Found problems: 85335
2018 Argentina National Olympiad, 1
Let $p$ a prime number and $r$ the remainder of the division of $p$ by $210$. It is known that $r$ is a composite number and can be written as the sum of two non-zero perfect squares. Find all primes less than $2018$ that satisfy these conditions.
2017 Hong Kong TST, 4
Consider the sequences with 2016 terms formed by the digits 1, 2, 3, and 4. Find the number of those sequences containing an even number of the digit 1.
1994 Mexico National Olympiad, 3
$ABCD$ is a parallelogram. Take $E$ on the line $AB$ so that $BE = BC$ and $B$ lies between $A$ and $E$. Let the line through $C$ perpendicular to $BD$ and the line through $E$ perpendicular to $AB$ meet at $F$. Show that $\angle DAF = \angle BAF$.
2007 Vietnam National Olympiad, 3
Let ABCD be trapezium that is inscribed in circle (O) with larger edge BC. P is a point lying outer segment BC. PA cut (O) at N(that means PA isn't tangent of (O)), the circle with diameter PD intersect (O) at E, DE meet BC at N. Prove that MN always pass through a fixed point.
IV Soros Olympiad 1997 - 98 (Russia), grade6
[b]p1.[/b] The numerator of the fraction was increased by 20%. By what percentage should its denominator be reduced so that the resulting fraction doubles?
[b]p2.[/b] From point $O$ on the plane there are four rays $OA$, $OB$, $OC$ and $OD$ (not necessarily in that order). It is known that $\angle AOB =40^o$, $\angle BOC = 70^o$, $\angle COD = 80^o$. What values can the angle between rays $OA$ and $OD$ take? (The angle between the rays is from $0^o$ to $180^o$.)
[b]p3.[/b] Three equal circles have a common interior point. Prove that there is a circle of the same radius containing the centers of these three circles.
[b]p4.[/b] Two non-leap years are consecutive. The first one has more Mondays than Wednesdays. Which of the seven days of the week will occur most often in the second year?
[b]p5.[/b] The difference between two four-digit numbers is $7$. How much can the sums of their digits differ?
[b]p6.[/b] The numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written on the board. In one move you can increase any of the numbers by $3$ or $5$. What is the minimum number of moves you need to make for all the numbers to become equal?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
1984 IMO Longlists, 38
Determine all continuous functions $f: \mathbb R \to \mathbb R$ such that
\[f(x + y)f(x - y) = (f(x)f(y))^2, \quad \forall(x, y) \in\mathbb{R}^2.\]
1996 IMO Shortlist, 8
Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that
\[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0.
\]
1995 Poland - First Round, 10
Prove that the equation $x^x = y^3+z^3$ has infinitely many solutions in positive integers $x,y,z$.
2021 Romanian Master of Mathematics Shortlist, G4
Let $ABC$ be an acute triangle, let $H$ and $O$ be its orthocentre and circumcentre, respectively,
and let $S$ and $T$ be the feet of the altitudes from $B$ to $AC$ and from $C$ to $AB$, respectively.
Let $M$ be the midpoint of the segment $ST$, and let $N$ be the midpoint of the segment $AH$. The line
through $O$, parallel to $BC$, crosses the sides $AC$ and $AB$ at $F$ and $G$, respectively. The line $NG$
meets the circle $BGO$ again at $K$, and the line $NF$ meets the circle $CFO$ again at $L$. Prove that
the triangles $BCM$ and $KLN$ are similar.
2016 Czech-Polish-Slovak Match, 2
Let $m,n > 2$ be even integers. Consider a board of size $m \times n$ whose every cell is colored either black or white. The Guesser does not see the coloring of the board but may ask the Oracle some questions about it. In particular, she may inquire about two adjacent cells (sharing an edge) and the Oracle discloses whether the two adjacent cells have the same color or not. The Guesser eventually wants to fi nd the parity of the number of adjacent cell-pairs whose colors are diff erent. What is the minimum number of inquiries the Guesser needs to make so that she is guaranteed to find her answer?(Czech Republic)
2012 Harvard-MIT Mathematics Tournament, 8
Let $x_1=y_1=x_2=y_2=1$, then for $n\geq 3$ let $x_n=x_{n-1}y_{n-2}+x_{n-2}y_{n-1}$ and $y_n=y_{n-1}y_{n-2}-x_{n-1}x_{n-2}$. What are the last two digits of $|x_{2012}|?$
1981 Putnam, A2
Two distinct squares of the $8\times8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common.
Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$ there exist twoadjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.
2013 Bundeswettbewerb Mathematik, 4
Two players $A$ and $B$ play the following game taking alternate moves. In each move, a player writes one digit on the blackboard. Each new digit is written either to the right or left of the sequence of digits already written on the blackboard. Suppose that $A$ begins the game and initially the blackboard was empty. $B$ wins the game if ,after some move of $B$, the sequence of digits written in the blackboard represents a perfect square. Prove that $A$ can prevent $B$ from winning.
2018 AMC 10, 14
A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list?
$\textbf{(A)}\ 202\qquad\textbf{(B)}\ 223\qquad\textbf{(C)}\ 224\qquad\textbf{(D)}\ 225\qquad\textbf{(E)}\ 234$
2009 ISI B.Stat Entrance Exam, 1
Two train lines intersect each other at a junction at an acute angle $\theta$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\alpha$ at a station on the other line. It subtends an angle $\beta (<\alpha)$ at the same station, when its rear is at the junction. Show that
\[\tan\theta=\frac{2\sin\alpha\sin\beta}{\sin(\alpha-\beta)}\]
2004 China Team Selection Test, 2
Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$
2014 Contests, 2
Let $ABCD$ be a parallelogram. On side $AB$, point $M$ is taken so that $AD = DM$. On side $AD$ point $N$ is taken so that $AB = BN$. Prove that $CM = CN$.
2024 Assara - South Russian Girl's MO, 2
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]
1951 Miklós Schweitzer, 13
Of how many terms does the expansion of a determinant of order $ 2n$ consist if those and only those elements $ a_{ik}$ are non-zero for which $ i\minus{}k$ is divisible by $ n$?
2017 Brazil Undergrad MO, 4
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers in which $\lim_{n\to\infty} a_n = 0$ such that there is a constant $c >0$ so that for all $n \geq 1$, $|a_{n+1}-a_n| \leq c\cdot a_n^2$. Show that exists $d>0$ with $na_n \geq d, \forall n \geq 1$.
2006 China Second Round Olympiad, 14
Let $2006$ be expressed as the sum of five positive integers $x_1, x_2, x_3, x_4, x_5$, and $S=\sum_{1\le i<j\le 5}x_ix_j$.
$ \textbf{(A)}$ What value of $x_1, x_2, x_3, x_4, x_5$ maximizes $S$?
$ \textbf{(A)}$ Find, with proof, the value of $x_1, x_2, x_3, x_4, x_5$ which minimizes of $S$ if $|x_i-x_j|\le 2$ for any $1\le i$, $j\le 5$.
2022 IMO Shortlist, A5
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.
1990 National High School Mathematics League, 8
Point $A(2,0)$. $P(\sin(2t-\frac{\pi}{3}),\cos(2t-\frac{\pi}{3}))$ is a moving point. When $t$ changes from $\frac{\pi}{12}$ to $\frac{\pi}{4}$, area swept by segment $AP$ is________.
2024 Sharygin Geometry Olympiad, 19
A triangle $ABC$, its circumcircle, and its incenter $I$ are drawn on the plane. Construct the circumcenter of $ABC$ using only a ruler.
2014 India National Olympiad, 4
Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.