Found problems: 85335
Kyiv City MO 1984-93 - geometry, 1987.9.4
Inscribe a triangle in a given circle, if its smallest side is known, as well as the point of intersection of altitudes lying outside the circle.
2017 IMO Shortlist, C1
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
2010 China Team Selection Test, 1
Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings:
(1) $\sum_{v\in V} f(v)=|E|$;
(2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$.
Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.
2017 QEDMO 15th, 8
For a function $f: R\to R $ , $ f (2017)> 0$ as well as $f (x^2 + yf (z)) = xf (x) + zf (y)$ for all $x,y,z \in R$ is known. What is the value of $f (-42)$?
2009 Tournament Of Towns, 4
We only know that the password of a safe consists of $7$ different digits. The safe will open if we enter $7$ different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than $7$ attempts?
[i](5 points for Juniors and 4 points for Seniors)[/i]
2020 Ukrainian Geometry Olympiad - December, 3
Given convex $1000$-gon. Inside this polygon, $1020$ points are chosen so that no $3$ of the $2020$ points do not lie on one line. Polygon is cut into triangles so that these triangles have vertices only those specified $2020$ points and each of these points is the vertex of at least one of cutting triangles. How many such triangles were formed?
2010 Contests, 1
Solve in the integers the diophantine equation
$$x^4-6x^2+1 = 7 \cdot 2^y.$$
PEN N Problems, 14
One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.
1999 AMC 8, 14
In trapezoid $ABCD$ , the sides $AB$ and $CD$ are equal. The perimeter of $ABCD$ is
[asy]
draw((0,0)--(4,3)--(12,3)--(16,0)--cycle);
draw((4,3)--(4,0),dashed);
draw((3.2,0)--(3.2,.8)--(4,.8));
label("$A$",(0,0),SW);
label("$B$",(4,3),NW);
label("$C$",(12,3),NE);
label("$D$",(16,0),SE);
label("$8$",(8,3),N);
label("$16$",(8,0),S);
label("$3$",(4,1.5),E);[/asy]
$ \text{(A)}\ 27\qquad\text{(B)}\ 30\qquad\text{(C)}\ 32\qquad\text{(D)}\ 34\qquad\text{(E)}\ 48 $
2007 iTest Tournament of Champions, 1
Find the remainder when $3^{2007}$ is divided by $2007$.
1992 IMTS, 5
Let $T = (a,b,c)$ be a triangle with sides $a,b$ and $c$ and area $\triangle$. Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a, b' = h_b, c' = h_c$) and denote its area by $\triangle '$. Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes of $T'$, and denote its area by $\triangle ''$. Given that $\triangle ' = 30$ and $\triangle '' = 20$, find $\triangle$.
2023 Putnam, B5
Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$ such that $\pi(\pi(k)) \equiv m k(\bmod n)$ for all $k \in\{1,2, \ldots, n\}$.
2012 Turkey Team Selection Test, 3
Two players $A$ and $B$ play a game on a $1\times m$ board, using $2012$ pieces numbered from $1$ to $2012.$ At each turn, $A$ chooses a piece and $B$ places it to an empty place. After $k$ turns, if all pieces are placed on the board increasingly, then $B$ wins, otherwise $A$ wins. For which values of $(m,k)$ pairs can $B$ guarantee to win?
2003 China Western Mathematical Olympiad, 4
Given that the sum of the distances from point $ P$ in the interior of a convex quadrilateral $ ABCD$ to the sides $ AB, BC, CD, DA$ is a constant, prove that $ ABCD$ is a parallelogram.
2012 Iran MO (3rd Round), 1
We've colored edges of $K_n$ with $n-1$ colors. We call a vertex rainbow if it's connected to all of the colors. At most how many rainbows can exist?
[i]Proposed by Morteza Saghafian[/i]
2019 Thailand Mathematical Olympiad, 8
Let $ABC$ be a triangle such that $AB\ne AC$ and $\omega$ be the circumcircle of this triangle.
Let $I$ be the center of the inscribed circle of $ABC$ which touches $BC$ at $D$.
Let the circle with diameter $AI$ meets $\omega$ again at $K$.
If the line $AI$ intersects $\omega$ again at $M$, show that $K, D, M$ are collinear.
1977 Putnam, A5
Prove that $$\binom{pa}{pb}=\binom{a}{b} (\text{mod } p)$$ for all integers $p,a,$ and $b$ with $p$ a prime, $p>0,$ and $a>b>0.$
Kvant 2022, M2688
Let $T_a, T_b$ and $T_c$ be the tangent points of the incircle $\Omega$ of the triangle $ABC$ with the sides $BC, CA$ and $AB{}$ respectively. Let $X, Y$ and $Z{}$ be points on the circle $\Omega$ such that $A{}$ lies on the ray $YX$, $B{}$ lies on the ray $ZY$ and $C{}$ lies on the ray $XZ$. Let $P{}$ be the intersection point of the segments $ZX$ and $T_bT_c$, and similarly $Q=XY \cap T_cT_a$ and $R=YZ\cap T_aT_b$. Prove that the points $A, B$ and $C{}$ lie on the lines $RP, PQ$ and $QR{}$, respectively.
[i]Proposed by L. Shatunov (11th grade student)[/i]
2015 Vietnam Team selection test, Problem 6
Find the smallest positive interger number $n$ such that there exists $n$ real numbers $a_1,a_2,\ldots,a_n$ satisfied three conditions as follow:
a. $a_1+a_2+\cdots+a_n>0$;
b. $a_1^3+a_2^3+\cdots+a_n^3<0$;
c. $a_1^5+a_2^5+\cdots+a_n^5>0$.
2003 All-Russian Olympiad Regional Round, 9.5
$100$ people came to the party. Then those who had no acquaintances among those who came left. Among those who remained, then those who had exactly $1$ friend , also left. Then those who had exactly $2$, $3$, $4$,$ . .$ , $99$ acquaintances among those remaining at the time of their departure did the same..What is the largest number of people left at the end?
2009 International Zhautykov Olympiad, 2
Find all real $ a$, such that there exist a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ satisfying the following inequality:
\[ x\plus{}af(y)\leq y\plus{}f(f(x))
\]
for all $ x,y\in\mathbb{R}$
2020 BMT Fall, 7
Circle $\Gamma$ has radius $10$, center $O$, and diameter $\overline{AB}$. Point $C$ lies on $\Gamma$ such that $AC = 12$. Let $P$ be the circumcenter of $\vartriangle AOC$. Line $AP$ intersects $\Gamma$ at $Q$, where $Q$ is different from $A$. Then the value of $\frac{AP}{AQ}$ can be expressed in the form $\frac{m}{n}$, where m and $n$ are relatively prime positive integers. Compute $m + n$.
2008 Czech and Slovak Olympiad III A, 3
Find all pairs of integers $(a,b)$ such that $a^2+ab+1\mid b^2+ab+a+b-1$.
2015 Online Math Open Problems, 8
Determine the number of sequences of positive integers $1 = x_0 < x_1 < \dots < x_{10} = 10^{5}$ with the property that for each $m=0,\dots,9$ the number $\tfrac{x_{m+1}}{x_m}$ is a prime number.
[i]Proposed by Evan Chen[/i]
2000 Italy TST, 4
On a mathematical competition $ n$ problems were given. The final results showed that:
(i) on each problem, exactly three contestants scored $ 7$ points;
(ii) for each pair of problems, exactly one contestant scored $ 7$ points on both problems.
Prove that if $ n \geq 8$, then there is a contestant who got $ 7$ points on each problem. Is this statement necessarily true if $ n \equal{} 7$?