Found problems: 85335
2015 AMC 8, 23
Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are 2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4, and 4.5. If a slip with 2 goes into cup $E$ and a slip with 3 goes into cup $B$, then the slip with 3.5 must go into what cup?
$
\textbf{(A) } A \qquad
\textbf{(B) } B \qquad
\textbf{(C) } C \qquad
\textbf{(D) } D \qquad
\textbf{(E) } E
$
Kvant 2019, M2547
The circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ are externally tangent at the point $T$. The circle $\omega_3$ centered at $O_3$ is tangent to the line $AB$ (the external common tangent of $\omega_1$ and $\omega_2$) at $D$ and externally tangent to $\omega_1$ and to $\omega_2$. The line $TD$ intersects again at $\omega_1$. Prove that $O_1 C \parallel AB$.
[I]Proposed by V. Rastorguev[/I]
2020 BMT Fall, Tie 5
The polynomial $f(x) = x^3 + rx^2 + sx + t$ has $r, s$, and $t$ as its roots (with multiplicity), where $f(1)$ is rational and $ t \ne 0$. Compute $|f(0)|$.
2008 Brazil Team Selection Test, 2
Find all polynomials $P (x)$ with complex coefficients such that $$P (x^2) = P (x) · P (x + 2)$$
for any complex number $x.$
2024 Durer Math Competition Finals, 2
For every subset $\mathcal{P}$ of the plane let $S(\mathcal{P})$ denote the set of circles and lines that intersect $\mathcal{P}$ in at least three points. Find all sets $\mathcal{P}$ consisting of 2024 points such that for any two distinct elements of $S(\mathcal{P}),$ their intersection points all belong to $\mathcal{P}{}.$
2022 Middle European Mathematical Olympiad, 3
Let $n$ be a positive integer. There are $n$ purple and $n$ white cows queuing in a line in some order. Tim wishes to sort the cows by colour, such that all purple cows are at the front of the line. At each step, he is only allowed to swap two adjacent groups of equally many consecutive cows. What is the minimal number of steps Tim needs to be able to fulfill his wish, regardless of the initial alignment of the cows?
2001 239 Open Mathematical Olympiad, 5
The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ K $ be the midpoint of the chord cut by the line $ AB $ on circles $ S_3 $. Prove that $ \angle CKA = \angle DKA $.
2021 Vietnam TST, 3
Let $ABC$ be a triangle and $N$ be a point that differs from $A,B,C$. Let $A_b$ be the reflection of $A$ through $NB$, and $B_a$ be the reflection of $B$ through $NA$. Similarly, we define $B_c, C_b, A_c, C_a$. Let $m_a$ be the line through $N$ and perpendicular to $B_cC_b$. Define similarly $m_b, m_c$.
a) Assume that $N$ is the orthocenter of $\triangle ABC$, show that the respective reflection of $m_a, m_b, m_c$ through the bisector of angles $\angle BNC, \angle CNA, \angle ANB$ are the same line.
b) Assume that $N$ is the nine-point center of $\triangle ABC$, show that the respective reflection of $m_a, m_b, m_c$ through $BC, CA, AB$ concur.
1987 All Soviet Union Mathematical Olympiad, 441
Ten sportsmen have taken part in a table-tennis tournament (each pair has met once only, no draws). Let $xi$ be the number of $i$-th player victories, $yi$ -- losses. Prove that $$x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2$$
2017 Pakistan TST, Problem 3
Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for all distinct $x,y,z$
$f(x)^2-f(y)f(z)=f(x^y)f(y)f(z)[f(y^z)-f(z^x)]$
2018 Poland - Second Round, 4
Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Circle of diameter $BC$ is tangent to line $AD$. Prove, that circle of diameter $AD$ is tangent to line $BC$.
IV Soros Olympiad 1997 - 98 (Russia), 10.5
At the base of the triangular pyramid $ABCD$ lies a regular triangle $ABC$ such that $AD = BC$. All plane angles at vertex $B$ are equal to each other. What might these angles be equal to?
2004 Federal Math Competition of S&M, 3
If $a,b,c$ are positive numbers such that $abc = 1$, prove the inequality
$\frac{1}{\sqrt{b+\frac{1}{a}+\frac{1}{2}}} + \frac{1}{\sqrt{c+\frac{1}{b}+\frac{1}{2}}} + \frac{1}{\sqrt{a+\frac{1}{c}+\frac{1}{2}}} \geq \sqrt{2}$
2005 South africa National Olympiad, 6
Consider the increasing sequence $1,2,4,5,7,9,10,12,14,16,17,19,\dots$ of positive integers, obtained by concatenating alternating blocks $\{1\},\{2,4\},\{5,7,9\},\{10,12,14,16\},\dots$ of odd and even numbers. Each block contains one more element than the previous one and the first element in each block is one more than the last element of the previous one. Prove that the $n$-th element of the sequence is given by \[2n-\Big\lfloor\frac{1+\sqrt{8n-7}}{2}\Big\rfloor.\]
(Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.)
2021 USA TSTST, 3
Find all positive integers $k > 1$ for which there exists a positive integer $n$ such that $\tbinom{n}{k}$ is divisible by $n$, and $\tbinom{n}{m}$ is not divisible by $n$ for $2\leq m < k$.
[i]Merlijn Staps[/i]
2009 India National Olympiad, 4
All the points in the plane are colored using three colors.Prove that there exists a triangle with vertices having the same color such that [i]either[/i] it is isosceles [i]or[/i] its angles are in geometric progression.
1986 IMO Longlists, 25
Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations:
\[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\]
Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$
1995 IMC, 2
Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$,
we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.
2010 Indonesia TST, 2
Consider a polynomial with coefficients of real numbers $ \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3\plus{}9a^2d\minus{}7abc \le 0.\]
[i]Hery Susanto, Malang[/i]
1988 AMC 12/AHSME, 26
Suppose that $p$ and $q$ are positive numbers for which \[ \log_{9}(p) = \log_{12}(q) = \log_{16}(p+q) \] What is the value of $\frac{q}{p}$?
$\textbf{(A)}\ \frac{4}{3}\qquad\textbf{(B)}\ \frac{1+\sqrt{3}}{2}\qquad\textbf{(C)}\ \frac{8}{5}\qquad\textbf{(D)}\ \frac{1+\sqrt{5}}{2}\qquad\textbf{(E)}\ \frac{16}{9} $
2003 Belarusian National Olympiad, 6
a) A positive integer is called [i]nice [/i] if it can be represented as an arithmetic mean of some (not necessarily distinct) positive integers each being a nonnegative power of $2$.
Prove that all positive integers are nice.
b) A positive integer is called [i]ugly [/i] if it can not be represented as an arithmetic mean of some pairwise distinct positive integers each being a nonnegative power of $2$.
Prove that there exist infinitely many ugly positive integers.
(A. Romanenko, D. Zmeikov)
1976 IMO, 3
A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where $[x]$ denotes the smallest integer $\leq x)$
2024 Vietnam National Olympiad, 1
For each real number $x$, let $\lfloor x \rfloor$ denote the largest integer not exceeding $x$.
A sequence $\{a_n \}_{n=1}^{\infty}$ is defined by $a_n = \frac{1}{4^{\lfloor -\log_4 n \rfloor}}, \forall n \geq 1.$ Let $b_n = \frac{1}{n^2} \left( \sum_{k=1}^n a_k - \frac{1}{a_1+a_2} \right), \forall n \geq 1.$
a) Find a polynomial $P(x)$ with real coefficients such that $b_n = P \left( \frac{a_n}{n} \right), \forall n \geq 1$.
b) Prove that there exists a strictly increasing sequence $\{n_k \}_{k=1}^{\infty}$ of positive integers such that $$\lim_{k \to \infty} b_{n_k} = \frac{2024}{2025}.$$
2024 Turkey MO (2nd Round), 1
Let $n\ge3$ be a positive integer. Each edge of a complete graph $K_n$ is assigned a real number satisfying the following conditions:
$(i)$ For any three vertices, the numbers assigned to two of the edges among them are equal, and the number on the third edge is strictly greater.
$(ii) $ The weight of a vertex is defined as the sum of the numbers assigned to the edges emanating from that vertex. The weights of all vertices are equal.
Find all possible values of $n$.
2016 Oral Moscow Geometry Olympiad, 1
The line passing through the center $I$ of the inscribed circle of a triangle $ABC$, perpendicular to $AI$ and intersects sides $AB$ and $AC$ at points $C'$ and $B'$, respectively. In the triangles $BC'I$ and $CB'I$, the altitudes $C'C_1$ and $B'B_1$ were drawn, respectively. Prove that the midpoint of the segment $B_1C_1$ lies on a straight line passing through point $I$ and perpendicular to $BC$.