Find $n$ such that the mean of $\frac74$, $\frac65$, and $\frac1n$ is $1$.

A weight of $11111$ grams is placed in the left pan of a balance. Weights are added one at a time, the first weighing $1$ gram, and each subsequent one weighing twice as much as the preceding one. Each weight may be added to either pan. After a while, equilibrium is achieved. Is the $16$ gram weight placed in the left pan or the right pan? ( AV Kalinin)

Prove that if $ a,b,c$ are positive real numbers, then: $ \frac{9}{a\plus{}b\plus{}c} \le 2 \left( \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \right) \le \frac{1}{a}\plus{}\frac{1}{b}\plus{}\frac{1}{c}.$

Let $n>2$ be an even integer. In the squares of a board of $n \times n$, pieces must be placed so that in each column the number of chips is even and different from zero, and in each row the number of chips is odd. Determine the fewest number of checkers to place on the board to satisfy this rule. To show a configuration with that number of tokens and explain why with fewer tokens the rule.

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Let $ n\ge 2 $ be a natural number and $ A $ be a subset of $ \{1,2,\ldots ,n\} $ having the property that $ x+y $ belongs to $ A $ for any choosing of $ x,y $ such that $ x+y\le n. $ Prove that the arithmetic mean of the elements of $ A $ is at least $ \frac{n+1}{2} . $

Construct triangle given all lenght of it altitudes. Please, do it elementary with Euclidian geometry (no trigonometry or coordinate geometry).

Let $n\in \mathbb{N},\ n\ge 2$. For any matrix $A\in \mathcal{M}_n(\mathbb{C})$, let $m(A)$ be the number of non-zero minors of $A$. Prove that: a)$m(I_n)=2^n-1$; b)If $A\in \mathcal{M}_n(\mathbb{C})$ is non-singular, then $m(A)\ge 2^n-1$. [i]Marius Ghergu[/i]

Ann and Beto play with a two pan balance scale. They have $2023$ dumbbells labeled with their weights, which are the numbers $1, 2, \dots, 2023$, with none of them repeating themselves. Each player, in turn, chooses a dumbbell that was not yet placed on the balance scale and places it on the pan with the least weight at the moment. If the scale is balanced, the player places it on any pan. Ana starts the game, and they continue in this way alternately until all the dumbbells are placed. Ana wins if at the end the scale is balanced, otherwise Beto win. Determine which of the players has a winning strategy and describe the strategy.

The sum of all but one of the interior angles of a convex polygon equals $2570^\circ$. The remaining angle is $\textbf{(A)} \ 90^\circ \qquad \textbf{(B)} \ 105^\circ \qquad \textbf{(C)} \ 120^\circ \qquad \textbf{(D)} \ 130^\circ \qquad \textbf{(E)} \ 144^\circ$

Let $ABC$ be a triangle and points $P,Q,R$ be on the sides $AB,BC,AC$, respectively. Now, let $A',B',C'$ be on the segments $PR,QP,RQ$ in a way that $AB||A'B'$ , $BC||B'C'$ and $AC||A'C'$. Prove that: \[ \frac{AB}{A'B'}=\frac{S_{PQR}}{S_{A'B'C'}}. \] Where $S_{XYZ}$ is the surface of the triangle $XYZ$.

Call a quadruple of positive integers $(a, b, c, d)$ fruitful if there are infinitely many integers $m$ such that $\text{gcd} (am + b, cm + d) = 2019$. Find all possible values of $|ad-bc|$ over fruitful quadruples $(a, b, c, d)$.

In each one of the given $2019$ boxes, there are $2019$ stones numbered as $1,2,...,2019$ with total mass of $1$ kilogram. In all situations satisfying these conditions, if one can pick stones from different boxes with different numbers, with total mass of at least 1 kilogram, in $k$ different ways, what is the maximal of $k$?

There are 2017 points in a plane. For each pair of these points we mark the middle of the segment they form when connected. What’s the least number of marked points?

The median $CM$ and the altitude $AH$ of an acute-angled triangle $ABC$ meet at point $O$. A point $D$ lies outside the triangle in such a way that $AOCD$ is a parallelogram. Find the length of $BD$, if $MO= a$, $OC = b$.

There are $ 63$ points arbitrarily on the circle $ \mathcal{C}$ with its diameter being $ 20$. Let $ S$ denote the number of triangles whose vertices are three of the $ 63$ points and the length of its sides is no less than $ 9$. Fine the maximum of $ S$.

Prove that if $A{}$ is an $n\times n$ matrix with complex entries such that $A+A^*=A^2A^*$ then $A=A^*$. (Here, we denote by $M^*$ the conjugate transpose $\overline{M}^t$ of the matrix $M{}$).

Wendy takes Honors Biology at school, a smallish class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarn would intersect if possible. If all possible positions of the students around the table are equally likely, let $m/n$ be the probability that the yarns intersect, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Triangle $ \triangle ABC$ is given. Points $ D$ i $ E$ are on line $ AB$ such that $ D \minus{} A \minus{} B \minus{} E, AD \equal{} AC$ and $ BE \equal{} BC$. Bisector of internal angles at $ A$ and $ B$ intersect $ BC,AC$ at $ P$ and $ Q$, and circumcircle of $ ABC$ at $ M$ and $ N$. Line which connects $ A$ with center of circumcircle of $ BME$ and line which connects $ B$ and center of circumcircle of $ AND$ intersect at $ X$. Prove that $ CX \perp PQ$.

Let m and n be positive integers. Alice wishes to walk from the point $(0, 0)$ to the point $(m,n)$ in increments of $(1, 0)$ and $(0, 1)$, and Bob wishes to walk from the point $(0,1)$ to the point $(m, n + 1)$ in increments of$ (1, 0)$ and $(0,1)$. Find (with proof) the number of ways for Alice and Bob to get to their destinations if their paths never pass through the same point (even at different times).

Let $a$ be a positive integer such that $5^{1994}-1\mid a$. Prove that the expression of $a$ in base $5$ contains at least $1994$ nonzero digits.

Consider the ellipsoid$$\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{b^2}=1$$($a$ and $b > 0$) and the ellipse $E$ which is the intersection of the ellipsoid with the plane of equation$$mx + ny + pz = 0$$where the point $P = [m, n, p]$ is a random point from the unit sphere $(m^2 + n^2 + p^2 = 1)$. Consider the random variable $A_E$ the area of the ellipse $E$. If the point $P$ is chosen with uniform distribution with respect to the area on the unit sphere, what is the expectation of $A_E$ ?

Find the largest positive integer $n$ such that the following equations have integer solutions in $x, y_{1}, y_{2}, ... , y_{n}$ : $(x+1)^{2}+y_{1}^{2}= (x+2)^{2}+y_{2}^{2}= ... = (x+n)^{2}+y_{n}^{2}.$

The set of values of $m$ for which $mx^2-6mx+5m+1>0$ for all real $x$ is $\textbf{(A)}~m<\frac14$ $\textbf{(B)}~m\ge0$ $\textbf{(C)}~0\le m\le\frac14$ $\textbf{(D)}~0\le m<\frac14$

Ann made a 3-step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 5-step staircase? $ \textbf{(A) }9\qquad\textbf{(B) }18\qquad\textbf{(C) }20\qquad\textbf{(D) }22\qquad\textbf{(E) }24 $ [asy] size(150); defaultpen(linewidth(0.8)); path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45); for(int i=0;i<=2;i=i+1) { for(int j=0;j<=3-i;j=j+1) { filldraw(shift((i,j))*h,black); filldraw(shift((j,i))*v,black); } }[/asy]