Found problems: 85335
2016 Azerbaijan JBMO TST, 3
All cells of the $m\times n$ table are colored either white or black such that all corner cells of any rectangle containing the cells of this table with sides greater than one cell are not the same color. For values $m = 2, 3, 4,$ find all $n$ such that the mentioned coloring is possible.
2008 Indonesia TST, 3
Let $n$ be an arbitrary positive integer.
(a) For every positive integers $a$ and $b$, show that $gcd(n^a + 1, n^b + 1) \le n^{gcd(a,b)} + 1$.
(b) Show that there exist infinitely many composite pairs ($a, b)$, such that each of them is not a multiply of the other number and equality holds in (a).
2021 CCA Math Bonanza, T3
For any real number $x$, we let $\lfloor x \rfloor$ be the unique integer $n$ such that $n \leq x < n+1$. For example. $\lfloor 31.415 \rfloor = 31$. Compute \[2020^{2021} - \left\lfloor\frac{2020^{2021}}{2021} \right \rfloor (2021).\]
[i]2021 CCA Math Bonanza Team Round #3[/i]
2021 JHMT HS, 4
There is a unique differentiable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ satisfying $f(x) + (f(x))^3 = x + x^7$ for all real $x.$ The derivative of $f(x)$ at $x = 2$ can be expressed as a common fraction $a/b.$ Compute $a + b.$
2006 Croatia Team Selection Test, 3
Let $ABC$ be a triangle for which $AB+BC = 3AC$. Let $D$ and $E$ be the points of tangency of the incircle with the sides $AB$ and $BC$ respectively, and let $K$ and $L$ be the other endpoints of the diameters originating from $D$ and $E.$ Prove that $C , A, L$, and $K$ lie on a circle.
2021 Moldova EGMO TST, 2
In triangle $ABC$ point $M$ is on side $AB$ such that $AM:AB=3:4$ and point $P$ is on side $BC$ such that $CP:CB=3:8$. Point $N$ is symmetric to $A$ with respect to point $P$. Prove that lines $MN$ and $AC$ are parallel.
2016 Kosovo National Mathematical Olympiad, 1
Find all three digit numbers such that the square of that number is equal to the sum of their digits in power of $5$ .
2012 CHMMC Fall, Individual
[b]p1.[/b] How many nonzero digits are in the number $(5^{94} + 5^{92})(2^{94} + 2^{92})$?
[b]p2.[/b] Suppose $A$ is a set of $2013$ distinct positive integers such that the arithmetic mean of any subset of $A$ is also an integer. Find an example of $A$.
[b]p3.[/b] How many minutes until the smaller angle formed by the minute and hour hands on the face of a clock is congruent to the smaller angle between the hands at $5:15$ pm? Round your answer to the nearest minute.
[b]p4.[/b] Suppose $a$ and $b$ are positive real numbers, $a + b = 1$, and $$1 +\frac{a^2 + 3b^2}{2ab}=\sqrt{4 +\frac{a}{b}+\frac{3b}{a}}.$$ Find $a$.
[b]p5.[/b] Suppose $f(x) = \frac{e^x- 12e^{-x}}{ 2}$ . Find all $x$ such that $f(x) = 2$.
[b]p6.[/b] Let $P_1$, $P_2$,$...$,$P_n$ be points equally spaced on a unit circle. For how many integer $n \in \{2, 3, ... , 2013\}$ is the product of all pairwise distances: $\prod_{1\le i<j\le n} P_iP_j$ a rational number?
Note that $\prod$ means the product. For example, $\prod_{1\le i\le 3} i = 1\cdot 2 \cdot 3 = 6$.
[b]p7.[/b] Determine the value $a$ such that the following sum converges if and only if $r \in (-\infty, a)$ :
$$\sum^{\infty}_{n=1}(\sqrt{n^4 + n^r} - n^2).$$
Note that $\sum^{\infty}_{n=1}\frac{1}{n^s}$ converges if and only if $s > 1$.
[b]p8.[/b] Find two pairs of positive integers $(a, b)$ with $a > b$ such that $a^2 + b^2 = 40501$.
[b]p9.[/b] Consider a simplified memory-knowledge model. Suppose your total knowledge level the night before you went to a college was $100$ units. Each day, when you woke up in the morning you forgot $1\%$ of what you had learned. Then, by going to lectures, working on the homework, preparing for presentations, you had learned more and so your knowledge level went up by $10$ units at the end of the day.
According to this model, how long do you need to stay in college until you reach the knowledge level of exactly $1000$?
[b]p10.[/b] Suppose $P(x) = 2x^8 + x^6 - x^4 +1$, and that $P$ has roots $a_1$, $a_2$, $...$ , $a_8$ (a complex number $z$ is a root of the polynomial $P(x)$ if $P(z) = 0$). Find the value of $$(a^2_1-2)(a^2_2-2)(a^2_3-2)...(a^2_8-2).$$
[b]p11.[/b] Find all values of $x$ satisfying $(x^2 + 2x-5)^2 = -2x^2 - 3x + 15$.
[b]p12.[/b] Suppose $x, y$ and $z$ are positive real numbers such that
$$x^2 + y^2 + xy = 9,$$
$$y^2 + z^2 + yz = 16,$$
$$x^2 + z^2 + xz = 25.$$
Find $xy + yz + xz$ (the answer is unique).
[b]p13.[/b] Suppose that $P(x)$ is a monic polynomial (i.e, the leading coefficient is $1$) with $20$ roots, each distinct and of the form $\frac{1}{3^k}$ for $k = 0,1,2,..., 19$. Find the coefficient of $x^{18}$ in $P(x)$.
[b]p14.[/b] Find the sum of the reciprocals of all perfect squares whose prime factorization contains only powers of $3$, $5$, $7$ (i.e. $\frac{1}{1} + \frac{1}{9} + \frac{1}{25} + \frac{1}{419} + \frac{1}{811} + \frac{1}{215} + \frac{1}{441} + \frac{1}{625} + ...$).
[b]p15.[/b] Find the number of integer quadruples $(a, b, c, d)$ which also satisfy the following system of equations:
$$1+b + c^2 + d^3 =0,$$ $$a + b^2 + c^3 + d^4 =0,$$ $$a^2 + b^3 + c^4 + d^5 =0,$$ $$a^3+b^4+c^5+d^6 =0.$$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2009 Kazakhstan National Olympiad, 4
Let $a,b,c,d $-reals positive numbers. Prove inequality:
$\frac{a^2+b^2+c^2}{ab+bc+cd}+\frac{b^2+c^2+d^2}{bc+cd+ad}+\frac{a^2+c^2+d^2}{ab+ad+cd}+\frac{a^2+b^2+d^2}{ab+ad+bc} \geq 4$
1967 AMC 12/AHSME, 33
[asy]
fill(circle((4,0),4),grey);
fill((0,0)--(8,0)--(8,-4)--(0,-4)--cycle,white);
fill(circle((7,0),1),white);
fill(circle((3,0),3),white);
draw((0,0)--(8,0),black+linewidth(1));
draw((6,0)--(6,sqrt(12)),black+linewidth(1));
MP("A", (0,0), W); MP("B", (8,0), E); MP("C", (6,0), S); MP("D",(6,sqrt(12)), N);
[/asy]
In this diagram semi-circles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, so that they are mutually tangent. If $\overline{CD} \bot \overline{AB}$, then the ratio of the shaded area to the area of a circle with $\overline{CD}$ as radius is:
$\textbf{(A)}\ 1:2\qquad
\textbf{(B)}\ 1:3\qquad
\textbf{(C)}\ \sqrt{3}:7\qquad
\textbf{(D)}\ 1:4\qquad
\textbf{(E)}\ \sqrt{2}:6$
2024 May Olympiad, 4
Let $ABCD$ be a convex quadrilateral and let $M$, $N$, $P$ and $Q$ be the midpoints of the sides $AB$, $CD$, $BC$ and $DA$ respectively. The line $MN$ intersects the segments $AP$ and $CQ$ at points $X$ and $Y$, respectively. Suppose that $MX = NY$. Prove that $\text{area}(ABCD) = 4 \cdot \text{area}(BXDY).$
1989 Poland - Second Round, 1
Solve the equation
$$ tg 7x - \sin 6x=\cos 4x - ctg 7x.$$
2020 China National Olympiad, 4
Find the largest positive constant $C$ such that the following is satisfied: Given $n$ arcs (containing their endpoints) $A_1,A_2,\ldots ,A_n$ on the circumference of a circle, where among all sets of three arcs $(A_i,A_j,A_k)$ $(1\le i< j< k\le n)$, at least half of them has $A_i\cap A_j\cap A_k$ nonempty, then there exists $l>Cn$, such that we can choose $l$ arcs among $A_1,A_2,\ldots ,A_n$, whose intersection is nonempty.
2011 N.N. Mihăileanu Individual, 2
Let be three real numbers $ x,y,z>1 $ that satisfy $ xyz=8. $ Prove that:
$$ \left( \sqrt{\log_2 x} +\sqrt{\log_2 y} \right)\cdot \left( \sqrt{\log_2 y} +\sqrt{\log_2 z} \right)\cdot \left( \sqrt{\log_2 z} +\sqrt{\log_2 x} \right)\le 8 $$
[i]Gabriela Constantinescu[/i]
Novosibirsk Oral Geo Oly IX, 2022.7
Altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through point $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.
2004 German National Olympiad, 5
Prove that for four positive real numbers $a,b,c,d$ the following inequality holds and find all equality cases:
$$a^3 +b^3 +c^3 +d^3 \geq a^2 b +b^2 c+ c^2 d +d^2 a.$$
2020 Romanian Master of Mathematics Shortlist, N1
Determine all pairs of positive integers $(m, n)$ for which there exists a bijective function \[f : \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_m \times \mathbb{Z}_n\]such that the vectors $f(\mathbf{v}) + \mathbf{v}$, as $\mathbf{v}$ runs through all of $\mathbb{Z}_m \times \mathbb{Z}_n$, are pairwise distinct.
(For any integers $a$ and $b$, the vectors $[a, b], [a + m, b]$ and $[a, b + n]$ are treated as equal.)
[i]Poland, Wojciech Nadara[/i]
2009 Puerto Rico Team Selection Test, 1
By the time a party is over, $ 28$ handshakes have occurred. If everyone shook everyone else's hand once, how many people attended the party?
2004 Thailand Mathematical Olympiad, 2
Let $a$ and $b$ be real numbers such that $$\begin{cases} a^6 - 3a^2b^4 = 3 \\ b^6 - 3a^4b^2 = 3\sqrt2.\end{cases}$$ What is the value of $a^4 + b^4$ ?
2006 Serbia Team Selection Test, 2
$$problem 2$$:A point $P$ is taken in the interior of a right triangle$ ABC$ with $\angle C = 90$ such hat
$AP = 4, BP = 2$, and$ CP = 1$. Point $Q$ symmetric to $P$ with respect to $AC$ lies on
the circumcircle of triangle $ABC$. Find the angles of triangle $ABC$.
2009 South East Mathematical Olympiad, 1
Find all pairs ($x,y$) of integers such that $x^2-2xy+126y^2=2009$.
2011 India IMO Training Camp, 3
Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that
\[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\]
for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that
\[a_{k+2011}=a_{k+2011^{2011}}.\]
2004 Argentina National Olympiad, 2
Determine all positive integers $a,b,c,d$ such that$$\begin{cases} a<b \\ a^2c =b^2d \\ ab+cd =2^{99}+2^{101} \end{cases}$$
2025 239 Open Mathematical Olympiad, 7
Given a $2025 \times 2025$ board and $k$ chips lying on the table. Initially, the board is empty. It is allowed to place a chip from the table on any free square $A$ if two conditions are met simultaneously:
– the cell next to $A$ from below has a chip (or $A$ is on the bottom edge of the board);
– the cell next to $A$ on the left has a chip (or $A$ is on the left edge of the board).
In addition, it is allowed to remove any chip from the board and put it on the table. At what minimum $k$ can a chip appear in the upper-right corner of the board?
1994 Moldova Team Selection Test, 3
Triangles $MAB{}$ and $MA_1B_1{}$ are similar and have the same orientation. Prove that the circumcircles of these triangles cointain the intersection point of lines $AA_1{}$ and $BB_1{}$.