2\cos\theta-x(1-x)+(1-x)^2\sin\theta>0$ for all $x\in[0,1]$, find the range value of $\theta$.

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

In a group of $n$ people, each one knows exactly three others. They are seated around a table. We say that the seating is $perfect$ if everyone knows the two sitting by their sides. Show that, if there is a perfect seating $S$ for the group, then there is always another perfect seating which cannot be obtained from $S$ by rotation or reflection.

Let $(a_n)_1^{\infty}$ be a sequence such that $a_n \le a_{n+m} \le a_n + a_m$ for all positive integers $n$ and $m$. Prove that $\frac{a_n}{n}$ has a limit as $n$ approaches infinity.

Prove that there is no real function $ f(x)$ satisfying $ f\left(f(x)\right) \equal{} x^2 \minus{} 2$ for all real number $ x$.

[b]6.1[/b] The student bought a briefcase, a fountain pen and a book. If the briefcase cost 5 times cheaper, the fountain pen was 2 times cheaper, and the book was 2 1/2 times cheaper cheaper, then the entire purchase would cost 2 rubles. If the briefcase was worth 2 times cheaper, a fountain pen is 4 times cheaper, and a book is 3 times cheaper, then the whole the purchase would cost 3 rubles. How much does it really cost? ´ [b]6.2.[/b] Which number is greater: $$\underbrace{888...88}_{19 \, digits} \cdot \underbrace{333...33}_{68 \, digits} \,\,\, or \,\,\, \underbrace{444...44}_{19 \, digits} \cdot \underbrace{666...67}_{68 \, digits} \, ?$$ [b]6.3[/b] Distance between Luga and Volkhov 194 km, between Volkhov and Lodeynoye Pole 116 km, between Lodeynoye Pole and Pskov 451 km, between Pskov and Luga 141 km. What is the distance between Pskov and Volkhov? [b]6.4 [/b] There are $4$ objects in pairs of different weights. How to use a pan scale without weights Using five weighings, arrange all these objects in order of increasing weights? [b]6.5 [/b]. Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams. [b]6.6 [/b] In task 6.1, determine what is more expensive: a briefcase or a fountain pen. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].

Consider a $6 \times 6$ grid. Define a [i]diagonal[/i] to be the six squares whose coordinates $(i,j)$ ($1 \le i,j \le 6)$ satisfy $i-j \equiv k \pmod 6$ for some $k=0,1,\dots,5$. Hence there are six diagonals. Determine if it is possible to fill it with the numbers $1,2,\dots,36$ (each exactly once) such that each row, each column, and each of the six diagonals has the same sum.

Inside the parallelogram $ABCD$ points $M, N, K$ and $L{}$ are on sides $AB, BC, CD{}$ and $DA$, respectively. Let $O_1, O_2, O_3$ and $O_4$ be the circumcenters of triangles repesctively $MBN, NCK, KDL$ and $LAM{}$. Prove that the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.

Given $n$ real numbers $0 < t_1 \leq t_2 \leq \cdots \leq t_n < 1$, prove that \[(1-t_n^2) \left( \frac{t_1}{(1-t_1^2)^2}+\frac{t_2}{(1-t_2^3)^2}+\cdots +\frac{t_n}{(1-t_n^{n+1})^2} \right) < 1.\]

Let $\phi = \frac{1+\sqrt5}{2}$. Prove that a positive integer appears in the list $$\lfloor \phi \rfloor , \lfloor 2 \phi \rfloor, \lfloor 3\phi \rfloor ,... , \lfloor n\phi \rfloor , ... $$ if and only if it appears exactly twice in the list $$\lfloor 1/ \phi \rfloor , \lfloor 2/ \phi \rfloor, \lfloor 3/\phi \rfloor , ... ,\lfloor n/\phi \rfloor , ... $$

Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.

A telephone number has the form $ ABC \minus{} DEF \minus{} GHIJ$, where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is, $ A > B > C$, $ D > E > F$, and $ G > H > I > J$. Furthermore, $ D$, $ E$, and $ F$ are consecutive even digits; $ G$, $ H$, $ I$, and $ J$ are consecutive odd digits; and $ A \plus{} B \plus{} C \equal{} 9$. Find $ A$. $ \textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8$

Let $a, b, c$ be integers satisfying $ab + bc + ca = 1.$ Prove that $(1+ a^2 )(1+ b^2 )(1+ c^2 )$ is a perfect square.

Suppose that $ a_1$, $ a_2$, ..., $ a_n$ are positive real numbers such that $ a_1 \leq a_2 \leq \dots \leq a_n$. Let \[ {{a_1 \plus{} a_2 \plus{} \dots \plus{} a_n} \over n} \equal{} m; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{a_1^2 \plus{} a_2^2 \plus{} \dots \plus{} a_n^2} \over n} \equal{} 1. \] Suppose that, for some $ i$, we know $ a_i \leq m$. Prove that: \[ n \minus{} i \geq n \left(m \minus{} a_i\right)^2 \]

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]

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]