Found problems: 85335
2013 Waseda University Entrance Examination, 1
Given a parabola $C: y^2=4px\ (p>0)$ with focus $F(p,\ 0)$. Let two lines $l_1,\ l_2$ passing through $F$ intersect orthogonaly each other,
$C$ intersects with $l_1$ at two points $P_1,\ P_2$ and $C$ intersects with $l_2$ at two points $Q_1,\ Q_2$. Answer the following questions.
(1) Set the equation of $l_1$ as $x=ay+p$ and let the coordinates of $P_1,\ P_2$ as $(x_1,\ y_1),\ (x_2,\ y_2)$, respectively. Express $y_1+y_2,\ y_1y_2$ in terms of $a,\ p$.
(2) Show that $\frac{1}{P_1P_2}+\frac{1}{Q_1Q_2}$ is constant regardless of way of taking $l_1,\ l_2$.
1985 Spain Mathematical Olympiad, 5
Find the equation of the circle in the complex plane determined by the roots of the equation $z^3 +(-1+i)z^2+(1-i)z+i= 0$.
2015 Israel National Olympiad, 7
The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number.
[list=a]
[*] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$.
[*] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$.
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2013 Bosnia and Herzegovina Junior BMO TST, 3
Let $M$ and $N$ be touching points of incircle with sides $AB$ and $AC$ of triangle $ABC$, and $P$ intersection point of line $MN$ and angle bisector of $\angle ABC$. Prove that $\angle BPC =90 ^{\circ}$
1978 IMO Longlists, 21
A circle touches the sides $AB,BC, CD,DA$ of a square at points $K,L,M,N$ respectively, and $BU, KV$ are parallel lines such that $U$ is on $DM$ and $V$ on $DN$. Prove that $UV$ touches the circle.
2012 All-Russian Olympiad, 1
Initially, there are $111$ pieces of clay on the table of equal mass. In one turn, you can choose several groups of an equal number of pieces and push the pieces into one big piece for each group. What is the least number of turns after which you can end up with $11$ pieces no two of which have the same mass?
2000 Moldova National Olympiad, Problem 6
Find all nonnegative integers $n$ for which $n^8-n^2$ is not divisible by $72$.
2010 Indonesia TST, 4
How many natural numbers $(a,b,n)$ with $ gcd(a,b)=1$ and $ n>1 $ such that the equation \[ x^{an} +y^{bn} = 2^{2010} \] has natural numbers solution $ (x,y) $
2018 Balkan MO Shortlist, N5
Let $x,y$ be positive integers. If for each positive integer $n$ we have that $$(ny)^2+1\mid x^{\varphi(n)}-1.$$
Prove that $x=1$.
[i](Silouanos Brazitikos, Greece)[/i]
2017 Auckland Mathematical Olympiad, 2
Two players take turns to write natural numbers on a board. The rules forbid writing numbers greater than $p$ and also divisors of previously written numbers. The player who has no move loses. Determine which player has a winning strategy for $p = 10$ and describe this strategy.
1969 Dutch Mathematical Olympiad, 3
Given a quadrilateral $ABCD$ with $AB = BD = DC$ and $AC = BC$. On $BC$ lies point $E$ such that $AE = AB$. Prove that $ED = EB$.
2022 Math Prize for Girls Problems, 4
Determine the largest integer $n$ such that $n < 103$ and $n^3 - 1$ is divisible by $103$.
2011 National Olympiad First Round, 20
$100$ students participate in an exam with $5$ questions. Every question is answered by exactly $50$ students. What is the least possible value of number of students who answered at most $2$ questions?
$\textbf{(A)}\ 21 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 16 \qquad\textbf{(E)}\ \text{None}$
2005 Balkan MO, 2
Find all primes $p$ such that $p^2-p+1$ is a perfect cube.
2008 AMC 12/AHSME, 15
Let $ k\equal{}2008^2\plus{}2^{2008}$. What is the units digit of $ k^2\plus{}2^k$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 8$
1993 National High School Mathematics League, 7
Equation $(1-\text{i})x^2+(\lambda+\text{i})x+(1+\text{i}\lambda)=0(\lambda\in\mathbb{R})$ has two imaginary roots, then the range value of $\lambda$ is________.
2018 China Western Mathematical Olympiad, 2
Let $n \geq 2$ be an integer. Positive reals $x_1, x_2, \cdots, x_n$ satisfy $x_1x_2 \cdots x_n = 1$.
Show: $$\{x_1\} + \{x_2\} + \cdots + \{x_n\} < \frac{2n-1}{2}$$
Where $\{x\}$ denotes the fractional part of $x$.
2016 Sharygin Geometry Olympiad, 3
A trapezoid $ABCD$ and a line $\ell$ perpendicular to its bases $AD$ and $BC$ are given. A point $X$ moves along $\ell$. The perpendiculars from $A$ to $BX$ and from $D$ to $CX$ meet at point $Y$ . Find the locus of $Y$ .
by D.Prokopenko
2023 Polish MO Finals, 5
Give a prime number $p>2023$. Let $r(x)$ be the remainder of $x$ modulo $p$. Let $p_1<p_2< \ldots <p_m$ be all prime numbers less that $\sqrt[4]{\frac{1}{2}p}$. Let $q_1, q_2, \ldots, q_n$ be the inverses modulo $p$ of $p_1, p_2, \ldots p_n$. Prove that for every integers $0 < a,b < p$, the sets
$$\{r(q_1), r(q_2), \ldots, r(q_m)\}, ~~ \{r(aq_1+b), r(aq_2+b), \ldots, r(aq_m+b)\}$$
have at most $3$ common elements.
2005 India National Olympiad, 1
Let $M$ be the midpoint of side $BC$ of a triangle $ABC$. Let the median $AM$ intersect the incircle of $ABC$ at $K$ and $L,K$ being nearer to $A$
than $L$. If $AK = KL = LM$, prove that the sides of triangle $ABC$ are in the ratio $5 : 10 : 13$ in some order.
2006 National Olympiad First Round, 27
If $x,y,z$ are positive real numbers such that $xy+yz+zx=5$, $x^2+y^2+z^2-xyz$ cannot be $\underline{\hspace{1cm}}$.
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 3\sqrt 3
\qquad\textbf{(E)}\ \text{None of above}
$
2019 AMC 10, 23
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers ($4$, $5$, $6$), then Tadd must say the next four numbers ($7$, $8$, $9$, $10$), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. What is the $2019$th number said by Tadd?
$ \textbf{(A)}\ 5743
\qquad\textbf{(B)}\ 5885
\qquad\textbf{(C)}\ 5979
\qquad\textbf{(D)}\ 6001
\qquad\textbf{(E)}\ 6011
$
Russian TST 2019, P1
A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold:
[list=1]
[*] each triangle from $T$ is inscribed in $\omega$;
[*] no two triangles from $T$ have a common interior point.
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Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.
1985 Traian Lălescu, 2.2
Find all square roots of integers, namely $ p, $ such that $ \left(\frac{p}{2}\right)^2 <3<\left(\frac{p+1}{2}\right)^2. $
1993 Vietnam Team Selection Test, 3
Let's consider the real numbers $x_1, x_2, x_3, x_4$ satisfying the condition
\[ \dfrac{1}{2}\le x_1^2+x_2^2+x_3^2+x_4^2\le 1 \]
Find the maximal and the minimal values of expression:
\[ A = (x_1 - 2 \cdot x_2 + x_3)^2 + (x_2 - 2 \cdot x_3 + x_4)^2 + (x_2 - 2 \cdot x_1)^2 + (x_3 - 2 \cdot x_4)^2 \]