Found problems: 85335
2015 HMMT Geometry, 3
Let $ABCD$ be a quadrilateral with $\angle BAD = \angle ABC = 90^{\circ}$, and suppose $AB=BC=1$, $AD=2$. The circumcircle of $ABC$ meets $\overline{AD}$ and $\overline{BD}$ at point $E$ and $F$, respectively. If lines $AF$ and $CD$ meet at $K$, compute $EK$.
1954 Moscow Mathematical Olympiad, 269
a) Given $100$ numbers $a_1, ..., a_{100}$ such that $\begin{cases}
a_1 - 3a_2 + 2a_3 \ge 0, \\
a_2 - 3a_3 + 2a_4 \ge 0, \\
a_3 - 3a_4 + 2a_5 \ge 0, \\
... \\
a_{99} - 3a_{100} + 2a_1 \ge 0, \\
a_{100} - 3a_1 + 2a_2 \ge 0 \end{cases}$
prove that the numbers are equal.
b) Given numbers $a_1=1, ..., a_{100}$ such that $\begin{cases}
a_1 - 4a_2 + 3a_3 \ge 0, \\
a_2 - 4a_3 + 3a_4 \ge 0, \\
a_3 - 4a_4 + 3a_5 \ge 0, \\
... \\
a_{99} - 4a_{100} + 3a_1 \ge 0, \\
a_{100} - 4a_1 + 3a_2 \ge 0 \end{cases}$
Find $a_2, a_3, ... , a_{100}.$
2024 Belarus Team Selection Test, 3.3
Olya and Tolya are playing a game on $[0,1]$ segment. In the beginning it is white. In the first round Tolya chooses a number $0 \leq l \leq 1$, and then Olya chooses a subsegment of $[0,1]$ of length $l$ and recolors every its point to the opposite color(white to black, black to white). In the next round players change roles, etc. The game lasts $2024$ rounds. Let $L$ be the sum of length of white segments after the end of the game. If $L > \frac{1}{2}$ Olya wins, otherwise Tolya wins. Which player has a strategy to guarantee his win?
[i]A. Naradzetski[/i]
2018 Finnish National High School Mathematics Comp, 2
The sides of triangle $ABC$ are $a = | BC |, b = | CA |$ and $c = | AB |$. Points $D, E$ and $F$ are the points on the sides $BC, CA$ and $AB$ such that $AD, BE$ and $CF$ are the angle bisectors of the triangle $ABC$. Determine the lengths of the segments $AD, BE$, and $CF$ in terms of $a, b$, and $c$.
2007 Thailand Mathematical Olympiad, 18
Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.
2000 Putnam, 3
Let $f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt)$, where each $a_j$ is areal and $a_N$ is not equal to $0$.
Let $N_k$ denote the number of zeroes (including multiplicites) of $\dfrac{d^k f}{dt^k}$. Prove that \[ N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N. \] [color=green][Only zeroes in [0, 1) should be counted.][/color]
2018 Brazil Team Selection Test, 4
In a triangle $ABC$, points $H, L, K$ are chosen on the sides $AB, BC, AC$, respectively, so that $CH \perp AB$, $HL \parallel AC$ and $HK \parallel BC$. In the triangle $BHL$, let $P, Q$ be the feet of the heights from the vertices $B$ and $H$. In the triangle $AKH$, let $R, S$ be the feet of the heights from the vertices $A$ and $H$. Show that the four points $P, Q, R, S$ are collinear.
2007 ISI B.Stat Entrance Exam, 10
Let $A$ be a set of positive integers satisfying the following properties:
(i) if $m$ and $n$ belong to $A$, then $m+n$ belong to $A$;
(ii) there is no prime number that divides all elements of $A$.
(a) Suppose $n_1$ and $n_2$ are two integers belonging to $A$ such that $n_2-n_1 >1$. Show that you can find two integers $m_1$ and $m_2$ in $A$ such that $0< m_2-m_1 < n_2-n_1$
(b) Hence show that there are two consecutive integers belonging to $A$.
(c) Let $n_0$ and $n_0+1$ be two consecutive integers belonging to $A$. Show that if $n\geq n_0^2$ then $n$ belongs to $A$.
2000 Flanders Math Olympiad, 3
Let $p_n$ be the $n$-th prime. ($p_1=2$)
Define the sequence $(f_j)$ as follows:
- $f_1=1, f_2=2$
- $\forall j\ge 2$: if $f_j = kp_n$ for $k<p_n$ then $f_{j+1}=(k+1)p_n$
- $\forall j\ge 2$: if $f_j = p_n^2$ then $f_{j+1}=p_{n+1}$
(a) Show that all $f_i$ are different
(b) from which index onwards are all $f_i$ at least 3 digits?
(c) which integers do not appear in the sequence?
(d) how many numbers with less than 3 digits appear in the sequence?
2021-IMOC qualification, A1
Prove that if positive reals $x,y$ satisfy $x+y= 3$, $x,y \ge 1$ then $$9(x- 1)(y- 1) + (y^2 + y+ 1)(x + 1) + (x^2-x+ 1)(y- 1) \ge 9$$
VII Soros Olympiad 2000 - 01, 9.1
Draw on the plane a set of points whose coordinates $(x,y)$ satisfy the equation $x^3 + y^3 = x^2y^2 + xy$.
2024 LMT Fall, 3
Two distinct positive even integers sum to $8$. Find the larger of the two integers.
2005 Today's Calculation Of Integral, 43
Evaluate
\[\int_0^{\frac{\pi}{2}} \cos ^ {2004}x\cos 2004x\ dx\]
2014 China Western Mathematical Olympiad, 5
Given a positive integer $m$, Prove that there exists a positive integers $n_0$ such that all first digits after the decimal points of $\sqrt{n^2+817n+m}$ in decimal representation are equal, for all integers $n>n_0$.
2006 China Girls Math Olympiad, 2
Let $O$ be the intersection of the diagonals of convex quadrilateral $ABCD$. The circumcircles of $\triangle{OAD}$ and $\triangle{OBC}$ meet at $O$ and $M$. Line $OM$ meets the circumcircles of $\triangle{OAB}$ and $\triangle{OCD}$ at $T$ and $S$ respectively.
Prove that $M$ is the midpoint of $ST$.
2011 AMC 12/AHSME, 4
In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His errorneous product was $161$. What is the correct value of the product of $a$ and $b$?
$ \textbf{(A)}\ 116 \qquad
\textbf{(B)}\ 161 \qquad
\textbf{(C)}\ 204 \qquad
\textbf{(D)}\ 214 \qquad
\textbf{(E)}\ 224 $
2018 South Africa National Olympiad, 4
Let $ABC$ be a triangle with circumradius $R$, and let $\ell_A, \ell_B, \ell_C$ be the altitudes through $A, B, C$ respectively. The altitudes meet at $H$. Let $P$ be an arbitrary point in the same plane as $ABC$. The feet of the perpendicular lines through $P$ onto $\ell_A, \ell_B, \ell_C$ are $D, E, F$ respectively. Prove that the areas of $DEF$ and $ABC$ satisfy the following equation:
$$
\operatorname{area}(DEF) = \frac{{PH}^2}{4R^2} \cdot \operatorname{area}(ABC).
$$
2013 Saudi Arabia IMO TST, 4
Determine whether it is possible to place the integers $1, 2,...,2012$ in a circle in such a way that the $2012$ products of adjacent pairs of numbers leave pairwise distinct remainders when divided by $2013$.
2016 Romanian Master of Mathematics Shortlist, C3
A set $S=\{ s_1,s_2,...,s_k\}$ of positive real numbers is "polygonal" if $k\geq 3$ and there is a non-degenerate planar $k-$gon whose side lengths are exactly $s_1,s_2,...,s_k$; the set $S$ is multipolygonal if in every partition of $S$ into two subsets,each of which has at least three elements, exactly one of these two subsets in polygonal. Fix an integer $n\geq 7$.
(a) Does there exist an $n-$element multipolygonal set, removal of whose maximal element leaves a multipolygonal set?
(b) Is it possible that every $(n-1)-$element subset of an $n-$element set of positive real numbers be multipolygonal?
2020 BMT Fall, 27
Estimate the number of $1$s in the hexadecimal representation of $2020!$.
If $E$ is your estimate and $A$ is the correct answer, you will receive $\max (25 - 0.5|A - E|, 0)$ points, rounded to the nearest integer.
1993 Miklós Schweitzer, 1
There are n points in the plane with the property that the distance between any two points is at least 1. Prove that for sufficiently large n , the number of pairs of points whose distance is in $[ t_1 , t_1 + 1] \cup [ t_2 , t_2 + 1]$ for some $t_1, t_2$ , is at most $[\frac{n^2}{3}]$ , and the bound is sharp.
2010 Contests, 4
Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that
\[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]
2016 SGMO, Q3
In Simoland there are $2017n$ cities arranged in a $2017\times n$ lattice grid. There are $2016$ MRT (train) tracks and each track can only go north and east, or south and east. Suppose that all the tracks together pass through all the cities. Determine the largest possible value of $n$.
2019 India PRMO, 16
Let $N$ denote the number of all natural numbers $n$ such that $n$ is divisible by a prime $p> \sqrt{n}$ and $p<20$. What is the value of $N$ ?
2020 MBMT, 28
Consider the system of equations $$a + 2b + 3c + \ldots + 26z = 2020$$ $$b + 2c + 3d + \ldots + 26a = 2019$$ $$\vdots$$ $$y + 2z + 3a + \ldots + 26x = 1996$$ $$z + 2a + 3b + \ldots + 26y = 1995$$ where each equation is a rearrangement of the first equation with the variables cycling and the coefficients staying in place. Find the value of $$z + 2y + 3x + \dots + 26a.$$
[i]Proposed by Joshua Hsieh[/i]