Found problems: 85335
1986 Kurschak Competition, 1
Any two members of a club with $3n+1$ people plays ping-pong, tennis or chess with each other. Everyone has exactly $n$ partners who plays ping-pong, $n$ who play tennis and $n$ who play chess.
Prove that we can choose three members of the club who play three different games amongst each other.
1993 APMO, 2
Find the total number of different integer values the function \[ f(x) = [x] + [2x] + [\frac{5x}{3}] + [3x] + [4x] \] takes for real numbers $x$ with $0 \leq x \leq 100$.
1987 AMC 8, 14
A computer can do $10,000$ additions per second. How many additions can it do in one hour?
$\text{(A)}\ 6\text{ million} \qquad \text{(B)}\ 36\text{ million} \qquad \text{(C)}\ 60\text{ million} \qquad \text{(D)}\ 216\text{ million} \qquad \text{(E)}\ 360\text{ million}$
2014 AIME Problems, 2
An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.
2004 China National Olympiad, 2
Let $c$ be a positive integer. Consider the sequence $x_1,x_2,\ldots$ which satisfies $x_1=c$ and, for $n\ge 2$,
\[x_n=x_{n-1}+\left\lfloor\frac{2x_{n-1}-(n+2)}{n}\right\rfloor+1\]
where $\lfloor x\rfloor$ denotes the largest integer not greater than $x$. Determine an expression for $x_n$ in terms of $n$ and $c$.
[i]Huang Yumin[/i]
2013 IFYM, Sozopol, 6
Prove that for each natural number $k$ there exists a natural number $n(k)$, such that for each $m\geq n(k)$ and each set $M$ of $m$ points in the plane, there can be chosen $k$ triangles, so that each has an angle greater than $120^\circ$.
2007 Iran Team Selection Test, 1
In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position
[i]By Sam Nariman[/i]
2022 AMC 12/AHSME, 3
Five rectangles, $A$, $B$, $C$, $D$, and $E$, are arranged in a square as shown below. These rectangles have dimensions $1\times6$, $2\times4$, $5\times6$, $2\times7$, and $2\times3$, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle?
[asy]
fill((3,2.5)--(3,4.5)--(5.3,4.5)--(5.3,2.5)--cycle,mediumgray);
draw((0,0)--(7,0)--(7,7)--(0,7)--(0,0));
draw((3,0)--(3,4.5));
draw((0,4.5)--(5.3,4.5));
draw((5.3,7)--(5.3,2.5));
draw((7,2.5)--(3,2.5));
[/asy]
$\textbf{(A) }A\qquad\textbf{(B) }B \qquad\textbf{(C) }C \qquad\textbf{(D) }D\qquad\textbf{(E) }E$
2021 USAJMO, 4
Carina has three pins, labeled $A, B$, and $C$, respectively, located at the origin of the coordinate plane. In a [i]move[/i], Carina may move a pin to an adjacent lattice point at distance $1$ away. What is the least number of moves that Carina can make in order for triangle $ABC$ to have area 2021?
(A lattice point is a point $(x, y)$ in the coordinate plane where $x$ and $y$ are both integers, not necessarily positive.)
2006 Polish MO Finals, 2
Tetrahedron $ABCD$ in which $AB=CD$ is given. Sphere inscribed in it is tangent to faces $ABC$ and $ABD$ respectively in $K$ and $L$. Prove that if points $K$ and $L$ are centroids of faces $ABC$ and $ABD$ then tetrahedron $ABCD$ is regular.
2023 All-Russian Olympiad, 1
Given are two monic quadratics $f(x), g(x)$ such that $f, g, f+g$ have two distinct real roots. Suppose that the difference of the roots of $f$ is equal to the difference of the roots of $g$. Prove that the difference of the roots of $f+g$ is not bigger than the above common difference.
2000 Romania National Olympiad, 1
Let be two natural primes $ 1\le q \le p. $ Prove that $ \left( \sqrt{p^2+q} +p\right)^2 $ is irrational and its fractional part surpasses $ 3/4. $
1992 India Regional Mathematical Olympiad, 2
If $\frac{1}{a} + \frac{1}{b} = \frac{1}{c}$, where $a,b,c$ are positive integers with no common factor, prove that $(a +b)$ is a square.
2017 Princeton University Math Competition, A4/B6
Let the sequence $a_{1}, a_{2}, \cdots$ be defined recursively as follows: $a_{n}=11a_{n-1}-n$. If all terms of the sequence are positive, the smallest possible value of $a_{1}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
2001 Denmark MO - Mohr Contest, 2
If there is a natural number $n$ such that the number $n!$ has exactly $11$ zeros at the end?
(With $n!$ is denoted the number $1\cdot 2\cdot 3 \cdot ... (n - )1 \cdot n$).
2024 Sharygin Geometry Olympiad, 1
Bisectors $AI$ and $CI$ meet the circumcircle of triangle $ABC$ at points $A_1, C_1$ respectively.
The circumcircle of triangle $AIC_1$ meets $AB$ at point $C_0$; point $A_0$ is defined similarly.
Prove that $A_0, A_1, C_0, C_1$ are collinear.
2025 Harvard-MIT Mathematics Tournament, 13
A number is [i]upwards[/i] if its digits in base $10$ are nondecreasing when read from left to right. Compute the number of positive integers less than $10^6$ that are both upwards and multiples of $11.$
1986 Polish MO Finals, 6
$ABC$ is a triangle. The feet of the perpendiculars from $B$ and $C$ to the angle bisector at $A$ are $K, L$ respectively. $N$ is the midpoint of $BC$, and $AM$ is an altitude. Show that $K,L,N,M$ are concyclic.
2002 National Olympiad First Round, 19
How many positive integers $A$ are there such that if we append $3$ digits to the rightmost of decimal representation of $A$, we will get a number equal to $1+2+\cdots + A$?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 2002
\qquad\textbf{e)}\ \text{None of above}
$
2007 Greece JBMO TST, 3
Let $ABCD$ be a rectangle with $AB=a >CD =b$. Given circles $(K_1,r_1) , (K_2,r_2)$ with $r_1<r_2$ tangent externally at point $K$ and also tangent to the sides of the rectangle, circle $(K_1,r_1)$ tangent to both $AD$ and $AB$, circle $(K_2,r_2)$ tangent to both $AB$ and $BC$. Let also the internal common tangent of those circles pass through point $D$.
(i) Express sidelengths $a$ and $b$ in terms of $r_1$ and $r_2$.
(ii) Calculate the ratios $\frac{r_1}{r_2}$ and $\frac{a}{b}$ .
(iii) Find the length of $DK$ in terms of $r_1$ and $r_2$.
2021/2022 Tournament of Towns, P4
Let us call a 1×3 rectangle a tromino. Alice and Bob go to different rooms, and each divides a 20 × 21 board into trominos. Then they compare the results, compute how many trominos are the same in both splittings, and Alice pays Bob that number of dollars.
What is the maximal amount Bob may guarantee to himself no matter how Alice plays?
2015 Iran Team Selection Test, 3
$a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ are $2n$ positive real numbers such that $a_1,a_2,\cdots ,a_n$ aren't all equal. And assume that we can divide $a_1,a_2,\cdots ,a_n$ into two subsets with equal sums.similarly $b_1,b_2,\cdots ,b_n$ have these two conditions. Prove that there exist a simple $2n$-gon with sides $a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ and parallel to coordinate axises Such that the lengths of horizontal sides are among $a_1,a_2,\cdots ,a_n$ and the lengths of vertical sides are among $b_1,b_2,\cdots ,b_n$.(simple polygon is a polygon such that it doesn't intersect itself)
2018 Polish Junior MO First Round, 2
Inside parallelogram $ABCD$ is point $P$, such that $PC = BC$. Show that line $BP$ is perpendicular to line which connects middles of sides of line segments $AP$ and $CD$.
2015 Saudi Arabia IMO TST, 1
Let $S$ be a positive integer divisible by all the integers $1, 2,...,2015$ and $a_1, a_2,..., a_k$ numbers in $\{1, 2,...,2015\}$ such that $2S \le a_1 + a_2 + ... + a_k$. Prove that we can select from $a_1, a_2,..., a_k$ some numbers so that the sum of these selected numbers is equal to $S$.
Lê Anh Vinh
2022 Romania Team Selection Test, 3
Let $ABC$ be a triangle and let its incircle $\gamma$ touch the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $P$ be a point strictly in the interior of $\gamma.$ The segments $PA,PB,PC$ cross $\gamma$ at $A_0,B_0,C_0$ respectively. Let $S_A,S_B,S_C$ be the centres of the circles $PEF,PFD,PDE$ respectively and let $T_A,T_B,T_C$ be the centres of the circles $PB_0C_0,PC_0A_0,PA_0B_0$ respectively. Prove that $S_AT_A, S_BT_B$ and $S_CT_C$ are concurrent.