Found problems: 85335
2012 Serbia National Math Olympiad, 3
We are given $n>1$ piles of coins. There are two different types of coins: real and fake coins; they all look alike, but coins of the same type have the same mass, while the coins from different types have different masses. Coins that belong to the same pile are of the same type. We know the mass of real coin.
Find the minimal number of weightings on digital scale that we need in order to conclude: which piles consists of which type of coins and also the mass of fake coin.
(We assume that every pile consists from infinite number of coins.)
2004 Uzbekistan National Olympiad, 3
Given a sequence {$a_n$} such that $a_1=2$ and for all positive integer $n\geq 2$ $a_{n+1}=\frac{a_n^4+9}{16a_n}$. Prove that $\frac {4}{5}<a_n<\frac {5}{4}$
2007 Princeton University Math Competition, 8
Points $P_1, P_2, P_3,$ and $P_4$ are $(0,0), (10, 20), (5, 15),$ and $(12, -6)$, respectively. For what point $P \in \mathbb{R}^2$ is the sum of the distances from $P$ to the other $4$ points minimal?
2008 Tuymaada Olympiad, 6
Let $ ABCD$ be an isosceles trapezoid with $ AD \parallel BC$. Its diagonals $ AC$ and $ BD$ intersect at point $ M$. Points $ X$ and $ Y$ on the segment $ AB$ are such that $ AX \equal{} AM$, $ BY \equal{} BM$. Let $ Z$ be the midpoint of $ XY$ and $ N$ is the point of intersection of the segments $ XD$ and $ YC$. Prove that the line $ ZN$ is parallel to the bases of the trapezoid.
[i]Author: A. Akopyan, A. Myakishev[/i]
2020 CHMMC Winter (2020-21), 2
Find the smallest positive integer $k$ such that there is exactly one prime number of the form $kx + 60$ for the integers $0 \le x \le 10$.
2022 USEMO, 2
A function $\psi \colon {\mathbb Z} \to {\mathbb Z}$ is said to be [i]zero-requiem[/i] if for any positive integer $n$ and any integers $a_1$, $\ldots$, $a_n$ (not necessarily distinct), the sums $a_1 + a_2 + \dots + a_n$ and $\psi(a_1) + \psi(a_2) + \dots + \psi(a_n)$ are not both zero.
Let $f$ and $g$ be two zero-requiem functions for which $f \circ g$ and $g \circ f$ are both the identity function (that is, $f$ and $g$ are mutually inverse bijections). Given that $f+g$ is [i]not[/i] a zero-requiem function, prove that $f \circ f$ and $g \circ g$ are both zero-requiem.
[i]Sutanay Bhattacharya[/i]
2016 Purple Comet Problems, 28
Find the sum of all the possible values of xy such that x and y are positive integers satisfying $(x^2 + 1)(y^2 + 1) + 2(x -y)(1 - xy) = 4(1 + xy) + 140$.
1969 IMO Longlists, 71
$(YUG 3)$ Let four points $A_i (i = 1, 2, 3, 4)$ in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by $S.$ What is the exact placement of the points $A_i$ if $S = 180^{\circ}$?
2008 Iran MO (3rd Round), 3
For each $ c\in\mathbb C$, let $ f_c(z,0)\equal{}z$, and $ f_c(z,n)\equal{}f_c(z,n\minus{}1)^2\plus{}c$ for $ n\geq1$.
a) Prove that if $ |c|\leq\frac14$ then there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is bounded.
b) Prove that if $ c>\frac14$ is a real number there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is unbounded.
2013 IFYM, Sozopol, 6
The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that
$f(x+1)\leq f(2x+1)$ and $f(3x+1)\geq f(6x+1)$ for $\forall$ $x\in \mathbb{R}$.
If $f(3)=2$, prove that there exist at least 2013 distinct values of $x$, for which $f(x)=2$.
2023 ISL, C1
Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps.
[list=1]
[*]select a $2\times 2$ square in the grid;
[*]flip the coins in the top-left and bottom-right unit squares;
[*]flip the coin in either the top-right or bottom-left unit square.
[/list]
Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves.
[i]Thanasin Nampaisarn, Thailand[/i]
2010 Abels Math Contest (Norwegian MO) Final, 3
$ a)$
There are $ 25$ participants in a mathematics contest having four problems. Each problem is considered solved or not solved (that is, partial solutions are not possible). Show that either there are four contestants having solved the same problems (or not having solved any of them), or two contestants, one of which has solved exactly the problems that the other did not solve.
$ b)$
There are $ k$ sport clubs for the students of a secondary school. The school has $ 100$ students, and for any selection of three of them, there exists a club having at least one of them, but not all, as a member. What is the least possible value of $ k$?
2021 LMT Spring, A18
Points $X$ and $Y$ are on a parabola of the form $y=\frac{x^2}{a^2}$ and $A$ is the point $(x, y) = (0, a)$. Assume $XY$ passes through $A$ and hits the line $y=-a$ at a point $B$. Let $\omega$ be the circle passing through $(0, -a)$, $A$, and $B$. A point $P$ is chosen on $\omega$ such that $PA = 8$. Given that $X$ is between $A$ and $B$, $AX=2$, and $XB=10$, find $PX \cdot PY$.
[i]Proposed by Kevin Zhao[/i]
Estonia Open Junior - geometry, 2013.1.4
Inside a circle $c$ with the center $O$ there are two circles $c_1$ and $c_2$ which go through $O$ and are tangent to the circle $c$ at points $A$ and $B$ crespectively. Prove that the circles $c_1$ and $c_2$ have a common point which lies in the segment $AB$.
MathLinks Contest 2nd, 1.2
We call a permutation $\sigma$ of the first $n$ positive integers friendly if and only if the following conditions are fulfilled:
(1) $\sigma(k + 1) \in \{2\sigma(k), 2\sigma(k) - 1, 2\sigma(k) - n, 2\sigma(k) - n - 1\}, \forall k \in \{1, 2, ..., n - 1\}$
(2) $\sigma(1) \in \{2 \sigma(n), 2\sigma(n) - 1, 2\sigma(n) - n, 2\sigma(n) - n - 1\}$.
Find all positive integers $n$ for which there exists such a friendly permutation of the first $n$ positive integers.
2003 Moldova Team Selection Test, 1
Each side of an arbitrarly triangle is divided into $ 2002$ congruent segments. After that, each vertex is joined with all "division" points on the opposite side.
Prove that the number of the regions formed, in which the triangle is divided, is divisible by $ 6$.
[i]Proposer[/i]: [b]Dorian Croitoru[/b]
2018 AMC 10, 2
Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alica have?
$ \textbf{(A) }\text{ Liliane has } 20\%\text{ more soda than Alice.}$
$\textbf{(B) }\text{ Liliane has } 25\%\text{ more soda than Alice.}$
$\textbf{(C) }\text{ Liliane has } 45\%\text{ more soda than Alice.}$
$ \textbf{(D) }\text{ Liliane has } 75\%\text{ more soda than Alice.}$
$\textbf{(E) }\text{ Liliane has } 100\%\text{ more soda than Alice.}$
OIFMAT III 2013, 1
Find all four-digit perfect squares such that:
$\bullet$ All your figures are less than $9$.
$\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.
2013 Princeton University Math Competition, 7
Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$, where $n$ is either $2012$ or $2013$.
2011 HMNT, 6
Let $ABC$ be an equilateral triangle with $AB = 3$. Circle $\omega$ with diameter $1$ is drawn inside the triangle such that it is tangent to sides $AB$ and $AC$. Let $P$ be a point on $\omega$ and $ $ be a point on segment $BC$. Find the minimum possible length of the segment $PQ$.
2022 ITAMO, 2
Let $ABC$ be an acute triangle with $AB<AC$. Let then
• $D$ be the foot of the bisector of the angle in $A$,
• $E$ be the point on segment $BC$ (different from $B$) such that $AB=AE$,
• $F$ be the point on segment $BC$ (different from $B$) such that $BD=DF$,
• $G$ be the point on segment $AC$ such that $AB=AG$.
Prove that the circumcircle of triangle $EFG$ is tangent to line $AC$.
2014 ASDAN Math Tournament, 5
A positive integer $k$ is $2014$-ambiguous if the quadratics $x^2+kx+2014$ and $x^2+kx-2014$ both have two integer roots. Compute the number of integers which are $2014$-ambiguous.
2011 AMC 12/AHSME, 19
At a competition with $N$ players, the number of players given elite status is equal to \[2^{1+\lfloor\log_2{(N-1)}\rfloor} - N. \] Suppose that $19$ players are given elite status. What is the sum of the two smallest possible values of $N$?
$ \textbf{(A)}\ 38\qquad
\textbf{(B)}\ 90 \qquad
\textbf{(C)}\ 154 \qquad
\textbf{(D)}\ 406 \qquad
\textbf{(E)}\ 1024$
2007 F = Ma, 6
At time $t = 0$ a drag racer starts from rest at the origin and moves along a straight line with velocity given by $v = 5t^2$, where $v$ is in $\text{m/s}$ and $t$ in $\text{s}$. The expression for the displacement of the car from $t = 0$ to time $t$ is
$ \textbf{(A)}\ 5t^3 \qquad\textbf{(B)}\ 5t^3/3\qquad\textbf{(C)}\ 10t \qquad\textbf{(D)}\ 15t^2 \qquad\textbf{(E)}\ 5t/2 $
1998 Chile National Olympiad, 7
When rolling two normal dice, the set of possible outcomes of the sum of the points is $2, 3, 3, 4,4, 4,..., 11, 11,12$. Notice that this sequence can be obtained from the identity $$(x + x^2 + x^3 + x^4 + x^5 + x^6) (x + x^2 + x^3 + x^4 + x^5 + x^6) = x^2 + 2x^3 + 3x^4 +... + 2x^{11} + x^{12}.$$ Design a crazy pair of dice, that is, two other cubes, not necessarily the same, with a natural number indicated on each face, such that the set of possible results of the sum of its points is equal to of two normal dice.