Found problems: 85335
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 $BB_1{}$.
2013 Stanford Mathematics Tournament, 5
An unfair coin lands heads with probability $\tfrac1{17}$ and tails with probability $\tfrac{16}{17}$. Matt flips the coin repeatedly until he flips at least one head and at least one tail. What is the expected number of times that Matt flips the coin?