Found problems: 85335
2020 Taiwan TST Round 1, 3
Let $N>2^{5000}$ be a positive integer. Prove that if $1\leq a_1<\cdots<a_k<100$ are distinct positive integers then the number
\[\prod_{i=1}^{k}\left(N^{a_i}+a_i\right)\]
has at least $k$ distinct prime factors.
Note. Results with $2^{5000}$ replaced by some other constant $N_0$ will be awarded points depending on the value of $N_0$.
[i]Proposed by Evan Chen[/i]
2003 Manhattan Mathematical Olympiad, 3
Two players play the following game, using a round table $4$ feet in diameter, and a large pile of quarters. Each player can put in his turn one quarter on the table, but the one who cannot put a quarter (because there is no free space on the table) loses the game. Is there a winning strategy for the first or for the second player?
2002 All-Russian Olympiad, 2
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $O$. The circumcircles of triangles $AOB$ and $COD$ intersect again at $K$. Point $L$ is such that the triangles $BLC$ and $AKD$ are similar and equally oriented. Prove that if the quadrilateral $BLCK$ is convex, then it is tangent [has an incircle].
2023 CMIMC Integration Bee, 1
\[\int_2^0 x^2+3\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2017 Bosnia And Herzegovina - Regional Olympiad, 3
Does there exist positive integer $n$ such that sum of all digits of number $n(4n+1)$ is equal to $2017$
2020 USOJMO, 6
Let $n \geq 2$ be an integer. Let $P(x_1, x_2, \ldots, x_n)$ be a nonconstant $n$-variable polynomial with real coefficients. Assume that whenever $r_1, r_2, \ldots , r_n$ are real numbers, at least two of which are equal, we have $P(r_1, r_2, \ldots , r_n) = 0$. Prove that $P(x_1, x_2, \ldots, x_n)$ cannot be written as the sum of fewer than $n!$ monomials. (A monomial is a polynomial of the form $cx^{d_1}_1 x^{d_2}_2\ldots x^{d_n}_n$, where $c$ is a nonzero real number and $d_1$, $d_2$, $\ldots$, $d_n$ are nonnegative integers.)
[i]Proposed by Ankan Bhattacharya[/i]
2011 Saudi Arabia BMO TST, 4
Let $(F_n )_{n\ge o}$ be the sequence of Fibonacci numbers: $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ , for every $n \ge 0$.
Prove that for any prime $p \ge 3$, $p$ divides $F_{2p} - F_p$ .
1990 Balkan MO, 4
Find the least number of elements of a finite set $A$ such that there exists a function $f : \left\{1,2,3,\ldots \right\}\rightarrow A$ with the property: if $i$ and $j$ are positive integers and $i-j$ is a prime number, then $f(i)$ and $f(j)$ are distinct elements of $A$.
1987 Putnam, B4
Let $(x_1,y_1) = (0.8, 0.6)$ and let $x_{n+1} = x_n \cos y_n - y_n \sin y_n$ and $y_{n+1}= x_n \sin y_n + y_n \cos y_n$ for $n=1,2,3,\dots$. For each of $\lim_{n\to \infty} x_n$ and $\lim_{n \to \infty} y_n$, prove that the limit exists and find it or prove that the limit does not exist.
1998 VJIMC, Problem 1
Let $a$ and $d$ be two positive integers. Prove that there exists a constant $K$ such that every set of $K$ consecutive elements of the arithmetic progression $\{a+nd\}_{n=1}^\infty$ contains at least one number which is not prime.
1990 IMO Longlists, 3
The integer $ 9$ can be written as a sum of two consecutive integers: $ 9 \equal{} 4\plus{}5.$ Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $ 9 \equal{} 4\plus{}5 \equal{} 2\plus{}3\plus{}4.$ Is there an integer that can be written as a sum of $ 1990$ consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly $ 1990$ ways?
2005 Portugal MO, 2
Consider the triangles $[ABC]$ and $[EDC]$, right at $A$ and $D$, respectively. Show that, if $E$ is the midpoint of $[AC]$, then $AB <BD$.
[img]https://cdn.artofproblemsolving.com/attachments/c/3/75bc1bda1a22bcf00d4fe7680c80a81a9aaa4c.png[/img]
1973 Miklós Schweitzer, 5
Verify that for every $ x > 0$, \[ \frac{\Gamma'(x\plus{}1)}{\Gamma (x\plus{}1)} > \log x.\]
[i]P. Medgyessy[/i]
1978 IMO Longlists, 44
In $ABC$ with $\angle C = 60^{\circ}$, prove that
\[\frac{c}{a} + \frac{c}{b} \ge2.\]
2005 Postal Coaching, 10
On the sides $AB$ and $BC$ of triangle $ABC$, points $K$ and $M$ are chosen such that the quadrilaterals $AKMC$ and $KBMN$ are cyclic , where $N = AM \cap CK$ . If these quads have the same circumradii, find $\angle ABC$
2009 Today's Calculation Of Integral, 441
Evaluate $ \int_1^e \frac{(x^2\ln x\minus{}1)e^x}{x}\ dx.$
2010 Indonesia TST, 1
Let $ ABCD$ be a trapezoid such that $ AB \parallel CD$ and assume that there are points $ E$ on the line outside the segment $ BC$ and $ F$ on the segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Let $ I,J,K$ respectively be the intersection of line $ EF$ and line $ CD$, the intersection of line $ EF$ and line $ AB$, and the midpoint of segment $ EF$. Prove that $ K$ is on the circumcircle of triangle $ CDJ$ if and only if $ I$ is on the circumcircle of triangle $ ABK$.
[i]Utari Wijayanti, Bandung[/i]
1984 Tournament Of Towns, (066) A5
Let $p(n)$ be the number of partitions of the natural number $n$ into natural summands. The diversity of a partition is by definition the number of different summands in it. Denote by $q(n)$ the sum of the diversities of all the $p(n) $ partitions of $n$.
(For example, $p(4) = 5$, the five distinct partitions of $4$ being $4, 3 + 1, 2+2, 2 + 1 + 1, 1 + 1 + 1 + 1,$ and $g(4) =1 + 2+1+ 2+1 = 7$.)
Prove that, for all natural numbers $n$,
(a) $q(n)= 1 + P(1) + P(2) + p(3) + ...+ p(n -1)$,
(b) $q(n) < \sqrt{2n} p(n)$.
(AV Zelevinskiy, Moscow)
2023 Malaysian IMO Training Camp, 3
A sequence of reals $a_1, a_2, \cdots$ satisfies for all $m>1$, $$a_{m+1}a_{m-1}=a_m^2-a_1^2$$ Prove that for all $m>n>1$, the sequence satisfies the equation $$a_{m+n}a_{m-n}=a_m^2-a_n^2$$
[i]Proposed by Ivan Chan Kai Chin[/i]
2023 CCA Math Bonanza, L4.2
A mouse is on the below grid:
\begin{center}
\begin{asy}
unitsize(2cm);
filldraw(circle((0,0),0.07), black);
filldraw(circle((0,1),0.07), black);
filldraw(circle((1,0),0.07), black);
filldraw(circle((0.5,0.5),0.07), black);
filldraw(circle((1,1),0.07), black);
draw((0,0)--(1,0));
draw((0,0)--(0,1));
draw((1,0)--(1,1));
draw((0,1)--(1,1));
draw((0,1)--(0.5,0.5));
draw((1,0)--(0.5,0.5));
draw((1,1)--(0.5,0.5));
draw((0,0)--(0.5,0.5));
\end{asy}
\end{center}
The paths connecting each node are the possible paths the mouse can take to walk from a node to another node. Call a ``turn" the action of a walk from one node to another. Given the mouse starts off on an arbitrary node, what is the expected number of turns it takes for the mouse to return to its original node?
[i]Lightning 4.2[/i]
1975 Vietnam National Olympiad, 4
Find all terms of the arithmetic progression $-1, 18, 37, 56, ...$ whose only digit is $5$.
2022 Greece National Olympiad, 1
Let $ABC$ be a triangle such that $AB<AC<BC$. Let $D,E$ be points on the segment $BC$ such that $BD=BA$ and $CE=CA$. If $K$ is the circumcenter of triangle $ADE$, $F$ is the intersection of lines $AD,KC$ and $G$ is the intersection of lines $AE,KB$, then prove that the circumcircle of triangle $KDE$ (let it be $c_1$), the circle with center the point $F$ and radius $FE$ (let it be $c_2$) and the circle with center $G$ and radius $GD$ (let it be $c_3$) concur on a point which lies on the line $AK$.
2015 Ukraine Team Selection Test, 6
Given an acute triangle $ABC, H$ is the foot of the altitude drawn from the point $A$ on the line $BC, P$ and $K \ne H$ are arbitrary points on the segments $AH$ and$ BC$ respectively. Segments $AC$ and $BP$ intersect at point $B_1$, lines $AB$ and $CP$ at point $C_1$. Let $X$ and $Y$ be the projections of point $H$ on the lines $KB_1$ and $KC_1$, respectively. Prove that points $A, P, X$ and $Y$ lie on one circle.
1994 All-Russian Olympiad, 6
I'll post some nice combinatorics problems here, taken from the wonderful training book "Les olympiades de mathmatiques" (in French) written by Tarik Belhaj Soulami.
Here goes the first one:
Let $\mathbb{I}$ be a non-empty subset of $\mathbb{Z}$ and let $f$ and $g$ be two functions defined on $\mathbb{I}$. Let $m$ be the number of pairs $(x,\;y)$ for which $f(x) = g(y)$, let $n$ be the number of pairs $(x,\;y)$ for which $f(x) = f(y)$ and let $k$ be the number of pairs $(x,\;y)$ for which $g(x) = g(y)$. Show that \[2m \leq n + k.\]
2000 Swedish Mathematical Competition, 6
Solve \[\left\{ \begin{array}{l} y(x+y)^2 = 9 \\
y(x^3-y^3) = 7 \\
\end{array} \right.
\]