Found problems: 85335
1967 IMO Shortlist, 6
Solve the system of equations:
$
\begin{matrix}
|x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4.
\end{matrix}
$
2020 Malaysia IMONST 2, 3
Find all possible integer values of $n$ such that $12n^2 + 12n + 11$ is a $4$-digit number with equal digits.
Kvant 2023, M2770
2. A unit square paper has a triangle-shaped hole (vertices of the hole are not on the border of the paper). Prove that a triangle with area of $1 / 6$ can be cut from the remaining paper.
Alexandr Yuran
2024 Chile National Olympiad., 2
On a table, there are many coins and a container with two coins. Vale and Diego play the following game, where Vale starts and then Diego plays, alternating turns. If at the beginning of a turn the container contains \( n \) coins, the player can add a number \( d \) of coins, where \( d \) divides exactly into \( n \) and \( d < n \). The first player to complete at least 2024 coins in the container wins. Prove that there exists a strategy for Vale to win, no matter the decisions made by Diego.
2017 Putnam, B4
Evaluate the sum
\[\sum_{k=0}^{\infty}\left(3\cdot\frac{\ln(4k+2)}{4k+2}-\frac{\ln(4k+3)}{4k+3}-\frac{\ln(4k+4)}{4k+4}-\frac{\ln(4k+5)}{4k+5}\right)\]
\[=3\cdot\frac{\ln 2}2-\frac{\ln 3}3-\frac{\ln 4}4-\frac{\ln 5}5+3\cdot\frac{\ln 6}6-\frac{\ln 7}7-\frac{\ln 8}8-\frac{\ln 9}9+3\cdot\frac{\ln 10}{10}-\cdots.\]
(As usual, $\ln x$ denotes the natural logarithm of $x.$)
2018 Mediterranean Mathematics OIympiad, 2
Let $ABC$ be acute triangle. Let $E$ and $F$ be points on $BC$, such that angles $BAE$ and $FAC$ are equal. Lines $AE$ and $AF$ intersect cirumcircle of $ABC$ at points $M$ and $N$. On rays $AB$ and $AC$ we have points $P$ and $R$, such that angle $PEA$ is equal to angle $B$ and angle $AER$ is equal to angle $C$. Let $L$ be intersection of $AE$ and $PR$ and $D$ be intersection of $BC$ and $LN$. Prove that
$$\frac{1}{|MN|}+\frac{1}{|EF|}=\frac{1}{|ED|}.$$
2010 Indonesia TST, 2
A government’s land with dimensions $n \times n$ are going to be sold in phases. The land is divided into $n^2$ squares with dimension $1 \times 1$. In the first phase, $n$ farmers bought a square, and for each rows and columns there is only one square that is bought by a farmer. After one season, each farmer could buy one more square, with the conditions that the newly-bought square has a common side with the farmer’s land and it hasn’t been bought by other farmers. Determine all values of n such that the government’s land could not be entirely sold within $n$ seasons.
2024 SG Originals, Q1
Find all permutations $(a_1, a_2, \cdots, a_{2024})$ of $(1, 2, \cdots, 2024)$ such that there exists a polynomial $P$ with integer coefficients satisfying $P(i) = a_i$ for each $i = 1, 2, \cdots, 2024$.
2008 Hanoi Open Mathematics Competitions, 8
Consider a convex quadrilateral $ABCD$. Let $O$ be the intersection of $AC$ and $BD$; $M, N$ be the centroid of $\Delta AOB$ and $\Delta COD$ and $P, Q$ be orthocenter of $\Delta BOC$ and $\Delta DOA$, respectively.
Prove that $MN\bot PQ$.
1997 Tournament Of Towns, (524) 1
How many integers from $1$ to $1997$ have the sum of their digits divisible by $5$?
(AI Galochkin)
2004 Regional Olympiad - Republic of Srpska, 3
Determine all pairs of positive integers $(a,b)$, such that the roots of the equations \[x^2-ax+a+b-3=0,\]
\[x^2-bx+a+b-3=0,\] are also positive integers.
1981 AMC 12/AHSME, 5
In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel, and diagonal $BD$ and side $AD$ have equal length. If $m\angle DBC=110^\circ$ and $m\angle CBD =30^\circ$, then $m \angle ADB=$
$\text{(A)}\ 80^\circ \qquad \text{(B)}\ 90^\circ \qquad \text{(C)}\ 100^\circ \qquad \text{(D)}\ 110^\circ \qquad \text{(E)}\ 120^\circ$
2001 Bulgaria National Olympiad, 2
Find all real values $t$ for which there exist real numbers $x$, $y$, $z$ satisfying :
$3x^2 + 3xz + z^2 = 1$ ,
$3y^2 + 3yz + z^2 = 4$,
$x^2 - xy + y^2 = t$.
2001 CentroAmerican, 3
Find all the real numbers $ N$ that satisfy these requirements:
1. Only two of the digits of $ N$ are distinct from $ 0$, and one of them is $ 3$.
2. $ N$ is a perfect square.
2016 Japan Mathematical Olympiad Preliminary, 1
Calculate the value of $\sqrt{\dfrac{11^4+100^4+111^4}{2}}$ and answer in the form of an integer.
2012 IFYM, Sozopol, 7
A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Let $A_1 B_1 C_1 D_1$ be the image of $ABCD$ after rotation with center $O$ and angle $\alpha \in (0,90^\circ)$. The points $P,Q,R$ and $S$ are intersections of $AB$ and $A_1 B_1$, $BC$ and $B_1 C_1$, $CD$ and $C_1 D_1$, and $DA$ and $D_1 A_1$. Prove that $PQRS$ is a parallelogram.
2023 AMC 12/AHSME, 3
A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$?
$\textbf{(A)}~\frac{9}{25}\qquad\textbf{(B)}~\frac{1}{9}\qquad\textbf{(C)}~\frac{1}{5}\qquad\textbf{(D)}~\frac{25}{169}\qquad\textbf{(E)}~\frac{4}{25}$
2014 Contests, 2
The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$.
Note: In all the triangles the three vertices do not lie on a straight line.
2012 Vietnam National Olympiad, 3
Find all $f:\mathbb{R} \to \mathbb{R}$ such that:
(a) For every real number $a$ there exist real number $b$:$f(b)=a$
(b) If $x>y$ then $f(x)>f(y)$
(c) $f(f(x))=f(x)+12x.$
Fractal Edition 2, P4
Show that:
$$
1+\frac{1}{4}+\frac{1}{9}+\dots+\frac{1}{2023^2}+\frac{1}{2024^2} < 2.
$$
2014 PUMaC Team, 6
Find the sum of positive integer solutions of $x$ for $\dfrac{x^2}{1716-x}=p$, where $p$ is a prime. (If there are no solutions, answer $0$.)
2017 China Girls Math Olympiad, 3
Given $a_i\ge 0,x_i\in\mathbb{R},(i=1,2,\ldots,n)$. Prove that
$$((1-\sum_{i=1}^n a_i\cos x_i)^2+(1-\sum_{i=1}^n a_i\sin x_i)^2)^2\ge 4(1-\sum_{i=1}^n a_i)^3$$
2006 Stanford Mathematics Tournament, 2
In a given sequence $\{S_1,S_2,...,S_k\}$, for terms $n\ge3$, $S_n=\sum_{i=1}^{n-1} i\cdot S_{n-i}$. For example, if the first two elements are 2 and 3, respectively, the third entry would be $1\cdot3+2\cdot2=7$, and the fourth would be $1\cdot7+2\cdot3+3\cdot2=19$, and so on. Given that a sequence of integers having this form starts with 2, and the 7th element is 68, what is the second element?
2021 Baltic Way, 8
We are given a collection of $2^{2^k}$ coins, where $k$ is a non-negative integer. Exactly one coin is fake.
We have an unlimited number of service dogs. One dog is sick but we do not know which one.
A test consists of three steps: select some coins from the collection of all coins; choose a service dog; the dog smells all of the selected coins at once.
A healthy dog will bark if and only if the fake coin is amongst them. Whether the sick dog will bark or not is random. \\
Devise a strategy to find the fake coin, using at most $2^k+k+2$ tests, and prove that it works.
2010 HMNT, 5
Circle $O$ has chord $AB$. A circle is tangent to $O$ at $T$ and tangent to$ AB$ at $X$ such that $AX = 2XB$. What is $\frac{AT}{BT}$ ?