Found problems: 85335
2016 Thailand Mathematical Olympiad, 3
Determine all functions $f : R \to R$ satisfying $f (f(x)f(y) + f(y)f(z) + f(z)f(x))= f(x) + f(y) + f(z)$ for all real numbers $x, y, z$.
2023 Simon Marais Mathematical Competition, A2
Let $n$ be a positive integer and let $f_1(x), f_2(x) \dots f_n(x)$ be affine functions from $\mathbb{R}$ to $\mathbb{R}$ such that, amongst the graph of these functions, no two are parallel and no three are concurrent. Let $S$ be the set of all convex functions $g(x)$ from $\mathbb{R}$ to $\mathbb{R}$ such that for each $x \in \mathbb{R}$, there exists $i$ such that $g(x) = f_i(x)$.
Determine the largest and smallest possible values of $|S|$ in terms of $n$.
(A function $f(x)$ is affine if it is of form $f(x) = ax + b$ for some $a, b \in \mathbb{R}$. A function $g(x)$ is convex if $g(\lambda x + (1 - \lambda) y) \leq \lambda g(x) + (1-\lambda)g(y)$ for all $x, y \in \mathbb{R}$ and $0 \leq \lambda \leq 1$)
2011 Greece Junior Math Olympiad, 3
If the number $3n +1$, where n is integer, is multiple of $7$, find the possible remainders of the following divisions:
(a) of $n$ with divisor $7$,
(b) of $n^{m}$ with divisor $7$, for all values of the positive integer $m, m >1$.
2006 Princeton University Math Competition, 9
Consider the set of sequences $\{S_i\}$ that start with $S_0 = 12$, $S_1 = 21$, $S_2 = 28$, and for $n > 2$, $S_n$ is the sum of two (not necessarily distinct) $S_{k_n}$ and $S_{j_n}$ with $k_n, j_n < n$. Find the largest integer that cannot be found in any sequence $S_i$.
2023 Saint Petersburg Mathematical Olympiad, 5
Let $x_0, x_1, \ldots, x_{n-1}, x_n=x_0$ be reals and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. The numbers $y_i$ for $i=0,1, \ldots, n-1$ are chosen such that $y_i$ is between $x_i$ and $x_{i+1}$. Prove that $\sum_{i=0}^{n-1}(x_{i+1}-x_i)f(y_i)$ can attain both positive and negative values, by varying the $y_i$.
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?
2014 Bosnia And Herzegovina - Regional Olympiad, 3
Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$.
Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$.
[i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$.
Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.
2021 Dutch IMO TST, 2
Stekel and Prick play a game on an $ m \times n$ board, where $m$ and $n$ are positive are integers. They alternate turns, with Stekel starting. Spine bets on his turn, he always takes a pawn on a square where there is no pawn yet. Prick does his turn the same, but his pawn must always come into a square adjacent to the square that Spike just placed a pawn in on his previous turn. Prick wins like the whole board is full of pawns. Spike wins if Prik can no longer move a pawn on his turn, while there is still at least one empty square on the board. Determine for all pairs $(m, n)$ who has a winning strategy.
2010 Today's Calculation Of Integral, 573
Find the area of the figure bounded by three curves
$ C_1: y\equal{}\sin x\ \left(0\leq x<\frac {\pi}{2}\right)$
$ C_2: y\equal{}\cos x\ \left(0\leq x<\frac {\pi}{2}\right)$
$ C_3: y\equal{}\tan x\ \left(0\leq x<\frac {\pi}{2}\right)$.
2021 Durer Math Competition Finals, 2
In the country of Óxisz the minister of finance observed at the end of the tax census that the sum of properties of any two neighboring city counted in dinar is divisible by $1000$, and she also observed that the sum of properties of all cities is also divisible by $1000$. What is the least sum of properties of all cities if the map of the cities looks as follows?
[img]https://cdn.artofproblemsolving.com/attachments/0/5/274730ebfdd52d0c3642dfbd0596fe587eb211.png[/img]
[i]Remark: The cities may have non-integer properties, but it is also positive. On the map the points are the cities, and two cities are neighboring if there is a direct connection between them.[/i]
2010 Contests, 3
A teacher wants to divide the $2010$ questions she asked in the exams during the school year into three folders of $670$ questions and give each folder to a student who solved all $670$ questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations in which there are at most two students who did not solve any given question.
2010 Peru MO (ONEM), 1
In each of the $9$ small circles of the following figure we write positive integers less than $10$, without repetitions. In addition, it is true that the sum of the $5$ numbers located around each one of the $3$ circles is always equal to $S$. Find the largest possible value of $S$.
[img]https://cdn.artofproblemsolving.com/attachments/6/6/2db2c1ac7f45022606fb0099f24e6287977d10.png[/img]
2024 Sharygin Geometry Olympiad, 24
Let $SABC$ be a pyramid with right angles at the vertex $S$. Points $A', B', C'$ lie on the edges $SA, SB, SC$ respectively in such a way that the triangles $ABC$ and $A'B'C'$ are similar. Does this yield that the planes $ABC$ and $A'B'C'$ are parallel?
2010 ITAMO, 5
In the land of Cockaigne, people play the following solitaire. It starts from a finite string of zeros and ones, and are granted the following moves:
(i) cancel each two consecutive ones;
(ii) delete three consecutive zeros;
(iii) if the substring within the string is $01$, one may replace this by substring $100$.
The moves (i), (ii) and (iii) must be made one at a time. You win if you can reduce the string to a string formed by two digits or less.
(For example, starting from $0101$, one can win using move (iii) first in the last two digits, resulting in $01100$, then playing the move (i) on two 'ones', and finally the move (ii) on the three zeros, one will get the empty string.)
Among all the $1024$ possible strings of ten-digit binary numbers, how many are there from which it is not possible to win the solitary?
1996 National High School Mathematics League, 1
The sum of first $n$ items of squence $(a_n)$ : $S_n$ satisfies that $S_n=2a_n-1$, squence $(b_n)$ satisfies that $b_{k+1}=a_k+b_k$ for all $k=1,2,\cdots$. Find the sum of first $n$ items of $(b_n)$.
2016 Indonesia TST, 2
Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds:
\[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]
2020 HMIC, 3
Let $P_1P_2P_3P_4$ be a tetrahedron in $\mathbb{R}^3$ and let $O$ be a point equidistant from each of its vertices. Suppose there exists a point $H$ such that for each $i$, the line $P_iH$ is perpendicular to the plane through the other three vertices. Line $P_1H$ intersects the plane through $P_2, P_3, P_4$ at $A$, and contains a point $B\neq P_1$ such that $OP_1=OB$. Show that $HB=3HA$.
[i]Michael Ren[/i]
1996 Taiwan National Olympiad, 2
Let $0<a\leq 1$ be a real number and let $a\leq a_{i}\leq\frac{1}{a_{i}}\forall i=\overline{1,1996}$ are real numbers. Prove that for any nonnegative real numbers $k_{i}(i=1,2,...,1996)$ such that $\sum_{i=1}^{1996}k_{i}=1$ we have $(\sum_{i=1}^{1996}k_{i}a_{i})(\sum_{i=1}^{1996}\frac{k_{i}}{a_{i}})\leq (a+\frac{1}{a})^{2}$.
LMT Team Rounds 2010-20, A2 B6
$1001$ marbles are drawn at random and without replacement from a jar of $2020$ red marbles and $n$ blue marbles. Find the smallest positive integer $n$ such that the probability that there are more blue marbles chosen than red marbles is strictly greater than $\frac{1}{2}$.
[i]Proposed by Taiki Aiba[/i]
2001 National High School Mathematics League, 11
The range of function $y=x+\sqrt{x^2-3x+2}(x\in\mathbb{R})$ is________.
2024 District Olympiad, P4
Consider the functions $f,g:\mathbb{R}\to\mathbb{R}$ such that $f{}$ is continous. For any real numbers $a<b<c$ there exists a sequence $(x_n)_{n\geqslant 1}$ which converges to $b{}$ and for which the limit of $g(x_n)$ as $n{}$ tends to infinity exists and satisfies \[f(a)<\lim_{n\to\infty}g(x_n)<f(c).\][list=a]
[*]Give an example of a pair of such functions $f,g$ for which $g{}$ is discontinous at every point.
[*]Prove that if $g{}$ is monotonous, then $f=g.$
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2024 European Mathematical Cup, 4
Find all functions $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x+yf(x)) = xf(1+y)$
for all x, y positive reals.
1980 IMO, 3
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]
1974 All Soviet Union Mathematical Olympiad, 203
Given a function $f(x)$ on the segment $0\le x\le 1$. For all $x, f(x)\ge 0, f(1)=1$. For all the couples of $(x_1,x_2)$ such, that all the arguments are in the segment $$f(x_1+x_2)\ge f(x_1)+f(x_2).$$
a) Prove that for all $x$ holds $f(x) \le 2x$.
b) Is the inequality $f(x) \le 1.9x$ valid?
1993 Vietnam Team Selection Test, 3
Let $n$ points $A_1, A_2, \ldots, A_n$, ($n>2$), be considered in the space, where no four points are coplanar. Each pair of points $A_i, A_j$ are connected by an edge. Find the maximal value of $n$ for which we can paint all edges by two colors – blue and red such that the following three conditions hold:
[b]I.[/b] Each edge is painted by exactly one color.
[b]II.[/b] For $i = 1, 2, \ldots, n$, the number of blue edges with one end $A_i$ does not exceed 4.
[b]III.[/b] For every red edge $A_iA_j$, we can find at least one point $A_k$ ($k \neq i, j$) such that the edges $A_iA_k$ and $A_jA_k$ are blue.