Found problems: 85335
2012 Silk Road, 2
In each cell of the table $4 \times 4$, in which the lines are labeled with numbers $1,2,3,4$, and columns with letters $a,b,c,d$, one number is written: $0$ or $1$ . Such a table is called [i]valid [/i] if there are exactly two units in each of its rows and in each column. Determine the number of [i]valid [/i] tables.
1976 IMO Longlists, 23
Prove that in a Euclidean plane there are infinitely many concentric circles $C$ such that all triangles inscribed in $C$ have at least one irrational side.
2011 Harvard-MIT Mathematics Tournament, 5
Let $f(x) = x^2 + 6x + c$ for all real number s$x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots?
2013 Stanford Mathematics Tournament, 15
Suppose we climb a mountain that is a cone with radius $100$ and height $4$. We start at the bottom of the mountain (on the perimeter of the base of the cone), and our destination is the opposite side of the mountain, halfway up (height $z = 2$). Our climbing speed starts at $v_0=2$ but gets slower at a rate inversely proportional to the distance to the mountain top (so at height $z$ the speed $v$ is $(h-z)v_0/h$). Find the minimum time needed to get to the destination.
2017 India IMO Training Camp, 1
Let $a,b,c$ be distinct positive real numbers with $abc=1$. Prove that $$\sum_{\text{cyc}} \frac{a^6}{(a-b)(a-c)}>15.$$
2017 Miklós Schweitzer, 1
Can one divide a square into finitely many triangles such that no two triangles share a side? (The triangles have pairwise disjoint interiors and their union is the square.)
2007 Hanoi Open Mathematics Competitions, 4
List the numbers$\sqrt{2}; \sqrt[3]{3}; \sqrt[4]{4}; \sqrt[5]{5}; \sqrt[6]{6}.$ in order from greatest to least.
2024 Korea National Olympiad, 7
In an acute triangle $ABC$, let a line $\ell$ pass through the orthocenter and not through point $A$. The line $\ell$ intersects line $BC$ at $P(\neq B, C)$. A line passing through $A$ and perpendicular to $\ell$ meets the circumcircle of triangle $ABC$ at $R(\neq A)$. Let the feet of the perpendiculars from $A, B$ to $\ell$ be $A', B'$, respectively. Define line $\ell_1$ as the line passing through $A'$ and perpendicular to $BC$, and line $\ell_2$ as the line passing through $B'$ and perpendicular to $CA$. Prove that if $Q$ is the reflection of the intersection of $\ell_1$ and $\ell_2$ across $\ell$, then $\angle PQR = 90^{\circ}$.
2009 Baltic Way, 8
Determine all positive integers $n$ for which there exists a partition of the set
\[\{n,n+1,n+2,\ldots ,n+8\}\]
into two subsets such that the product of all elements of the first subset is equal to the product of all elements of the second subset.
1974 All Soviet Union Mathematical Olympiad, 203
Given a function $f(x)$ on the segment $0\le x\le 1$. For all $x, f(x)\ge 0, f(1)=1$. For all the couples of $(x_1,x_2)$ such, that all the arguments are in the segment $$f(x_1+x_2)\ge f(x_1)+f(x_2).$$
a) Prove that for all $x$ holds $f(x) \le 2x$.
b) Is the inequality $f(x) \le 1.9x$ valid?
2013 Princeton University Math Competition, 13
The equation $x^5-2x^4-1=0$ has five complex roots $r_1,r_2,r_3,r_4,r_5$. Find the value of \[\dfrac1{r_1^8}+\dfrac1{r_2^8}+\dfrac1{r_3^8}+\dfrac1{r_4^8}+\dfrac1{r_5^8}.\]
2019 China Team Selection Test, 2
Let $S$ be a set of positive integers, such that $n \in S$ if and only if $$\sum_{d|n,d<n,d \in S} d \le n$$
Find all positive integers $n=2^k \cdot p$ where $k$ is a non-negative integer and $p$ is an odd prime, such that $$\sum_{d|n,d<n,d \in S} d = n$$
2018 Junior Balkan Team Selection Tests - Moldova, 1
$a_1,a_2,...a_{2018}$ are positive numbers,and $a_{2018}^2+a_{2017}^2=a_{2016}^2-a_{2015}^2+a_{2014}^2-...+a_{2}^2-a_{1}^2.$ Prove that $A=a_1a_2...a_{2018}+2025$ is a difference of two squares
1998 Greece National Olympiad, 2
For a regular $n$-gon, let $M$ be the set of the lengths of the segments joining its vertices. Show that the sum of the squares of the elements of $M$ is greater than twice the area of the polygon.
2010 National Olympiad First Round, 12
How many integer quadruples $a,b,c,d$ are there such that $7$ divides $ab-cd$ where $0\leq a,b,c,d < 7$?
$ \textbf{(A)}\ 412
\qquad\textbf{(B)}\ 385
\qquad\textbf{(C)}\ 294
\qquad\textbf{(D)}\ 252
\qquad\textbf{(E)}\ \text{None}
$
2016 Iran Team Selection Test, 3
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
2021 Saudi Arabia BMO TST, 4
In the popular game of Minesweeper, some fields of an $a \times b$ board are marked with a mine and on all the remaining fields the number of adjacent fields that contain a mine is recorded. Two fields are considered adjacent if they share a common vertex. For which $k \in \{0, 1, 2, 3, 4, 5, 6, 7, 8\}$ is it possible for some $a$ and $b$ , $ab > 2021$, to create a board whose fields are covered in mines, except for $2021$ fields who are all marked with $k$?
2012 Belarus Team Selection Test, 1
For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$
[i]Proposed by Suhaimi Ramly, Malaysia[/i]
1992 IMO Longlists, 79
Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that
\[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\]
where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$
2009 AMC 10, 19
A particular $ 12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $ 1$, it mistakenly displays a $ 9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?
$ \textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac58\qquad \textbf{(C)}\ \frac34\qquad \textbf{(D)}\ \frac56\qquad \textbf{(E)}\ \frac {9}{10}$
2015 Belarus Team Selection Test, 1
A circle intersects a parabola at four distinct points. Let $M$ and $N$ be the midpoints of the arcs of the circle which are outside the parabola. Prove that the line $MN$ is perpendicular to the axis of the parabola.
I. Voronovich
1989 National High School Mathematics League, 15
For any positive integer $n$, $a_n>0$, and $\sum_{j=1}^{n}a_j^3=\left(\sum_{j=1}^{n}a_j\right)^2$. Prove that $a_n=n$
1998 May Olympiad, 3
Given a $4 \times 4$ grid board with each square painted a different color, you want to cut it into two pieces of equal area by making a single cut along the grid lines. In how many ways can it be done?
2012 Kazakhstan National Olympiad, 2
Let $ABCD$ be an inscribed quadrilateral, in which $\angle BAD<90$. On the rays $AB$ and $AD$ are selected points $K$ and $L$, respectively, such that$ KA = KD, LA = LB$. Let $N$ - the midpoint of $AC$.Prove that if $\angle BNC=\angle DNC $,so $\angle KNL=\angle BCD $
2016 Latvia Baltic Way TST, 1
$2016$ numbers written on the board: $\frac{1}{2016}, \frac{2}{2016}, \frac{3}{2016}, ..., \frac{2016}{2016}$. In one move, it is allowed to choose any two numbers $a$ and $b$ written on the board, delete them, and write the number $3ab - 2a - 2b + 2$ instead. Determine what number will remain written on the board after $2015$ moves.