Found problems: 85335
2016 India Regional Mathematical Olympiad, 3
$a, b, c, d$ are integers such that $ad + bc$ divides each of $a, b, c$ and $d$. Prove that $ad + bc =\pm 1$
2010 Tournament Of Towns, 2
At a circular track, $2n$ cyclists started from some point at the same time in the same direction with different constant speeds. If any two cyclists are at some point at the same time again, we say that they meet. No three or more of them have met at the same time. Prove that by the time every two cyclists have met at least once, each cyclist has had at least $n^2$ meetings.
2010 National Chemistry Olympiad, 19
Which process is exothermic?
$ \textbf{(A)}\hspace{.05in}\text{condensation} \qquad\textbf{(B)}\hspace{.05in}\text{fusion} \qquad\textbf{(C)}\hspace{.05in}\text{sublimation} \qquad\textbf{(D)}\hspace{.05in}\text{vaporization} \qquad $
2012 India PRMO, 4
The letters $R, M$, and $O$ represent whole numbers. If $R \times M \times O = 240$, $R \times O + M =46$ and $R + M \times O = 64$, what is the value of $R + M + O$?
2016 AIME Problems, 11
Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$, and $\left(P(2)\right)^2 = P(3)$. Then $P(\tfrac72)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
VI Soros Olympiad 1999 - 2000 (Russia), 10.1
Let's call the "Soros product" of two different numbers, $a$ and $b$, the number $a + b + ab$. Is it possible, based on numbers $1$ and $4$, after repeated application of this operation to the already obtained products, to obtain:
a) the number $1999$?
b) the number $2000$?
2005 Baltic Way, 5
Let $a$, $b$, $c$ be positive real numbers such that $abc=1$. Prove that
\[\frac a{a^{2}+2}+\frac b{b^{2}+2}+\frac c{c^{2}+2}\leq 1 \]
2013 Online Math Open Problems, 28
Let $n$ denote the product of the first $2013$ primes. Find the sum of all primes $p$ with $20 \le p \le 150$ such that
(i) $\frac{p+1}{2}$ is even but is not a power of $2$, and
(ii) there exist pairwise distinct positive integers $a,b,c$ for which \[ a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b) \] is divisible by $p$ but not $p^2$.
[i]Proposed by Evan Chen[/i]
2001 Turkey Team Selection Test, 3
For all integers $x,y,z$, let \[S(x,y,z) = (xy - xz, yz-yx, zx - zy).\] Prove that for all integers $a$, $b$ and $c$ with $abc>1$, and for every integer $n\geq n_0$, there exists integers $n_0$ and $k$ with $0<k\leq abc$ such that \[S^{n+k}(a,b,c) \equiv S^n(a,b,c) \pmod {abc}.\] ($S^1 = S$ and for every integer $m\geq 1$, $S^{m+1} = S \circ S^m.$
$(u_1, u_2, u_3) \equiv (v_1, v_2, v_3) \pmod M \Longleftrightarrow u_i \equiv v_i \pmod M (i=1,2,3).$)
2014 CHMMC (Fall), 3
Two players play a game on a pile of $n$ beans. On each player's turn, they may take exactly $1$, $4$, or $7$ beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many $n$ between $2014$ and $2050$ (inclusive) does the second player win?
EGMO 2017, 1
Let $ABCD$ be a convex quadrilateral with $\angle DAB=\angle BCD=90^{\circ}$ and $\angle ABC> \angle CDA$. Let $Q$ and $R$ be points on segments $BC$ and $CD$, respectively, such that line $QR$ intersects lines $AB$ and $AD$ at points $P$ and $S$, respectively. It is given that $PQ=RS$.Let the midpoint of $BD$ be $M$ and the midpoint of $QR$ be $N$.Prove that the points $M,N,A$ and $C$ lie on a circle.
2001 China Team Selection Test, 2
$a$ and $b$ are natural numbers such that $b > a > 1$, and $a$ does not divide $b$. The sequence of natural numbers $\{b_n\}_{n=1}^\infty$ satisfies $b_{n + 1} \geq 2b_n \forall n \in \mathbb{N}$. Does there exist a sequence $\{a_n\}_{n=1}^\infty$ of natural numbers such that for all $n \in \mathbb{N}$, $a_{n + 1} - a_n \in \{a, b\}$, and for all $m, l \in \mathbb{N}$ ($m$ may be equal to $l$), $a_m + a_l \not\in \{b_n\}_{n=1}^\infty$?
1981 Romania Team Selection Tests, 6.
In the plane of traingle $ABC$ we consider a variable point $M$ which moves on line $MA$ towards $A$. Halfway there, it stops and starts moving in a straight line line towards $B$. Halfway there, it stops and starts moving in a straight line towards $C$, and halfway there it stops and starts moving in a straight line towards $A$, and so on. Show that $M$ will get as close as we want to the vertices of a fixed triangle with area $\text{area}(ABC)/7$.
2010 Contests, 4
(a) Determine all pairs $(x, y)$ of (real) numbers with $0 < x < 1$ and $0 <y < 1$ for which $x + 3y$ and $3x + y$ are both integer. An example is $(x,y) =( \frac{8}{3}, \frac{7}{8}) $, because $ x+3y =\frac38 +\frac{21}{8} =\frac{24}{8} = 3$ and $ 3x+y = \frac98 + \frac78 =\frac{16}{8} = 2$.
(b) Determine the integer $m > 2$ for which there are exactly $119$ pairs $(x,y)$ with $0 < x < 1$ and $0 < y < 1$ such that $x + my$ and $mx + y$ are integers.
Remark: if $u \ne v,$ the pairs $(u, v)$ and $(v, u)$ are different.
2023 India National Olympiad, 3
Let $\mathbb N$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f:\mathbb N\to \mathbb N$ satisfying:
[list]
[*] for any $x,a\in\mathbb N$, the quantity $\frac{f(x+a)-f(x)}{a}$ is an integer if and only if $a=1$;
[*] for all $x\in \mathbb N$, we have $|f(x)-cx|<2023$.
[/list]
[i]Proposed by Sutanay Bhattacharya[/i]
Putnam 1939, A7
Do either $(1)$ or $(2)$:
$(1)$ Let $C_a$ be the curve $(y - a^2)^2 = x^2(a^2 - x^2).$ Find the curve which touches all $C_a$ for $a > 0.$ Sketch the solution and at least two of the $C_a.$
$(2)$ Given that $(1 - hx)^{-1}(1 - kx)^{-1} = \sum_{i\geq0}a_i x^i,$ prove that $(1 + hkx)(1 - hkx)^{-1}(1 - h^2x)^{-1}(1 - k^2x)^{-1} = \sum_{i\geq0} a_i^2 x^i.$
2020 Estonia Team Selection Test, 1
Let $a_1, a_2,...$ a sequence of real numbers.
For each positive integer $n$, we denote $m_n =\frac{a_1 + a_2 +... + a_n}{n}$.
It is known that there exists a real number $c$ such that for any different positive integers $i, j, k$: $(i - j) m_k + (j - k) m_i + (k - i) m_j = c$.
Prove that the sequence $a_1, a_2,..$ is arithmetic
2015 Romania Team Selection Tests, 3
If $k$ and $n$ are positive integers , and $k \leq n$ , let $M(n,k)$ denote the least common multiple of the numbers $n , n-1 , \ldots , n-k+1$.Let $f(n)$ be the largest positive integer $ k \leq n$ such that $M(n,1)<M(n,2)<\ldots <M(n,k)$ . Prove that :
[b](a)[/b] $f(n)<3\sqrt{n}$ for all positive integers $n$ .
[b](b)[/b] If $N$ is a positive integer , then $f(n) > N$ for all but finitely many positive integers $n$.
2019 Bangladesh Mathematical Olympiad, 4
$A$ is a positive real number.$n$ is positive integer number.Find the set of possible values of the infinite sum $x_0^n+x_1^n+x_2^n+...$ where $x_0,x_1,x_2...$ are all positive real numbers so that the infinite series $x_0+x_1+x_2+...$ has sum $A$.
2006 Estonia National Olympiad, 3
The sequence $ (F_n)$ of Fibonacci numbers satisfies $ F_1 \equal{} 1, F_2 \equal{} 1$ and $ F_n \equal{} F_{n\minus{}1} \plus{}F_{n\minus{}2}$ for all $ n \ge 3$. Find all pairs of positive integers $ (m, n)$, such that $ F_m . F_n \equal{} mn$.
2010 Thailand Mathematical Olympiad, 3
Let $\vartriangle ABC$ be a scalene triangle with $AB < BC < CA$. Let $D$ be the projection of $A$ onto the angle bisector of $\angle ABC$, and let $E$ be the projection of $A$ onto the angle bisector of $\angle ACB$. The line $DE$ cuts sides $AB$ and AC at $M$ and $N$, respectively. Prove that $$\frac{AB+AC}{BC} =\frac{DE}{MN} + 1$$
2014 Iran Team Selection Test, 6
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC$ at $D$.
let $X$ is a point on arc $BC$ from circumcircle of triangle $ABC$ such that if $E,F$ are feet of perpendicular from $X$ on $BI,CI$ and $M$ is midpoint of $EF$ we have $MB=MC$.
prove that $\widehat{BAD}=\widehat{CAX}$
2023 Paraguay Mathematical Olympiad, 1
In the following sequence of numbers, each term, starting with the third, is obtained by adding three times the previous term plus twice the previous term to the previous one:
$$a_1, a_2, 78, a_4, a_5, 3438, a_7, a_8,…$$
As seen in the sequence, the third term is $78$ and the sixth term is $3438$. What is the value of the term $a_7$?
2014 Romania Team Selection Test, 1
Let $ABC$ a triangle and $O$ his circumcentre.The lines $OA$ and $BC$ intersect each other at $M$ ; the points $N$ and $P$ are defined in an analogous way.The tangent line in $A$ at the circumcircle of triangle $ABC$ intersect $NP$ in the point $X$ ; the points $Y$ and $Z$ are defined in an analogous way.Prove that the points $X$ , $Y$ and $Z$ are collinear.
2010 Contests, 1
There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors.
P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.