Found problems: 85335
2021 ISI Entrance Examination, 5
Let $a_0, a_1,\dots, a_{19} \in \mathbb{R}$ and $$P(x) = x^{20} + \sum_{i=0}^{19}a_ix^i, x \in \mathbb{R}.$$ If $P(x)=P(-x)$ for all $x \in \mathbb{R}$, and $$P(k)=k^2,$$ for $k=0, 1, 2, \dots, 9$ then find $$\lim_{x\rightarrow 0} \frac{P(x)}{\sin^2x}.$$
1997 Hungary-Israel Binational, 3
Let $ ABC$ be an acute angled triangle whose circumcenter is $ O$. The three diameters of the circumcircle that pass through $ A$, $ B$, and $ C$, meet the opposite sides $ BC$, $ CA$, and $ AB$ at the points $ A_1$, $ B_1$ and $ C_1$, respectively. The circumradius of $ ABC$ is of length $ 2P$, where $ P$ is a prime number. The lengths of $ OA_1$, $ OB_1$, $ OC_1$ are integers. What are the lengths of the sides of the triangle?
2002 Croatia National Olympiad, Problem 4
Let $(a_n)_{n\in\mathbb N}$ be an increasing sequence of positive integers. A term $a_k$ in the sequence is said to be good if it a sum of some other terms (not necessarily distinct). Prove that all terms of the sequence, apart from finitely many of them, are good.
2015 Romania Masters in Mathematics, 2
For an integer $n \geq 5,$ two players play the following game on a regular $n$-gon. Initially, three consecutive vertices are chosen, and one counter is placed on each. A move consists of one player sliding one counter along any number of edges to another vertex of the $n$-gon without jumping over another counter. A move is legal if the area of the triangle formed by the counters is strictly greater after the move than before. The players take turns to make legal moves, and if a player cannot make a legal move, that player loses. For which values of $n$ does the player making the first move have a winning strategy?
2009 Harvard-MIT Mathematics Tournament, 7
Paul fi lls in a $7\times7$ grid with the numbers $1$ through $49$ in a random arrangement. He then erases his work and does the same thing again, to obtain two diff erent random arrangements of the numbers in the grid. What is the expected number of pairs of numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements?
2022 Bulgarian Spring Math Competition, Problem 10.4
Find the smallest odd prime $p$, such that there exist coprime positive integers $k$ and $\ell$ which satisfy
\[4k-3\ell=12\quad \text{ and }\quad \ell^2+\ell k +k^2\equiv 3\text{ }(\text{mod }p)\]
1990 AMC 12/AHSME, 27
Which of these triples could [u]not[/u] be the lengths of the three altitudes of a triangle?
$ \textbf{(A)}\ 1,\sqrt{3},2 \qquad\textbf{(B)}\ 3,4,5 \qquad\textbf{(C)}\ 5,12,13 \qquad\textbf{(D)}\ 7,8,\sqrt{113} \qquad\textbf{(E)}\ 8,15,17 $
Indonesia MO Shortlist - geometry, g4
Given an acute triangle $ABC$ with $AB <AC$. Points $P$ and $Q$ lie on the angle bisector of $\angle BAC$ so that $BP$ and $CQ$ are perpendicular on that angle bisector. Suppose that point $E, F$ lie respectively at sides $AB$ and $AC$ respectively, in such a way that $AEPF$ is a kite. Prove that the lines $BC, PF$, and $QE$ intersect at one point.
2002 Vietnam Team Selection Test, 1
Find all triangles $ABC$ for which $\angle ACB$ is acute and the interior angle bisector of $BC$ intersects the trisectors $(AX, (AY$ of the angle $\angle BAC$ in the points $N,P$ respectively, such that $AB=NP=2DM$, where $D$ is the foot of the altitude from $A$ on $BC$ and $M$ is the midpoint of the side $BC$.
2019 Kazakhstan National Olympiad, 4
Find all positive integers $n,k,a_1,a_2,...,a_k$ so that $n^{k+1}+1$ is divisible by $(na_1+1)(na_2+1)...(na_k+1)$
1997 Estonia National Olympiad, 3
Each diagonal of a convex pentagon is parallel to one of its sides. Prove that the ratio of the length of each diagonal to the length of the corresponding parallel side is the same, and find this ratio.
1992 Baltic Way, 12
Let $ N$ denote the set of natural numbers. Let $ \phi: N\rightarrow N$ be a bijective function and assume that there exists a finite limit
\[ \lim_{n\rightarrow\infty}\frac{\phi(n)}{n}\equal{}L.
\] What are the possible values of $ L$?
2021 Moldova Team Selection Test, 2
Prove that if $p$ and $q$ are two prime numbers, such that
$$p+p^2+p^3+...+p^q=q+q^2+q^3+...+q^p,$$
then $p=q$.
2013 BMT Spring, 1
A time is called [i]reflexive [/i] if its representation on an analog clock would still be permissible if the hour and minute hand were switched. In a given non-leap day ($12:00:00.00$ a.m. to $11:59:59.99$ p.m.), how many times are reflexive?
1966 Kurschak Competition, 3
Do there exist two infinite sets of non-negative integers such that every non-negative integer can be uniquely represented in the form $a + b$ with $a$ in $A$ and $b$ in $B$?
2009 QEDMO 6th, 5
Let $p$ be a prime number and let further $p + 1$ rational numbers $a_0,...,a_p$ with the following property given: If one removes any of the $p + 1$ numbers, then the remaining may be split in at least two groups , which all have the same mean value (for different distant numbers, however, these mean values may be different). Prove that all $p + 1$ numbers are equal.
2020 Germany Team Selection Test, 2
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.
(Australia)
2010 Contests, 1
Consider a triangle $ABC$ such that $\angle A = 90^o, \angle C =60^o$ and $|AC|= 6$. Three circles with centers $A, B$ and $C$ are pairwise tangent in points on the three sides of the triangle.
Determine the area of the region enclosed by the three circles (the grey area in the figure).
[asy]
unitsize(0.2 cm);
pair A, B, C;
real[] r;
A = (6,0);
B = (6,6*sqrt(3));
C = (0,0);
r[1] = 3*sqrt(3) - 3;
r[2] = 3*sqrt(3) + 3;
r[3] = 9 - 3*sqrt(3);
fill(arc(A,r[1],180,90)--arc(B,r[2],270,240)--arc(C,r[3],60,0)--cycle, gray(0.7));
draw(A--B--C--cycle);
draw(Circle(A,r[1]));
draw(Circle(B,r[2]));
draw(Circle(C,r[3]));
dot("$A$", A, SE);
dot("$B$", B, NE);
dot("$C$", C, SW);
[/asy]
2013 Today's Calculation Of Integral, 880
For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and
let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows.
(1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$.
(2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$
(3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.
1975 Vietnam National Olympiad, 4
Find all terms of the arithmetic progression $-1, 18, 37, 56, ...$ whose only digit is $5$.
2021 MIG, 10
If $k$ raisins are distributed evenly to eleven children, four raisins would be left over. How many raisins would be left over if $3k$ raisins were distributed instead?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }7$
2021 Pan-African, 3
Let $(a_i)_{i\in \mathbb{N}}$ and $(p_i)_{i\in \mathbb{N}}$ be two sequences of positive integers such that the following conditions hold:
$\bullet ~~a_1\ge 2$.
$\bullet~~ p_n$ is the smallest prime divisor of $a_n$ for every integer $n\ge 1$
$\bullet~~ a_{n+1}=a_n+\frac{a_n}{p_n}$ for every integer $n\ge 1$
Prove that there is a positive integer $N$ such that $a_{n+3}=3a_n$ for every integer $n>N$
2017 China Team Selection Test, 5
Given integer $m\geq2$,$x_1,...,x_m$ are non-negative real numbers,prove that:$$(m-1)^{m-1}(x_1^m+...+x_m^m)\geq(x_1+...+x_m)^m-m^mx_1...x_m$$and please find out when the equality holds.
2001 USAMO, 4
Let $P$ be a point in the plane of triangle $ABC$ such that the segments $PA$, $PB$, and $PC$ are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to $PA$. Prove that $\angle BAC$ is acute.
1991 All Soviet Union Mathematical Olympiad, 551
A sequence of positive integers is constructed as follows. If the last digit of $a_n$ is greater than $5$, then $a_{n+1}$ is $9a_n$. If the last digit of $a_n$ is $5$ or less and an has more than one digit, then $a_{n+1}$ is obtained from $a_n$ by deleting the last digit. If $a_n$ has only one digit, which is $5$ or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?